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On the unsteady unidirectional flows generated by impulsive motion of a boundary or sudden application of a pressure gradient
International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
On the unsteady unidirectional !ows generated by impulsive motion of a boundary or sudden application of a pressure gradient M. Emin Erdo*gan Makina Fakultesi, 80191, Gumussuyu, Istanbul, Turkey I˙.T.U., Received 9 July 2000; received in revised form 15 March 2001
Abstract Some exact solutions of the time-dependent Navier–Stokes equations are discussed for !ows impulsively started from rest by the motion of a boundary or two boundaries or by sudden application of a pressure gradient. It is shown that the expressions of the quantities such as velocity, !ux and skin friction obtained in a series form for large times can also be used to 4nd their values for small times or vice versa. Furthermore, conditions for which the values of the quantities at large times are nearly the same with those at small times are derived. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Unsteady Couette !ow; Unsteady Poiseuille !ow; Stokes’s 4rst problem; Viscous !uid
1. Introduction The governing equations for !uid mechanics are the Navier–Stokes equations. This nonlinear set of partial di=erential equations has no general solution, and a few number of exact solutions were found. Exact solutions are very important for many reasons. They provide a standard for checking the accuracies of many approximate methods such as numerical and empirical. Although computer techniques make the complete numerical integration of the Navier–Stokes equations feasible, the accuracy of the results can be established by a comparison with an exact solution. Many attempts to collect the exact solution of the Navier–Stokes equations E-mail address: [email protected] (M. Emin Erdo*gan).
have been done. The only comprehensive review is that due to Berker [1]. However, there are many new exact solutions which have been published in journals or in review articles. Two recent reviews, one for unsteady !ows [2] and the other for steady !ows [3] of viscous !uids have been published by Wang. An exact solution is de4ned as a solution of the Navier–Stokes equations and the continuity equation. An exact solution may be in a closed form or in a series form or in a form expressed by a numerical method. Most of the exact solutions for unsteady !ows are in series forms. They may be slowly convergent or rapidly convergent. However, it is possible to replace a rapidly convergent series with a slowly convergent one. Then this provides very important facilities. Steady or unsteady !ows, one can be obtained as a result of several e=ects. This can be: some kind
0020-7462/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 1 ) 0 0 0 3 5 - X
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of motion of the boundaries, application of a body force, wall that applies a tangential stress on the !uid or application of a pressure gradient. One or two of these e=ects can be applied together to the !uid. If the !uid is initially at rest, the motion of the !uid may eventually become steady or remain unsteady. This fact depends on the boundary condition and the kind of e=ects exerted on the !uid to set it in motion. In this paper, the !ows are obtained as a result of an impulsive motion of a boundary or sudden application of a pressure gradient, therefore they are time-dependent !ows. The examination of these !ows is not only done to characterize them but also to follow their development in time. A comparison of the values of the quantities such as velocity, !ux and skin friction on a boundary obtained for small times after the start with those obtained for large times can be given. It can then be shown under which conditions the values of the quantities at small times are nearly the same with those at large times. In this paper, unsteady !ows considered are Stokes’s 4rst problem, unsteady Couette !ow, unsteady !ow between two parallel plates which are suddenly moved in the same or in the opposite direction, unsteady Poiseuille !ow and unsteady generalized Couette !ow. The solutions for these !ows are in the form of series in terms of time. These series may be slowly convergent for small times and rapidly convergent for large times or vice versa. However, it can be shown that the series is slowly convergent for large times, if necessary condition is established, this series can also be used for large times without any diKculty. This is very important, because, sometimes it can be diKcult to obtain the solution for small times but it can be easy to obtain it for large times, then the solution obtained for large times can be used for small times. The opposite can also be true. 2. Flow due to impulsive motion of a plane wall The !ow over a plane wall which is initially at rest and is suddenly moved in its own plane with a constant velocity is termed Stokes’s 4rst problem or Rayleigh-type !ow [4,5]. The !uid stays in the region y ¿ 0 and the x-axis is chosen as the plane
wall. The governing equation is @u @2 u = 2; @t @y
(1)
where u(y; t) is the velocity, is the kinematic viscosity of the !uid, y is the coordinate and t is the time. The boundary and the initial conditions are u(0; t) = U
for t¿0;
u(y; 0) = 0
for y¿0;
u→0
for y → ∞;
(2)
where U is the velocity of the plate at y = 0. The solution of Eq. (1) subject to the boundary and the initial conditions is y u (3) = erfc √ U 2 t where erfc x is the complementary error function which is de4ned as ∞ 2 [exp(− 2 )] d erfc x = √ x and can be found in tabulated form [6]. The expression for velocity given by Eq. (3) can be obtained by the similarity method or the Laplace transform method or the sine transform method. Tokuda [7] has shown that the Rayleigh-type !ow develops due to unsteady boundary layer !ow past a !at plate. The !ow in this section is considered for a later use. The frictional force per unit area exerted by the !uid on the plate is 1=2 @u = − U : (4)
@y y=0 t Eq. (4) shows that the skin friction decreases as t −1=2 . The source of the vorticity is the rigid wall located at y = 0, and the vorticity di=uses into the !uid. The total vorticity in the !uid can be written as ∞ ! dy = U: 0
This exact result can also be obtained by the distribution of vorticity in the boundary layer where U is the external !ow velocity.
M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
It is useful to consider the !ow 4eld in two regions: y(t) and y(t) where (t) = (t)1=2 . In the 4rst region, the aforementioned conditions on the plate are satis4ed but those at in4nity are not satis4ed. However, in the second case, the conditions on the plate are not satis4ed but those at in4nity are satis4ed. If one considers (t) as a characteristic length and U is a characteristic velocity, then, the variation of u=U with y=(t) gives a single curve which is called the similarity curve. If one takes =U as a characteristic length and =U 2 as a characteristic time, the variation of u=U with y=(=U ) for various values of t=(=U 2 ) illustrates the development of the velocity distribution with time. It will be seen later that the unsteady Couette !ow has the same development near the moving plate. The reason is that there is no e=ect on the !ow due to the stationary boundary at early times. 3. Unsteady Couette ow When two parallel plane walls are in relative motion, a simple shearing motion with linear pro4le is obtained and this !ow is termed as the Couette !ow. In practice, one plate is held stationary and the other is moved. Suppose that the !uid is bounded by the two rigid boundaries at y = 0 and y = h and is initially at rest. Then the !uid motion starts suddenly due to the constant velocity of the upper plate in its own plane, the lower plate being held stationary. The governing equation is Eq. (1), and the boundary and the initial conditions are u(0; t) = 0
for all t;
u(h; t) = U
for t¿0;
u(y; 0) = 0
for 0 6 y¡h;
(5)
where h is the distance between two boundaries and it will be considered as a length scale of the !ow. In this case there is no similarity solution as in the case of the !at plate considered previously. The solution of Eq. (1) subject to boundary conditions (5) may be obtained by the Laplace transform method. Here, the solution is obtained by a method which can be applied to all unsteady unidirectional and two-dimensional !ows. Since the velocity distribu-
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tion in the case of the steady state is Uy=h, then, the velocity can be written as y u = U − f(y; t); h where f(y; t) satis4es the following di=erential equation, @2 f @f = 2 @t @y and the boundary and the initial conditions are y f(0; t) = 0; f(h; t) = 0; f(y; 0) = U : h The 4rst two of the boundary conditions for f(y; t) suggests a solution in the following form 2 2 2 n y f= ; An e−n t=h sin h n=1 where the values of An can be obtained by the third condition for f(y; t). Thus the velocity distribution is given by n y u y 2 (−1)n −n2 2 t=h2 = + e : (6) sin U h n=1 n h If the upper plate is held stationary and the lower one moves at a constant speed, then, the velocity distribution becomes n y y 2 1 −n2 2 t=h2 u =1 − − e : sin U h n=1 n h This expression for velocity is obtained from Eq. (6), by replacing y with h − y. The rapidity with which terms of this series expansion tend to zero increases with n. The 4rst term n = 1 survives longest and therefore this 4rst term dominates the series. Eq. (6) is obtained for large times. Although the series in Eq. (6) is slowly convergent for small times, it may also be used for small times, if the number of terms which are considered is suKcient. The volume !ux Q across a plane normal to the !ow and per unit width of this plane is h u dy Q= 0
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and inserting Eq. (6), one 4nds Q 8 =1− 2 (2n + 1)−2 Uh=2 n=0 ×exp[ − (2n + 1)2 2 t=h2 ]:
(7)
The series in Eq. (7) is valid for all t. For t = 0, since 2 1 = ; 2 (2n + 1) 8 n=0 Q becomes zero and when t goes to in4nity Q becomes Uh=2 which corresponds to that of the steady-state case. The dimensionless number t=h2 may be taken as a non-dimensional time. The values of Q for various values of t=h2 are given below: t=h2
0.01
0.05
0.1
0.5
1.0
Q=Uh=2 0.2258 0.5041 0.6979 0.9942 0.999958 The required time for Q to attain the asymptotic value is short and it is about t=h2 = 1. The frictional force per unit area exerted by the !uid on the moving plate at y = h is
U @u −n2 2 t=h2 = 1+2 e : h = @y y=h h n=1 (8)
The series in Eq. (8) is valid for all t. When t goes to in4nity the frictional force becomes U=h which corresponds to the steady-state case. The skin friction at the initial times is in4nite as in the case of the impulsive motion of a !at plate. This is an expected result. For small values of time, the !ow near the moving plate has a character similar to that of the impulsive motion of a plane wall. The values of the skin friction for various values of t=h2 are given below: t=h2
0.05
0.1
0.5
1.0
h =( U=h) 2.523130 1.784286 1.014384 1.000104 The required time to attain the asymptotic value of the skin friction is about t=h2 = 1.
The expressions given by Eqs. (6), (7) and (8) obtained for large times can also be used for small times. Although for small times, namely, t=h2 1, the series in these expressions are slowly convergent, the accurated results can be obtained by keeping suKcient number of terms in the series. This can be realized easily. For example, in Eq. (6), for t=h2 = 0:001, to obtain the value of u=U to 14 decimal places, the required number of terms is about 54. This shows that the expressions of the quantities such as velocity, !ux and skin friction obtained for large times can also be used to 4nd their values for small times. 3.1. Solutions for small times The series solution of velocity, !ux and skin friction for large times are slowly convergent for t=h2 1, namely for small times. Therefore, to obtain the accurate values of these quantities using the series expansion for large times, suKcient number of terms must be calculated. In order to 4nd the number of terms to be calculated the Laplace transform method is used. For small times the Laplace transform method is very useful [8]. The Laplace transform of u(y; t) is de4ned by the equation ∞ u= ue−st dt: 0
Then Eq. (1) and boundary condition (5) take the following forms s u − u = 0; u(0) = 0;
u(h) =
U ; s
where primes denotes di=erentiation with respect to y. The solution subject to boundary conditions is u sinh qy = ; U sinh qh where q = (s=)1=2 . This equation is exact and the inverse of it gives Eq. (6) which is obtained for large times. In order to obtain valid expressions for small times the method given in [8] is used. Since small t corresponds to large s, then, uO can be
M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
written as u e−q[(2n+1)h−y] e−q[(2n+1)h+y] = : − U s s 0 0
(9)
The inverse of QO can be obtained in the following form [8] 1=2 Q n+1 t =4 2 ierfc Uh=2 h (t=h2 )1=2 n=0
n (t=h2 )1=2
The inverse of uO can be found in the following form [8]
+
(2n + 1) − y=h u = erfc U 2(t=h2 )1=2 0
2n + 1 − 2 ierfc ; 2(t=h2 )1=2 n=0
(2n + 1) + y=h − erfc : 2 1=2 0
2(t=h )
It is clearly seen that for y=h1 and t=h2 1, Eq. (10) takes the form h−y u = erfc √ ; U 2 t which is the velocity distribution of the !ow due to impulsive motion of a plate near y = h. The series expansion in Eq. (10) is rapidly convergent for t=h2 1, but is slowly convergent for t=h2 1. However, it can also be used for t=h2 1. For example, for t=h2 = 10, in order to obtain accurate value of u=U to 14 decimal places, the required number of terms in the series in Eq. (10) is about 17. This shows that the expressions obtained for small times can also be used to obtain the values of the quantities such as velocity, !ux and skin friction for large times. For small times, the volume !ux Q across a plane normal to the !ow and per unit width of this plane can be obtained by Eq. (9). QO is the Laplace transform of Q and is given by QO =
0
h
u dy:
Inserting Eq. (9) into this equation, one 4nds QO e−(2n+2)hq e−2nhq = + U n=0 qs qs n=0 e−(2n+1)hq
−2
n=0
qs
:
ierfc
n=0
(10)
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where in erfc x =
x
∞
(11)
in−1 erfc d ;
0
i erfc x = erfc x are integrals of the complementary error function [8] which are obtained from tables. For t=h2 1, Eq. (11) can be written approximately as 1=2 t Q =4 : Uh=2 h2 For example, for t=h2 = 0:01, this gives the value 0.2257 accurate to four decimal places which was obtained for large times. It is easily shown that Eq. (11) which is obtained for small times, can also be used for large times without any diKculty. The frictional force (@u=@y)y=h per unit area exerted by the !uid on the moving plate can be obtained by Eq. (9). The Laplace transform of O h = (@u=@y)y=h is given as Oh = (@u=@y) y=h and inserting Eq. (9), one 4nds @u Oh = @y y=h
U e−2nqh e−(2n+2)qh = + : h n=0 s1=2 s1=2 n=0 The inverse of Oh can be obtained in the following form [8] @u h = @y y=h 2 2 2 2
U=h = e−n =t=h + e−(n+1) =t=h : ( t=h2 )1=2 n=0 n=0
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M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
The expression of h obtained for small times, can also be used for large times. For t=h2 = 1, one obtains h =( U=h) = 1:000103 accurate to six decimal places which can be compared with 1.000104 obtained for large times. The discussion shows that expressions obtained for small times can also be used for large times or vice versa. This is very important, because, sometimes it can be diKcult to 4nd the values of the quantities such as velocity, !ux and skin friction for small times or vice versa. Then, if the value of the quantity such as velocity can be found for either large or small times, this value can also be used for the other. The equality of two expressions for the skin friction yields
2 2 2 1 1 + 2 e−n =x ; 1 + 2 e−n x = √ nx n=1 n=1 for all x and this equality is tested numerically. A similar !ow to the Couette !ow can be obtained in two parallel plates which are stationary but the plane at the centerlines moves suddenly at a constant speed. The !uid is bounded by two boundaries at y = − b and at y = b. The governing equation is Eq. (1) and the boundary and the initial conditions are u( ± b; t) = 0
for all t;
u(0; t) = U
for t¿0;
u(y; 0) = 0
for y = 0;
where 2b is the distance between two parallel plates. It is clearly seen that there are two Couette !ows, one is in the region 0 6 y 6 b and the other is in the region −b 6 y 6 0. Thus, the velocity distribution can be given without diKculty in the following forms y 2 1 −n2 2 t=b2 n y u =1− − e sin U b n=1 n b for y ¿ 0 and u y 2 1 −n2 2 t=b2 n y =1+ + e sin U b n=1 n b for y 6 0:
However, using |y|, a single expression can be replaced. 4. Flow between two parallel plates suddenly set in motion with the same speed Suppose that the !uid is bounded by two rigid boundaries at y = − b and at y = b, and it is initially at rest and the !uid starts suddenly due to the motion at the same speed of the upper and the lower plates in their own planes. The governing equation is Eq. (1) and the boundary and the initial conditions are u( ± b; t) = U u(y; 0) = 0
t¿0;
for − b¡y¡b;
where 2b is the distance between two parallel plates. It is clearly seen that when t goes to in4nity u(y; t) tends to U . Since the velocity distribution in the case of the steady state is U , namely, the !uid moves eventually at the same speed of the plates, then, the velocity u(y; t) can be written as u = 1 − f(y; t); U where f(y; t) satis4es the following di=erential equation @2 f @f = 2 @t @y and the boundary and the initial conditions are f( ± b; t) = 0;
f(y; 0) = 1:
When t goes to in4nity f(y; t) tends to zero. The 4rst boundary condition suggests a solution in the following form 2 An e− n t cos n y; f= n=0
where n = (2n + 1) =2b. The values of An can be obtained by the second of the boundary conditions. Thus the velocity distribution is given by 4 (−1)n u =1− exp[ − (2n + 1)2 2 =4] U n=0 2n + 1 ×cos(2n + 1) =2;
(12)
M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
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The series in Eq. (13) is valid for all t. For t = 0, since 2 1 ; = (2n + 1)2 8 0 Q becomes zero. When t goes to in4nity Q tends to 2bU which corresponds to that of the steady-state case. t=b2 may be taken as a dimensionless time. The values of Q for various values of t=b2 are given below: t=b2
0.01
0.1
0.5
1.0
1.5
2.0
Q=2bU 0.1129 0.3568 0.7640 0.9313 0.9800 0.9942 Fig. 1. Flow between two parallel plates suddenly set in motion with the same speed. The variation of u=U with = y=b for various values of = t=b2 .
where = t=b2 and = y=b. They are nondimensional time and dimensionless distance. The 4rst term n = 0 survives longest and therefore this 4rst term dominates the series. Eq. (12) is obtained for large times, however, it will be seen later that it can be used for small times, if the number of terms which are considered is suKcient. The variation of u=U with y=b for various values of time, = t=b2 , is illustrated in Fig. 1. For small values of time, the !ow near the wall has the character of the Couette !ow and there is no e=ect due to the plates on the region near the centerline. The required time for u to attain the asymptotic value is short and it is about t=b2 = 2. The volume !ux Q across a plane normal to the !ow and per unit width of this plane is b u dy Q= −b
and inserting Eq. (12), one 4nds 8 Q =1− 2 (2n + 1)−2 0 2bU ×exp[ − (2n + 1)2 2 =4]:
The required time to attain the asymptotic value of Q is short and it is about t=b2 = 2. The frictional force per unit area exerted by the !uid on the plate at y = b is @u b = @y y=b =2
U exp[ − (2n + 1)2 2 =4] b 0
(14)
or b =( U=b) = C(t) where C(t) = 2 exp[(−2n + 1)2 2 =4] 0
which is dimensionless skin friction coeKcient. The values of the skin friction for various values of time are given below: t=b2 0.01
0.1
0.5
1.0
1.5
2.0
C(t) 5.6419 1.7796 0.5824 0.1696 0.0494 0.0144 The required time to attain the asymptotic value of the skin friction is about = 2. It will be seen later that the expressions given by Eqs. (12), (13) and (14) obtained for large times can also be used for small times. 4.1. Solutions for small times
(13)
The series solutions of velocity, !ux and skin friction for large times are slowly convergent
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M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
for t=b2 1, namely, for small times. For small times the Laplace transform method is very useful [8]. If the Laplace transform of u is u, O then, Eq. (1) and boundary conditions take the following forms s u − u = 0; U u( ± b) = ; s where primes denote di=erentiation with respect to y. The solution subject to boundary conditions is u cosh qy = ; U cosh qb where q = (s=)1=2 . This equation is exact and the inverse of it gives Eq. (12) which is obtained for large times. For small times, the method given in [8] is used. Since small t corresponds to large s, then, uO can be written as u e−q[(2n+1)b−y] = (−1)n U n=0 s +
n=0
(−1)n
e−q[(2n+1)b+y] : s
(15)
The inverse of uO can be found in the following form [8] (2n + 1) − u √ = (−1)n erfc U n=0 2 +
n=0
(−1)n erfc
(2n + 1) + √ : 2
(16)
The series expansions in Eq. (16) is rapidly convergent for t=b2 1, but is slowly convergent for t=b2 1. However, it can be easily shown that it can also be used for t=b2 1. For t=b2 = 0:01 and y=b = 0, Eq. (16) gives u=U = 2erfc5 = 3:074918 × 10−12 accurate to six decimal places. This shows the rapidity of the expansions in Eq. (16). This result can also be obtained by Eq. (12), number of terms considered in the series is about 100. However, this can be easily realized using a computer. For small times, the volume !ux Q across a plane normal to the !ow and per unit width of this plane
can be obtained by Eq. (15). QO is the Laplace transform of Q and is given by b u dy: QO = −b
Inserting Eq. (15) into this equation, one 4nds −2nbq −q(2n+2)b QO ne ne = ( − 1) − ( − 1) : 2U1=2 s3=2 s3=2 0 0 The inverse of QO can be obtained in the following form [8] n 1 Q = 21=2 √ + 2 (−1)n ierfc √ (17) 2bU n=1 where ierfc x =
x
∞
erfc d
is the integral of the complementary error function [8] which is obtained from tables. The expression for Q given in Eq. (17) obtained for small times, can also be used for large times. Indeed, for t=b2 = 1, this gives the value 0.9313 accurate to four decimal places which can be compared with 0.9313 obtained for large times. This shows that Eq. (17) which is obtained for small times can also be used for large times without any diKculty. The frictional force (@u=@y)y=b per unit area exerted by the !uid on the plate at y = b can be obtained by Eq. (15). The Laplace transform of O b = (@u=@y)y=b is given as Ob = (@u=@y) y=b and inserting Eq. (15), one 4nds e−q[(2n+1)b−y] @u = U (−1)n q Ob = @y y=b s n=0 −q[(2n+1)b+y] n e − (−1) q : s n=0 The inverse of Ob can be obtained in the following form [8] @u b = @y y=b 2
U=b 1+2 (−1)n e−n = : = ( )1=2 1
M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
The expression of b is obtained for small times, however, it can also be used for large times. For t=b2 = 1, it gives the value b =( U=b) = 0:1696 accurate to four decimal places which is exactly the same with that for large times. The discussion shows that the expressions obtained for small times can also be used for large times or vice versa. Then if the values of the quantities such as velocity, !ux and skin friction can be found for either large or small times, these values can also be used for the other. The equality of the two expressions for the skin friction yields 2 2 2 1 1 + 2 (−1)n e−n =x 2 e−(2n+1) =4 = √ x n=0 n=1 for all x and this equality is tested numerically. Suppose the !ow between two parallel plates which are suddenly moved with the same speed but in the opposite direction. The governing equation is Eq. (1) and the boundary and the initial conditions are u(b; t) = U
for t¿0;
u(−b; t) = − U
for t¿0;
u(0; t) = 0
for all t;
u(y; 0) = 0
for − b¡y¡b:
It is clearly seen that there are two Couette !ows, one is in the region 0 6 y 6 b and the other is in the region −b 6 y 6 0. Therefore, the velocity distribution can be obtained without diKculty in the following form y u y 2 (−1)n −n2 2 = + e sin n : U b 1 n b 5. Unsteady Poiseuille ow Flow of a viscous !uid between two parallel plates which are stationary is set in motion due to sudden application of a constant pressure gradient is termed as the Poiseuille !ow. Suppose that the
1099
!uid is bounded by two parallel plates at y = − b and y = b, and it is initially at rest and the !uid starts suddenly due to a constant pressure gradient. The governing equation, and the boundary and the initial conditions are @2 u 1 dp @u =− + 2; @t dx @y u( ± b; t) = 0 u(y; 0) = 0
(18)
for all t; for − b 6 y 6 b;
where 2b is the distance between two parallel plates. It is clearly seen that when t goes to in4nity u(y; t) tends to the velocity distribution in the following form y2 u =1 − 2 : 2 −(1=2 ) (dp=d x)b b Since the velocity distribution in the case of the steady-state is known, then, the velocity u(y; t) can be written as y2 u = 1 − 2 − f(y; t) 2 −(1=2 ) (dp=d x)b b where dp=d x is the constant pressure gradient and f(y; t) satis4es the following di=erential equation: @2 f @f = 2 @t @y and the boundary and the initial conditions are: f( ± b; t) = 0;
f(y; 0) = 1 −
y2 : b2
When t goes to in4nity f(y; t) tends to zero. The 4rst boundary condition suggests a solution in the following form 2 An e− n t cos n y; f= n=0
where n = (2n + 1) y=2b. The values of An can be obtained by the second condition for f(y; t). Thus
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M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
Inserting Eq. (19) into this equation one 4nds Q −(2b3 =3 )(dp=d x) 96 =1 − 4 (2n + 1)−4 exp[ − n2 t]: (20) n=0 The series in Eq. (20) is valid for all t. For t = 0, since 4 1 ; = (2n + 1)4 96 n=0 Q becomes zero. When t goes to in4nity Q tends to −(2b3 =3 )(dp=d x) which is the !ux for steady Poiseuille !ow. The values of Q for various values of t=b2 are given below: Fig. 2. Unsteady Poiseuille !ow. The variations of u=U with = y=b for various values of = t=b2 .
t=b2
0.01
0.05
0.1
0.5
1.0
Q(t)=Q(∞) 0.0279 0.1247 0.2287 0.7130 0.9164 the velocity distribution is given by u −(1=2 ) (dp=d x)b2 = 1 − 2 −
32 (−1)n 3 n=0 (2n + 1)3
×exp[ − (2n + 1)2 2 =4] cos(2n + 1) =2;
(19) where = y=b and = t=b2 . The 4rst term in the series n = 0 survives longest and therefore this 4rst term dominates the series. Eq. (19) is obtained for large times, however, it will be seen later that it can be used for small times. The variation of u=U with y=b for various values of t=b2 is illustrated in Fig. 2. It can be seen from the Fig. 2 that the required time to attain the asymptotic form of the velocity is about t=b2 = 1. The volume !ux Q across a plane normal to the !ow and per unit width of this plane is
Q=
b
−b
u dy:
The required time to attain the asymptotic value of Q is about t=b2 = 2. The frictional force per unit area exerted by the !uid on the plate at y = − b is @u b = @y y=−b dp 8 = − b 1− 2 (2n + 1)−2 dx n=0 ×exp[ − (2n + 1)2 2 =4] :
(21)
The values of the skin friction for various values of time are given below: t=b2
0.01
b =(− dp b) dx
0.1129 0.3568 0.7640 0.9313 0.9800 0.9942
0.1
0.5
1.0
1.5
2.0
The expression of the skin friction given by Eq. (21) is exactly the same with that of the volume !ux given by Eq. (13) for the !ow between two moving parallel plates. The required time to attain the asymptotic value of the skin friction is about t=b2 = 2. It will be seen later that the expressions
M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
given by Eqs. (19) – (21) obtained for large times can also be used for small times. 5.1. Solutions for small times The series solutions of velocity, !ux and skin friction are slowly convergent for t=b2 1, namely for small times. For small times the Laplace transform method is used [8]. If the Laplace transform of u is u, O then, Eq. (18) and boundary conditions take the following form 1 dp s ; u − u =
s d x u( ± b) = 0; where primes denote di=erentiation with respect to y. The solution subject to boundary condition is 1 u cosh qy = 2− 2 −(1=) (dp=d x) s s cosh qb where q = (s=)1=2 . This equation is exact and the inverse of it gives Eq. (19) which is obtained for large times. For small times the method given in [8] is used. Since small t corresponds to large s, then, uO can be written as u −(1=) (dp=d x) 1 e−q[(2n+1)b−y] = 2− (−1)n s s2 n=0 −q[(2n+1)b+y] ne + (−1) : (22) s2 n=0 The inverse of uO can be found in the following form [8] u −(1=2 ) (dp=d x)b2
(2n + 1) − √ = 2 1 − 4 (−1)n i2 erfc 2 0 (2n + 1) + 2 √ : (23) + i erfc 2 The series expansion in Eq. (23) is rapidly convergent for 1, but it is slowly convergent for 1. However, it can be easily shown that it can also be used for 1.
1101
For small times, the value of !ux Q across a plane normal to the !ow and per unit width of this plane can be obtained by Eq. (22). QO is the Laplace transform of Q and it is given as b QO = u dy: −b
Inserting Eq. (22) into this equation, one 4nds QO −(2=)(dp=d x)b 1 1=2 e−2nbq = 2− (−1)n 5=2
s
b
0
s
e−(2n+2)bq − (−1)n 5=2 0
s
:
The inverse of QO can be obtained in the following form [8] Q 3 −(2b =3 )(dp=d x) 4 1=2 = 3 1 − 3 n − 161=2 (−1)n i3 erfc 1=2 (24) 1 where in erfc x =
x
∞
in−1 erfc d ;
0
i erfc x = erfc x; are integrals of the complementary error function which are obtained from tables. The expression for Q, given in Eq. (24) obtained for small times, can also be used for large times. Indeed, for t=b2 = 1, this gives about 0.9159 accurate to four decimal places which can be compared with that of the large times. This shows that Eq. (24) which is obtained for small times can also be used for large times without any diKculty. The frictional force (@u=@y)y=b per unit area exerted by the !uid on the plate at y = − b can be obtained by Eq. (22). The Laplace transform of O b = (@u=@y)y=b is given as Ob = (@u=@y) y=b and
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inserting Eq. (22), one 4nds that @u
dp = − 1=2 Ob = @y y=−b d x −2nqb −q(2n+2)b ne ne × (−1) 3=2 − (−1) : s s3=2 n=0 n=0 The inverse of Ob can be obtained in the following form [8] @u dp = 2 − b 1=2 b = @y y=−b dx n 1 × √ + 2 (−1)n i erfc √ ; (25) 1 where i erfc x is the integral of the complementary error function [8] which is obtained from the tables. The expression of b is obtained for small times, however, it can also be used for large times. Indeed, for t=b2 = 1, Eq. (25) gives b =(−dp=d x)b = 0:9314 accurate to four decimal places which can be compared with that for large times. Thus, if the value of the skin friction can be found for one of small times or large times, this value can also be used for the other. The expressions given by Eqs. (19) – (21) obtained for large times can also be used for small times. Although for small times, namely, t=b2 1, the series in these expressions are slowly convergent, the accurate results can be obtained keeping suKcient number of terms in the series. However, this can be realized easily. For example, in Eq. (19), for t=b2 = 0:001 to obtain the value of u=U accurate to fourteen decimal places, the required number of terms is about 44. This shows that the expressions of the quantities such as velocity, !ux and skin friction obtained for small times can also be used to 4nd the values of them for large times. The expressions given by Eqs. (23) – (25) obtained for small times can also be used for large times. Although for large times, namely, t=b2 1, the series in these expressions are slowly convergent, the accurate results can be obtained keeping suKcient number of terms in the series. This can be realized without diKculty. Indeed, for example,
in Eq. (23), for t=b2 = 10, in order to obtain accurate value of u=U to fourteen decimal places, the required number of terms in the series in Eq. (23) is about 17. This shows that the expression obtained for small times can also be used to obtain the values of the quantities such as velocity, !ux and skin friction for large times. The opposite can also be true. 6. Unsteady generalized Couette ow The !ow between two parallel plates, one of which is at rest, the other moving in its own plane with a constant speed is called the simple Couette !ow. The !ow between two parallel plates produced by a constant pressure gradient in the direction of the !ow is called the two-dimensional Poiseuille !ow. The general case of the Couette !ow or the generalized Couette !ow is a superposition of the simple Couette !ow over the two-dimensional Poiseuille !ow [9]. This type of !ow is very important and there are many practical applications [10,11]. Suppose that the !uid is bounded by two parallel plates at y = 0 and y = h, and it is initially at rest. The !uid starts suddenly due to a pressure gradient and by the motion of the upper plate. The governing equation is Eq. (18), and the boundary and the initial condition are u(h; t) = U
for t¿0;
u(0; t) = 0
for all t;
u(y; 0) = 0
for 0 6 y¡h;
where h is the distance between two parallel plates. It is clearly seen that when t goes to in4nity u(y; t) tends to the velocity distribution in the following form y y2 u y = + − 2 ; U h h h where = (−h2 dp=d x)=2 U which is the dimensionless pressure gradient. Since the velocity distribution in the case of the steady-state is known, then, the velocity u(y; t) can be written as y y2 u y = + − 2 − f(y; t) U h h h
M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
1103
and f(y; t) satis4es the following di=erential equation @2 f @f = 2 @t @y and then the boundary and the initial conditions become f(h; t) = 0;
f(0; t) = 0; y y y2 f(y; 0) = + − 2 : h h h
When t goes to in4nity f(y; t) tends to zero. The 4rst two boundary conditions suggest a solution in the following form 2 2 2 n y : An e−n t=h sin f= h 1 The values of An can be obtained by the third boundary condition. Thus the velocity distribution is given by y 2 (−1)n y y2 u = + − + U h h h2 n=1 n n y 2 n −n2 2 t=h2 − 3 3 [1 − (−1) ] e sin : n h (26) If the upper plate is held stationary and the lower plate moves at a speed U , then the velocity distribution becomes 2 1 y y y2 u =1− + − + U h h h2 n=1 n 2 n y n −n2 2 t=h2 + 3 3 [1 − (−1) ] e sin : n h This expression for velocity is obtained from Eq. (26) by replacing y with h − y. Rapidity with which terms of this series expansion tend to zero increases with n. The 4rst term n = 1 survives longest and therefore this 4rst term dominates the series. Eq. (26) is obtained for large times, namely t=h2 1, however, the series in Eq. (26) is slowly convergent for small times, namely t=h2 1, can be used for small times, if the number of terms which are considered is suKcient. The variation of u=U with y=h for various values of and t=h2 is
Fig. 3. The unsteady generalized Couette !ow. The variation of u=U with = y=b for various values of = t=h2 . A: = 0:1; = − 3:309254; B: = 0:1, = − 0:419696; C: = 0:5; = − 3:003807; D: = 0:5; = − 0:991396; E: = ∞; = 0. A and C: zero !ux; B and D: separation at the bottom wall.
illustrated in Fig. 3. The values of correspond to the zero !ux and the separation at the lower wall. The volume !ux Q across a plane normal to the !ow and per unit width of this plane is h u dy Q= 0
and inserting Eq. (26) into this equation, one 4nds Q 4 1 = 1+ − 2 Uh=2 3 0 (2n + 1)2
2 2 2 4 × 1+ e−(2n+1) t=h : 2 2 (2n + 1) (27) The series in Eq. (27) is valid for all t. For t = 0, since 2 4 1 1 and ; = = (2n + 1)2 8 (2n + 1)4 96 0 0 Q becomes zero and when t goes to in4nity Q tends to Uh(1 + =3)=2 which corresponds to that of the steady-state case. t=h2 may be taken as a dimensionless number. The values of Q depends on and t=h2 . For a 4xed value of , the variation
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M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
of Q with t=h2 can be found. The case that Q is zero is very important. In the steady-state case for Q = 0 one 4nds = − 3. For t=h2 = 0:01; Q = 0 gives = − 6:641134 accurate to six decimal places. For t=h2 = 0:1; Q = 0 gives = − 3:309254; for t=h2 = 0:5; Q = 0 gives = − 3:003807: These values for Q = 0 show that the smallest value of | | corresponds to that of the steady-state case. The reason that | | is greater than 3 is that the e=ect of the pressure gradient is greater than that of the speed of the upper plate. The frictional force per unit area exerted by the !uid on the moving plate at y = h is
(@u=@y)y=h h = =1 −
U=h
U=h 2 2 2 n 1 − 2 2 [(−1) − 1] e−n +2 n 1 and on the stationary plate at y = 0 is
(@u=@y)y=0 0 = =1 + + 2 (−1)n
U=h
U=h 1 2 2 2 − 2 2 [1 − (−1)n ] e−n ; (28) n
where t=h2 is a non-dimensional time. To 4nd the value of the non-dimensional pressure gradient that corresponds to the separation at the lower plate is very important. It will well known that separation occurs when the shear stress at wall is zero. In the case of the steady-state, the separation at the lower plate occurs at = − 1. For = 0:1, the separation occurs at = − 0:419696 accurate to six decimal places. For = 0:5; the separation occurs at = − 0:991396. These values of show that the greatest value of | | corresponds to that of the steady state case. The reason that | | is smaller than 1 is that the e=ect of the speed of the upper plate is greater than that of the pressure gradient. The expressions given by Eqs. (26) – (28) obtained for large times can also be used for small times. Although for small times, the series in these expressions are slowly convergent, the accurate results can be obtained keeping suKcient number of terms in the series.
6.1. Solutions for small times The series solution of velocity, !ux and skin friction obtained for large times are slowly convergent for small times. Therefore, to obtain the accurate values of these quantities using the series expansion for large times, suKcient number of terms must be calculated. In order to 4nd the number of terms to be calculated the Laplace transform method is very useful [8]. The Laplace transform of u(y; t) is de4ned by the equation ∞ u= ue−st dt: 0
Then Eq. (18) and the boundary conditions take the following forms 1 dp s ; u − u =
s d x u(0) = 0;
u(h) =
U ; s
where primes denote di=erentiation with respect to y. The solution subject to boundary conditions is
sinh qy 2 u sinh qy = + 2 2 1− U sinh qh h s sinh qh sinh q(h − y) − ; sinh qh where q = (s=)1=2 . This equation is exact and the inverse of it gives Eq. (26) which is obtained for large times. For small times the method given in [8] is used. Since small t corresponds to large s, then, uO can be written as u e−q[(2n+1)h−y] e−q[(2n+1)h+y] = − U n=0 s s n=0 2 1 e−q[(2n+1)h−y] − + 2 h s2 n=0 s2 +
e−q[(2n+1)h+y] n=0
+
s2
e−q[(2n+2)h−y] n=0
s2
−
e−q[2nh+y] n=0
:
s2
(29)
M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
(2n + 1) − (2n + 1) + u √ √ = erfc − erfc U n=0 2 2 n=0 (2n + 1) − √ i2 erfc + 2 1 − 4 2 n=0
(2n + 1) + 2n + √ + i erfc − i2 erfc √ 2 2 n=0 n=0 (2n + 2) − 2 √ : (30) + i erfc 2 n=0 2
This series expansion in Eq. (30) is rapidly convergent for 1, but it is slowly convergent for 1. However, it can be easily shown that it can also be used for 1. For small times, the volume !ux Q across a plane normal to the !ow and per unit width of this plane can be obtained by Eq. (29). QO is the Laplace transform of Q and it is given as h O u dy: Q= 0
Inserting Eq. (29) into this equation, one 4nds e−q(2n+1)h QO e−2nqh e−(2n+2)qh = + −2 U n=0 qs qs qs n=0 n=0 e−2nqh e−(2n+2)qh 2 1 − 2 + + 2 h s2 qs2 qs2 n=0 n=0 e−(2n+1)qh −2 : qs2 n=0
The inverse of QO can be obtained in the following form [8] n n+1 Q = 41=2 i erfc √ + i erfc √ Uh=2 n=0 n=0 2n + 1 −2 i erfc √ + 4 n=0
The inverse of uO can be found in the following form [8]
× 1 − 16
1=2
n i3 erfc √ n=0
n+1 2n + 1 i erfc √ − 2 i3 erfc √ + 2 n=0 n=0
where n
i erfc x =
x
∞
1105
3
; (31)
in−1 erfc d ;
i0 erfc x = erfc x are integrals of the complementary error function [8] which are obtained from tables. It is easily shown that Eq. (31) which is obtained for small times can also be used for large times without any diKculty. For = 0:5; Q = 0 gives = − 3:011983 accurate to six decimal places which can be compared with that for large times. The frictional force per unit area exerted by the !uid on the stationary plate at y = 0 can be obtained by Eq. (29). The Laplace transform of O 0 = (@u|@y)y=0 is given as O0 = (@u=@y) y=0 and inserting Eq. (29) into the expression for O0 , one 4nds @u O0 = @y y=0 q 2 q −2nqh −(2n+1)qh e = U 2 + 2 e s h s2 n=0 n=0 q q −(2n+2)qh −(2n+1)qh + : e −2 e s2 s2 n=0 n=0 The inverse of O0 is obtained in the following form [8] 2 −(2n+1)2 =4 0 = e + 4 1=2
U=h ( )1=2 n=0 n n+1 × i erfc √ + i erfc √ n=0 n=0 2n + 1 −2 i erfc √ : 2 n=0 The expression of 0 is obtained for small times, however it can also be used for large times. It
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M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
is well known that separation occurs when the shear stress at the wall is zero. For = 0:1, separation occurs at = 0:4189 accurate to four decimal places and = 0:5 separation occurs at = 0:9836 accurate to four decimal places which can be compared with those of the values for large times. 7. Conclusions Some exact solutions of the Navier–Stokes equations are discussed. These are the Stokes’s 4rst problem, unsteady Couette !ow, unsteady !ow between two parallel plates which are suddenly moved in the same or in the opposite direction, unsteady Poiseuille !ow and unsteady generalized Couette !ow which is a superposition of the unsteady Couette !ow and the unsteady Poiseuille !ow. The solutions of these !ows are in the form of series in terms of time. These series may be rapidly convergent for small times or slowly convergent for large times or vice versa. The series is slowly convergent for large times, if necessary condition is established, this series can also be used for large times. The opposite can also be true. The series solution of velocity for the unsteady Couette !ow is rapidly convergent for large times but slowly convergent for small times. However, it can be used for small times. For example, for t=h2 = 0:001, to attain the value of u=U accurate to fourteen decimal places, the required number of terms is about 54. The series solution of velocity for the unsteady Couette !ow is rapidly convergent for small times but slowly convergent for large times. However, it can also be used for large times. For example, for t=h2 = 10, in order to obtain accurate value of u=U to fourteen decimal places, the required number of terms in the series is about 17. The series solution of velocity for the unsteady Poiseuille !ow is rapidly convergent for large times, but slowly convergent for small times. However, it can be used for small times. For example for t=b2 = 0:001 to obtain the value of the velocity accurate to fourteen decimal
places, the required number of terms in the series is about 44. The series solution of the velocity for the unsteady Poiseuille !ow is rapidly convergent for small times, but slowly convergent for large times. However it can be used for large times. For example, for t=b2 = 10, in order to obtain the accurate value of the velocity to fourteen decimal places, the required number of terms in the series is about 17. These examples show that the expression obtained for small times can also be used to obtain the values of the quantities such as velocity, !ux and skin friction for large times. The opposite can also be true. This is very important, because, sometimes it can be diKcult to obtain the solution for small times but it can be easy to obtain it for large times, then the solution obtained for large times can be used for small times. The opposite can also be true. References [1] R. Berker, IntQegration des equations du mouvement d’un !uid visqueux incompressible, in: Handbook of Fluid Dynamics, VIII=3, Springer, Berlin, 1963. [2] C.Y. Wang, Exact solutions of the unsteady Navier– Stokes equations, Appl. Mech. Rev. 42 (1969) s270–s282. [3] C.Y. Wang, Exact solutions of the steady-state Navier– Stokes equations, Annu. Rev. Fluid Mech. 23 (1991) 159–177. [4] M.E. Erdo*gan, A note on an unsteady !ow of a viscous !uid due to an oscillating plane wall, Int. J. Non-Linear Mech. 35 (2000) 1–6. [5] Y. Zeng, S. Weinbaum, Stokes problem for moving half-planes, J. Fluid Mech. 287 (1995) 59–74. [6] M. Abramovitz, I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. [7] N. Tokuda, On the impulsive motion of a !at plate in a viscous !uid, J. Fluid Mech. 33 (1968) 657– 672, 35 (1968) 828. [8] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, 2nd Edition, Oxford University Press, Oxford, 1973. [9] H. Schlichting, Boundary Layer Theory, 6th Edition, McGraw-Hill, New York, 1968. [10] M.E. Erdo*gan, I.T. GUurgUoze, Generalized Coutte !ow of a dipolar !uid between two parallel plates, J. Appl. Math. Phys. (ZAMP) 20 (1969) 785–790. [11] M.E. Erdo*gan, E=ect of the side walls in generalized Couette !ow, Appl. Mech. Eng. 3 (1998) 271–286.
On the unsteady unidirectional !ows generated by impulsive motion of a boundary or sudden application of a pressure gradient M. Emin Erdo*gan Makina Fakultesi, 80191, Gumussuyu, Istanbul, Turkey I˙.T.U., Received 9 July 2000; received in revised form 15 March 2001
Abstract Some exact solutions of the time-dependent Navier–Stokes equations are discussed for !ows impulsively started from rest by the motion of a boundary or two boundaries or by sudden application of a pressure gradient. It is shown that the expressions of the quantities such as velocity, !ux and skin friction obtained in a series form for large times can also be used to 4nd their values for small times or vice versa. Furthermore, conditions for which the values of the quantities at large times are nearly the same with those at small times are derived. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Unsteady Couette !ow; Unsteady Poiseuille !ow; Stokes’s 4rst problem; Viscous !uid
1. Introduction The governing equations for !uid mechanics are the Navier–Stokes equations. This nonlinear set of partial di=erential equations has no general solution, and a few number of exact solutions were found. Exact solutions are very important for many reasons. They provide a standard for checking the accuracies of many approximate methods such as numerical and empirical. Although computer techniques make the complete numerical integration of the Navier–Stokes equations feasible, the accuracy of the results can be established by a comparison with an exact solution. Many attempts to collect the exact solution of the Navier–Stokes equations E-mail address: [email protected] (M. Emin Erdo*gan).
have been done. The only comprehensive review is that due to Berker [1]. However, there are many new exact solutions which have been published in journals or in review articles. Two recent reviews, one for unsteady !ows [2] and the other for steady !ows [3] of viscous !uids have been published by Wang. An exact solution is de4ned as a solution of the Navier–Stokes equations and the continuity equation. An exact solution may be in a closed form or in a series form or in a form expressed by a numerical method. Most of the exact solutions for unsteady !ows are in series forms. They may be slowly convergent or rapidly convergent. However, it is possible to replace a rapidly convergent series with a slowly convergent one. Then this provides very important facilities. Steady or unsteady !ows, one can be obtained as a result of several e=ects. This can be: some kind
0020-7462/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 1 ) 0 0 0 3 5 - X
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M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
of motion of the boundaries, application of a body force, wall that applies a tangential stress on the !uid or application of a pressure gradient. One or two of these e=ects can be applied together to the !uid. If the !uid is initially at rest, the motion of the !uid may eventually become steady or remain unsteady. This fact depends on the boundary condition and the kind of e=ects exerted on the !uid to set it in motion. In this paper, the !ows are obtained as a result of an impulsive motion of a boundary or sudden application of a pressure gradient, therefore they are time-dependent !ows. The examination of these !ows is not only done to characterize them but also to follow their development in time. A comparison of the values of the quantities such as velocity, !ux and skin friction on a boundary obtained for small times after the start with those obtained for large times can be given. It can then be shown under which conditions the values of the quantities at small times are nearly the same with those at large times. In this paper, unsteady !ows considered are Stokes’s 4rst problem, unsteady Couette !ow, unsteady !ow between two parallel plates which are suddenly moved in the same or in the opposite direction, unsteady Poiseuille !ow and unsteady generalized Couette !ow. The solutions for these !ows are in the form of series in terms of time. These series may be slowly convergent for small times and rapidly convergent for large times or vice versa. However, it can be shown that the series is slowly convergent for large times, if necessary condition is established, this series can also be used for large times without any diKculty. This is very important, because, sometimes it can be diKcult to obtain the solution for small times but it can be easy to obtain it for large times, then the solution obtained for large times can be used for small times. The opposite can also be true. 2. Flow due to impulsive motion of a plane wall The !ow over a plane wall which is initially at rest and is suddenly moved in its own plane with a constant velocity is termed Stokes’s 4rst problem or Rayleigh-type !ow [4,5]. The !uid stays in the region y ¿ 0 and the x-axis is chosen as the plane
wall. The governing equation is @u @2 u = 2; @t @y
(1)
where u(y; t) is the velocity, is the kinematic viscosity of the !uid, y is the coordinate and t is the time. The boundary and the initial conditions are u(0; t) = U
for t¿0;
u(y; 0) = 0
for y¿0;
u→0
for y → ∞;
(2)
where U is the velocity of the plate at y = 0. The solution of Eq. (1) subject to the boundary and the initial conditions is y u (3) = erfc √ U 2 t where erfc x is the complementary error function which is de4ned as ∞ 2 [exp(− 2 )] d erfc x = √ x and can be found in tabulated form [6]. The expression for velocity given by Eq. (3) can be obtained by the similarity method or the Laplace transform method or the sine transform method. Tokuda [7] has shown that the Rayleigh-type !ow develops due to unsteady boundary layer !ow past a !at plate. The !ow in this section is considered for a later use. The frictional force per unit area exerted by the !uid on the plate is 1=2 @u = − U : (4)
@y y=0 t Eq. (4) shows that the skin friction decreases as t −1=2 . The source of the vorticity is the rigid wall located at y = 0, and the vorticity di=uses into the !uid. The total vorticity in the !uid can be written as ∞ ! dy = U: 0
This exact result can also be obtained by the distribution of vorticity in the boundary layer where U is the external !ow velocity.
M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
It is useful to consider the !ow 4eld in two regions: y(t) and y(t) where (t) = (t)1=2 . In the 4rst region, the aforementioned conditions on the plate are satis4ed but those at in4nity are not satis4ed. However, in the second case, the conditions on the plate are not satis4ed but those at in4nity are satis4ed. If one considers (t) as a characteristic length and U is a characteristic velocity, then, the variation of u=U with y=(t) gives a single curve which is called the similarity curve. If one takes =U as a characteristic length and =U 2 as a characteristic time, the variation of u=U with y=(=U ) for various values of t=(=U 2 ) illustrates the development of the velocity distribution with time. It will be seen later that the unsteady Couette !ow has the same development near the moving plate. The reason is that there is no e=ect on the !ow due to the stationary boundary at early times. 3. Unsteady Couette ow When two parallel plane walls are in relative motion, a simple shearing motion with linear pro4le is obtained and this !ow is termed as the Couette !ow. In practice, one plate is held stationary and the other is moved. Suppose that the !uid is bounded by the two rigid boundaries at y = 0 and y = h and is initially at rest. Then the !uid motion starts suddenly due to the constant velocity of the upper plate in its own plane, the lower plate being held stationary. The governing equation is Eq. (1), and the boundary and the initial conditions are u(0; t) = 0
for all t;
u(h; t) = U
for t¿0;
u(y; 0) = 0
for 0 6 y¡h;
(5)
where h is the distance between two boundaries and it will be considered as a length scale of the !ow. In this case there is no similarity solution as in the case of the !at plate considered previously. The solution of Eq. (1) subject to boundary conditions (5) may be obtained by the Laplace transform method. Here, the solution is obtained by a method which can be applied to all unsteady unidirectional and two-dimensional !ows. Since the velocity distribu-
1093
tion in the case of the steady state is Uy=h, then, the velocity can be written as y u = U − f(y; t); h where f(y; t) satis4es the following di=erential equation, @2 f @f = 2 @t @y and the boundary and the initial conditions are y f(0; t) = 0; f(h; t) = 0; f(y; 0) = U : h The 4rst two of the boundary conditions for f(y; t) suggests a solution in the following form 2 2 2 n y f= ; An e−n t=h sin h n=1 where the values of An can be obtained by the third condition for f(y; t). Thus the velocity distribution is given by n y u y 2 (−1)n −n2 2 t=h2 = + e : (6) sin U h n=1 n h If the upper plate is held stationary and the lower one moves at a constant speed, then, the velocity distribution becomes n y y 2 1 −n2 2 t=h2 u =1 − − e : sin U h n=1 n h This expression for velocity is obtained from Eq. (6), by replacing y with h − y. The rapidity with which terms of this series expansion tend to zero increases with n. The 4rst term n = 1 survives longest and therefore this 4rst term dominates the series. Eq. (6) is obtained for large times. Although the series in Eq. (6) is slowly convergent for small times, it may also be used for small times, if the number of terms which are considered is suKcient. The volume !ux Q across a plane normal to the !ow and per unit width of this plane is h u dy Q= 0
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and inserting Eq. (6), one 4nds Q 8 =1− 2 (2n + 1)−2 Uh=2 n=0 ×exp[ − (2n + 1)2 2 t=h2 ]:
(7)
The series in Eq. (7) is valid for all t. For t = 0, since 2 1 = ; 2 (2n + 1) 8 n=0 Q becomes zero and when t goes to in4nity Q becomes Uh=2 which corresponds to that of the steady-state case. The dimensionless number t=h2 may be taken as a non-dimensional time. The values of Q for various values of t=h2 are given below: t=h2
0.01
0.05
0.1
0.5
1.0
Q=Uh=2 0.2258 0.5041 0.6979 0.9942 0.999958 The required time for Q to attain the asymptotic value is short and it is about t=h2 = 1. The frictional force per unit area exerted by the !uid on the moving plate at y = h is
U @u −n2 2 t=h2 = 1+2 e : h = @y y=h h n=1 (8)
The series in Eq. (8) is valid for all t. When t goes to in4nity the frictional force becomes U=h which corresponds to the steady-state case. The skin friction at the initial times is in4nite as in the case of the impulsive motion of a !at plate. This is an expected result. For small values of time, the !ow near the moving plate has a character similar to that of the impulsive motion of a plane wall. The values of the skin friction for various values of t=h2 are given below: t=h2
0.05
0.1
0.5
1.0
h =( U=h) 2.523130 1.784286 1.014384 1.000104 The required time to attain the asymptotic value of the skin friction is about t=h2 = 1.
The expressions given by Eqs. (6), (7) and (8) obtained for large times can also be used for small times. Although for small times, namely, t=h2 1, the series in these expressions are slowly convergent, the accurated results can be obtained by keeping suKcient number of terms in the series. This can be realized easily. For example, in Eq. (6), for t=h2 = 0:001, to obtain the value of u=U to 14 decimal places, the required number of terms is about 54. This shows that the expressions of the quantities such as velocity, !ux and skin friction obtained for large times can also be used to 4nd their values for small times. 3.1. Solutions for small times The series solution of velocity, !ux and skin friction for large times are slowly convergent for t=h2 1, namely for small times. Therefore, to obtain the accurate values of these quantities using the series expansion for large times, suKcient number of terms must be calculated. In order to 4nd the number of terms to be calculated the Laplace transform method is used. For small times the Laplace transform method is very useful [8]. The Laplace transform of u(y; t) is de4ned by the equation ∞ u= ue−st dt: 0
Then Eq. (1) and boundary condition (5) take the following forms s u − u = 0; u(0) = 0;
u(h) =
U ; s
where primes denotes di=erentiation with respect to y. The solution subject to boundary conditions is u sinh qy = ; U sinh qh where q = (s=)1=2 . This equation is exact and the inverse of it gives Eq. (6) which is obtained for large times. In order to obtain valid expressions for small times the method given in [8] is used. Since small t corresponds to large s, then, uO can be
M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
written as u e−q[(2n+1)h−y] e−q[(2n+1)h+y] = : − U s s 0 0
(9)
The inverse of QO can be obtained in the following form [8] 1=2 Q n+1 t =4 2 ierfc Uh=2 h (t=h2 )1=2 n=0
n (t=h2 )1=2
The inverse of uO can be found in the following form [8]
+
(2n + 1) − y=h u = erfc U 2(t=h2 )1=2 0
2n + 1 − 2 ierfc ; 2(t=h2 )1=2 n=0
(2n + 1) + y=h − erfc : 2 1=2 0
2(t=h )
It is clearly seen that for y=h1 and t=h2 1, Eq. (10) takes the form h−y u = erfc √ ; U 2 t which is the velocity distribution of the !ow due to impulsive motion of a plate near y = h. The series expansion in Eq. (10) is rapidly convergent for t=h2 1, but is slowly convergent for t=h2 1. However, it can also be used for t=h2 1. For example, for t=h2 = 10, in order to obtain accurate value of u=U to 14 decimal places, the required number of terms in the series in Eq. (10) is about 17. This shows that the expressions obtained for small times can also be used to obtain the values of the quantities such as velocity, !ux and skin friction for large times. For small times, the volume !ux Q across a plane normal to the !ow and per unit width of this plane can be obtained by Eq. (9). QO is the Laplace transform of Q and is given by QO =
0
h
u dy:
Inserting Eq. (9) into this equation, one 4nds QO e−(2n+2)hq e−2nhq = + U n=0 qs qs n=0 e−(2n+1)hq
−2
n=0
qs
:
ierfc
n=0
(10)
1095
where in erfc x =
x
∞
(11)
in−1 erfc d ;
0
i erfc x = erfc x are integrals of the complementary error function [8] which are obtained from tables. For t=h2 1, Eq. (11) can be written approximately as 1=2 t Q =4 : Uh=2 h2 For example, for t=h2 = 0:01, this gives the value 0.2257 accurate to four decimal places which was obtained for large times. It is easily shown that Eq. (11) which is obtained for small times, can also be used for large times without any diKculty. The frictional force (@u=@y)y=h per unit area exerted by the !uid on the moving plate can be obtained by Eq. (9). The Laplace transform of O h = (@u=@y)y=h is given as Oh = (@u=@y) y=h and inserting Eq. (9), one 4nds @u Oh = @y y=h
U e−2nqh e−(2n+2)qh = + : h n=0 s1=2 s1=2 n=0 The inverse of Oh can be obtained in the following form [8] @u h = @y y=h 2 2 2 2
U=h = e−n =t=h + e−(n+1) =t=h : ( t=h2 )1=2 n=0 n=0
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M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
The expression of h obtained for small times, can also be used for large times. For t=h2 = 1, one obtains h =( U=h) = 1:000103 accurate to six decimal places which can be compared with 1.000104 obtained for large times. The discussion shows that expressions obtained for small times can also be used for large times or vice versa. This is very important, because, sometimes it can be diKcult to 4nd the values of the quantities such as velocity, !ux and skin friction for small times or vice versa. Then, if the value of the quantity such as velocity can be found for either large or small times, this value can also be used for the other. The equality of two expressions for the skin friction yields
2 2 2 1 1 + 2 e−n =x ; 1 + 2 e−n x = √ nx n=1 n=1 for all x and this equality is tested numerically. A similar !ow to the Couette !ow can be obtained in two parallel plates which are stationary but the plane at the centerlines moves suddenly at a constant speed. The !uid is bounded by two boundaries at y = − b and at y = b. The governing equation is Eq. (1) and the boundary and the initial conditions are u( ± b; t) = 0
for all t;
u(0; t) = U
for t¿0;
u(y; 0) = 0
for y = 0;
where 2b is the distance between two parallel plates. It is clearly seen that there are two Couette !ows, one is in the region 0 6 y 6 b and the other is in the region −b 6 y 6 0. Thus, the velocity distribution can be given without diKculty in the following forms y 2 1 −n2 2 t=b2 n y u =1− − e sin U b n=1 n b for y ¿ 0 and u y 2 1 −n2 2 t=b2 n y =1+ + e sin U b n=1 n b for y 6 0:
However, using |y|, a single expression can be replaced. 4. Flow between two parallel plates suddenly set in motion with the same speed Suppose that the !uid is bounded by two rigid boundaries at y = − b and at y = b, and it is initially at rest and the !uid starts suddenly due to the motion at the same speed of the upper and the lower plates in their own planes. The governing equation is Eq. (1) and the boundary and the initial conditions are u( ± b; t) = U u(y; 0) = 0
t¿0;
for − b¡y¡b;
where 2b is the distance between two parallel plates. It is clearly seen that when t goes to in4nity u(y; t) tends to U . Since the velocity distribution in the case of the steady state is U , namely, the !uid moves eventually at the same speed of the plates, then, the velocity u(y; t) can be written as u = 1 − f(y; t); U where f(y; t) satis4es the following di=erential equation @2 f @f = 2 @t @y and the boundary and the initial conditions are f( ± b; t) = 0;
f(y; 0) = 1:
When t goes to in4nity f(y; t) tends to zero. The 4rst boundary condition suggests a solution in the following form 2 An e− n t cos n y; f= n=0
where n = (2n + 1) =2b. The values of An can be obtained by the second of the boundary conditions. Thus the velocity distribution is given by 4 (−1)n u =1− exp[ − (2n + 1)2 2 =4] U n=0 2n + 1 ×cos(2n + 1) =2;
(12)
M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
1097
The series in Eq. (13) is valid for all t. For t = 0, since 2 1 ; = (2n + 1)2 8 0 Q becomes zero. When t goes to in4nity Q tends to 2bU which corresponds to that of the steady-state case. t=b2 may be taken as a dimensionless time. The values of Q for various values of t=b2 are given below: t=b2
0.01
0.1
0.5
1.0
1.5
2.0
Q=2bU 0.1129 0.3568 0.7640 0.9313 0.9800 0.9942 Fig. 1. Flow between two parallel plates suddenly set in motion with the same speed. The variation of u=U with = y=b for various values of = t=b2 .
where = t=b2 and = y=b. They are nondimensional time and dimensionless distance. The 4rst term n = 0 survives longest and therefore this 4rst term dominates the series. Eq. (12) is obtained for large times, however, it will be seen later that it can be used for small times, if the number of terms which are considered is suKcient. The variation of u=U with y=b for various values of time, = t=b2 , is illustrated in Fig. 1. For small values of time, the !ow near the wall has the character of the Couette !ow and there is no e=ect due to the plates on the region near the centerline. The required time for u to attain the asymptotic value is short and it is about t=b2 = 2. The volume !ux Q across a plane normal to the !ow and per unit width of this plane is b u dy Q= −b
and inserting Eq. (12), one 4nds 8 Q =1− 2 (2n + 1)−2 0 2bU ×exp[ − (2n + 1)2 2 =4]:
The required time to attain the asymptotic value of Q is short and it is about t=b2 = 2. The frictional force per unit area exerted by the !uid on the plate at y = b is @u b = @y y=b =2
U exp[ − (2n + 1)2 2 =4] b 0
(14)
or b =( U=b) = C(t) where C(t) = 2 exp[(−2n + 1)2 2 =4] 0
which is dimensionless skin friction coeKcient. The values of the skin friction for various values of time are given below: t=b2 0.01
0.1
0.5
1.0
1.5
2.0
C(t) 5.6419 1.7796 0.5824 0.1696 0.0494 0.0144 The required time to attain the asymptotic value of the skin friction is about = 2. It will be seen later that the expressions given by Eqs. (12), (13) and (14) obtained for large times can also be used for small times. 4.1. Solutions for small times
(13)
The series solutions of velocity, !ux and skin friction for large times are slowly convergent
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M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
for t=b2 1, namely, for small times. For small times the Laplace transform method is very useful [8]. If the Laplace transform of u is u, O then, Eq. (1) and boundary conditions take the following forms s u − u = 0; U u( ± b) = ; s where primes denote di=erentiation with respect to y. The solution subject to boundary conditions is u cosh qy = ; U cosh qb where q = (s=)1=2 . This equation is exact and the inverse of it gives Eq. (12) which is obtained for large times. For small times, the method given in [8] is used. Since small t corresponds to large s, then, uO can be written as u e−q[(2n+1)b−y] = (−1)n U n=0 s +
n=0
(−1)n
e−q[(2n+1)b+y] : s
(15)
The inverse of uO can be found in the following form [8] (2n + 1) − u √ = (−1)n erfc U n=0 2 +
n=0
(−1)n erfc
(2n + 1) + √ : 2
(16)
The series expansions in Eq. (16) is rapidly convergent for t=b2 1, but is slowly convergent for t=b2 1. However, it can be easily shown that it can also be used for t=b2 1. For t=b2 = 0:01 and y=b = 0, Eq. (16) gives u=U = 2erfc5 = 3:074918 × 10−12 accurate to six decimal places. This shows the rapidity of the expansions in Eq. (16). This result can also be obtained by Eq. (12), number of terms considered in the series is about 100. However, this can be easily realized using a computer. For small times, the volume !ux Q across a plane normal to the !ow and per unit width of this plane
can be obtained by Eq. (15). QO is the Laplace transform of Q and is given by b u dy: QO = −b
Inserting Eq. (15) into this equation, one 4nds −2nbq −q(2n+2)b QO ne ne = ( − 1) − ( − 1) : 2U1=2 s3=2 s3=2 0 0 The inverse of QO can be obtained in the following form [8] n 1 Q = 21=2 √ + 2 (−1)n ierfc √ (17) 2bU n=1 where ierfc x =
x
∞
erfc d
is the integral of the complementary error function [8] which is obtained from tables. The expression for Q given in Eq. (17) obtained for small times, can also be used for large times. Indeed, for t=b2 = 1, this gives the value 0.9313 accurate to four decimal places which can be compared with 0.9313 obtained for large times. This shows that Eq. (17) which is obtained for small times can also be used for large times without any diKculty. The frictional force (@u=@y)y=b per unit area exerted by the !uid on the plate at y = b can be obtained by Eq. (15). The Laplace transform of O b = (@u=@y)y=b is given as Ob = (@u=@y) y=b and inserting Eq. (15), one 4nds e−q[(2n+1)b−y] @u = U (−1)n q Ob = @y y=b s n=0 −q[(2n+1)b+y] n e − (−1) q : s n=0 The inverse of Ob can be obtained in the following form [8] @u b = @y y=b 2
U=b 1+2 (−1)n e−n = : = ( )1=2 1
M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
The expression of b is obtained for small times, however, it can also be used for large times. For t=b2 = 1, it gives the value b =( U=b) = 0:1696 accurate to four decimal places which is exactly the same with that for large times. The discussion shows that the expressions obtained for small times can also be used for large times or vice versa. Then if the values of the quantities such as velocity, !ux and skin friction can be found for either large or small times, these values can also be used for the other. The equality of the two expressions for the skin friction yields 2 2 2 1 1 + 2 (−1)n e−n =x 2 e−(2n+1) =4 = √ x n=0 n=1 for all x and this equality is tested numerically. Suppose the !ow between two parallel plates which are suddenly moved with the same speed but in the opposite direction. The governing equation is Eq. (1) and the boundary and the initial conditions are u(b; t) = U
for t¿0;
u(−b; t) = − U
for t¿0;
u(0; t) = 0
for all t;
u(y; 0) = 0
for − b¡y¡b:
It is clearly seen that there are two Couette !ows, one is in the region 0 6 y 6 b and the other is in the region −b 6 y 6 0. Therefore, the velocity distribution can be obtained without diKculty in the following form y u y 2 (−1)n −n2 2 = + e sin n : U b 1 n b 5. Unsteady Poiseuille ow Flow of a viscous !uid between two parallel plates which are stationary is set in motion due to sudden application of a constant pressure gradient is termed as the Poiseuille !ow. Suppose that the
1099
!uid is bounded by two parallel plates at y = − b and y = b, and it is initially at rest and the !uid starts suddenly due to a constant pressure gradient. The governing equation, and the boundary and the initial conditions are @2 u 1 dp @u =− + 2; @t dx @y u( ± b; t) = 0 u(y; 0) = 0
(18)
for all t; for − b 6 y 6 b;
where 2b is the distance between two parallel plates. It is clearly seen that when t goes to in4nity u(y; t) tends to the velocity distribution in the following form y2 u =1 − 2 : 2 −(1=2 ) (dp=d x)b b Since the velocity distribution in the case of the steady-state is known, then, the velocity u(y; t) can be written as y2 u = 1 − 2 − f(y; t) 2 −(1=2 ) (dp=d x)b b where dp=d x is the constant pressure gradient and f(y; t) satis4es the following di=erential equation: @2 f @f = 2 @t @y and the boundary and the initial conditions are: f( ± b; t) = 0;
f(y; 0) = 1 −
y2 : b2
When t goes to in4nity f(y; t) tends to zero. The 4rst boundary condition suggests a solution in the following form 2 An e− n t cos n y; f= n=0
where n = (2n + 1) y=2b. The values of An can be obtained by the second condition for f(y; t). Thus
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M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
Inserting Eq. (19) into this equation one 4nds Q −(2b3 =3 )(dp=d x) 96 =1 − 4 (2n + 1)−4 exp[ − n2 t]: (20) n=0 The series in Eq. (20) is valid for all t. For t = 0, since 4 1 ; = (2n + 1)4 96 n=0 Q becomes zero. When t goes to in4nity Q tends to −(2b3 =3 )(dp=d x) which is the !ux for steady Poiseuille !ow. The values of Q for various values of t=b2 are given below: Fig. 2. Unsteady Poiseuille !ow. The variations of u=U with = y=b for various values of = t=b2 .
t=b2
0.01
0.05
0.1
0.5
1.0
Q(t)=Q(∞) 0.0279 0.1247 0.2287 0.7130 0.9164 the velocity distribution is given by u −(1=2 ) (dp=d x)b2 = 1 − 2 −
32 (−1)n 3 n=0 (2n + 1)3
×exp[ − (2n + 1)2 2 =4] cos(2n + 1) =2;
(19) where = y=b and = t=b2 . The 4rst term in the series n = 0 survives longest and therefore this 4rst term dominates the series. Eq. (19) is obtained for large times, however, it will be seen later that it can be used for small times. The variation of u=U with y=b for various values of t=b2 is illustrated in Fig. 2. It can be seen from the Fig. 2 that the required time to attain the asymptotic form of the velocity is about t=b2 = 1. The volume !ux Q across a plane normal to the !ow and per unit width of this plane is
Q=
b
−b
u dy:
The required time to attain the asymptotic value of Q is about t=b2 = 2. The frictional force per unit area exerted by the !uid on the plate at y = − b is @u b = @y y=−b dp 8 = − b 1− 2 (2n + 1)−2 dx n=0 ×exp[ − (2n + 1)2 2 =4] :
(21)
The values of the skin friction for various values of time are given below: t=b2
0.01
b =(− dp b) dx
0.1129 0.3568 0.7640 0.9313 0.9800 0.9942
0.1
0.5
1.0
1.5
2.0
The expression of the skin friction given by Eq. (21) is exactly the same with that of the volume !ux given by Eq. (13) for the !ow between two moving parallel plates. The required time to attain the asymptotic value of the skin friction is about t=b2 = 2. It will be seen later that the expressions
M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
given by Eqs. (19) – (21) obtained for large times can also be used for small times. 5.1. Solutions for small times The series solutions of velocity, !ux and skin friction are slowly convergent for t=b2 1, namely for small times. For small times the Laplace transform method is used [8]. If the Laplace transform of u is u, O then, Eq. (18) and boundary conditions take the following form 1 dp s ; u − u =
s d x u( ± b) = 0; where primes denote di=erentiation with respect to y. The solution subject to boundary condition is 1 u cosh qy = 2− 2 −(1=) (dp=d x) s s cosh qb where q = (s=)1=2 . This equation is exact and the inverse of it gives Eq. (19) which is obtained for large times. For small times the method given in [8] is used. Since small t corresponds to large s, then, uO can be written as u −(1=) (dp=d x) 1 e−q[(2n+1)b−y] = 2− (−1)n s s2 n=0 −q[(2n+1)b+y] ne + (−1) : (22) s2 n=0 The inverse of uO can be found in the following form [8] u −(1=2 ) (dp=d x)b2
(2n + 1) − √ = 2 1 − 4 (−1)n i2 erfc 2 0 (2n + 1) + 2 √ : (23) + i erfc 2 The series expansion in Eq. (23) is rapidly convergent for 1, but it is slowly convergent for 1. However, it can be easily shown that it can also be used for 1.
1101
For small times, the value of !ux Q across a plane normal to the !ow and per unit width of this plane can be obtained by Eq. (22). QO is the Laplace transform of Q and it is given as b QO = u dy: −b
Inserting Eq. (22) into this equation, one 4nds QO −(2=)(dp=d x)b 1 1=2 e−2nbq = 2− (−1)n 5=2
s
b
0
s
e−(2n+2)bq − (−1)n 5=2 0
s
:
The inverse of QO can be obtained in the following form [8] Q 3 −(2b =3 )(dp=d x) 4 1=2 = 3 1 − 3 n − 161=2 (−1)n i3 erfc 1=2 (24) 1 where in erfc x =
x
∞
in−1 erfc d ;
0
i erfc x = erfc x; are integrals of the complementary error function which are obtained from tables. The expression for Q, given in Eq. (24) obtained for small times, can also be used for large times. Indeed, for t=b2 = 1, this gives about 0.9159 accurate to four decimal places which can be compared with that of the large times. This shows that Eq. (24) which is obtained for small times can also be used for large times without any diKculty. The frictional force (@u=@y)y=b per unit area exerted by the !uid on the plate at y = − b can be obtained by Eq. (22). The Laplace transform of O b = (@u=@y)y=b is given as Ob = (@u=@y) y=b and
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M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
inserting Eq. (22), one 4nds that @u
dp = − 1=2 Ob = @y y=−b d x −2nqb −q(2n+2)b ne ne × (−1) 3=2 − (−1) : s s3=2 n=0 n=0 The inverse of Ob can be obtained in the following form [8] @u dp = 2 − b 1=2 b = @y y=−b dx n 1 × √ + 2 (−1)n i erfc √ ; (25) 1 where i erfc x is the integral of the complementary error function [8] which is obtained from the tables. The expression of b is obtained for small times, however, it can also be used for large times. Indeed, for t=b2 = 1, Eq. (25) gives b =(−dp=d x)b = 0:9314 accurate to four decimal places which can be compared with that for large times. Thus, if the value of the skin friction can be found for one of small times or large times, this value can also be used for the other. The expressions given by Eqs. (19) – (21) obtained for large times can also be used for small times. Although for small times, namely, t=b2 1, the series in these expressions are slowly convergent, the accurate results can be obtained keeping suKcient number of terms in the series. However, this can be realized easily. For example, in Eq. (19), for t=b2 = 0:001 to obtain the value of u=U accurate to fourteen decimal places, the required number of terms is about 44. This shows that the expressions of the quantities such as velocity, !ux and skin friction obtained for small times can also be used to 4nd the values of them for large times. The expressions given by Eqs. (23) – (25) obtained for small times can also be used for large times. Although for large times, namely, t=b2 1, the series in these expressions are slowly convergent, the accurate results can be obtained keeping suKcient number of terms in the series. This can be realized without diKculty. Indeed, for example,
in Eq. (23), for t=b2 = 10, in order to obtain accurate value of u=U to fourteen decimal places, the required number of terms in the series in Eq. (23) is about 17. This shows that the expression obtained for small times can also be used to obtain the values of the quantities such as velocity, !ux and skin friction for large times. The opposite can also be true. 6. Unsteady generalized Couette ow The !ow between two parallel plates, one of which is at rest, the other moving in its own plane with a constant speed is called the simple Couette !ow. The !ow between two parallel plates produced by a constant pressure gradient in the direction of the !ow is called the two-dimensional Poiseuille !ow. The general case of the Couette !ow or the generalized Couette !ow is a superposition of the simple Couette !ow over the two-dimensional Poiseuille !ow [9]. This type of !ow is very important and there are many practical applications [10,11]. Suppose that the !uid is bounded by two parallel plates at y = 0 and y = h, and it is initially at rest. The !uid starts suddenly due to a pressure gradient and by the motion of the upper plate. The governing equation is Eq. (18), and the boundary and the initial condition are u(h; t) = U
for t¿0;
u(0; t) = 0
for all t;
u(y; 0) = 0
for 0 6 y¡h;
where h is the distance between two parallel plates. It is clearly seen that when t goes to in4nity u(y; t) tends to the velocity distribution in the following form y y2 u y = + − 2 ; U h h h where = (−h2 dp=d x)=2 U which is the dimensionless pressure gradient. Since the velocity distribution in the case of the steady-state is known, then, the velocity u(y; t) can be written as y y2 u y = + − 2 − f(y; t) U h h h
M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
1103
and f(y; t) satis4es the following di=erential equation @2 f @f = 2 @t @y and then the boundary and the initial conditions become f(h; t) = 0;
f(0; t) = 0; y y y2 f(y; 0) = + − 2 : h h h
When t goes to in4nity f(y; t) tends to zero. The 4rst two boundary conditions suggest a solution in the following form 2 2 2 n y : An e−n t=h sin f= h 1 The values of An can be obtained by the third boundary condition. Thus the velocity distribution is given by y 2 (−1)n y y2 u = + − + U h h h2 n=1 n n y 2 n −n2 2 t=h2 − 3 3 [1 − (−1) ] e sin : n h (26) If the upper plate is held stationary and the lower plate moves at a speed U , then the velocity distribution becomes 2 1 y y y2 u =1− + − + U h h h2 n=1 n 2 n y n −n2 2 t=h2 + 3 3 [1 − (−1) ] e sin : n h This expression for velocity is obtained from Eq. (26) by replacing y with h − y. Rapidity with which terms of this series expansion tend to zero increases with n. The 4rst term n = 1 survives longest and therefore this 4rst term dominates the series. Eq. (26) is obtained for large times, namely t=h2 1, however, the series in Eq. (26) is slowly convergent for small times, namely t=h2 1, can be used for small times, if the number of terms which are considered is suKcient. The variation of u=U with y=h for various values of and t=h2 is
Fig. 3. The unsteady generalized Couette !ow. The variation of u=U with = y=b for various values of = t=h2 . A: = 0:1; = − 3:309254; B: = 0:1, = − 0:419696; C: = 0:5; = − 3:003807; D: = 0:5; = − 0:991396; E: = ∞; = 0. A and C: zero !ux; B and D: separation at the bottom wall.
illustrated in Fig. 3. The values of correspond to the zero !ux and the separation at the lower wall. The volume !ux Q across a plane normal to the !ow and per unit width of this plane is h u dy Q= 0
and inserting Eq. (26) into this equation, one 4nds Q 4 1 = 1+ − 2 Uh=2 3 0 (2n + 1)2
2 2 2 4 × 1+ e−(2n+1) t=h : 2 2 (2n + 1) (27) The series in Eq. (27) is valid for all t. For t = 0, since 2 4 1 1 and ; = = (2n + 1)2 8 (2n + 1)4 96 0 0 Q becomes zero and when t goes to in4nity Q tends to Uh(1 + =3)=2 which corresponds to that of the steady-state case. t=h2 may be taken as a dimensionless number. The values of Q depends on and t=h2 . For a 4xed value of , the variation
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M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
of Q with t=h2 can be found. The case that Q is zero is very important. In the steady-state case for Q = 0 one 4nds = − 3. For t=h2 = 0:01; Q = 0 gives = − 6:641134 accurate to six decimal places. For t=h2 = 0:1; Q = 0 gives = − 3:309254; for t=h2 = 0:5; Q = 0 gives = − 3:003807: These values for Q = 0 show that the smallest value of | | corresponds to that of the steady-state case. The reason that | | is greater than 3 is that the e=ect of the pressure gradient is greater than that of the speed of the upper plate. The frictional force per unit area exerted by the !uid on the moving plate at y = h is
(@u=@y)y=h h = =1 −
U=h
U=h 2 2 2 n 1 − 2 2 [(−1) − 1] e−n +2 n 1 and on the stationary plate at y = 0 is
(@u=@y)y=0 0 = =1 + + 2 (−1)n
U=h
U=h 1 2 2 2 − 2 2 [1 − (−1)n ] e−n ; (28) n
where t=h2 is a non-dimensional time. To 4nd the value of the non-dimensional pressure gradient that corresponds to the separation at the lower plate is very important. It will well known that separation occurs when the shear stress at wall is zero. In the case of the steady-state, the separation at the lower plate occurs at = − 1. For = 0:1, the separation occurs at = − 0:419696 accurate to six decimal places. For = 0:5; the separation occurs at = − 0:991396. These values of show that the greatest value of | | corresponds to that of the steady state case. The reason that | | is smaller than 1 is that the e=ect of the speed of the upper plate is greater than that of the pressure gradient. The expressions given by Eqs. (26) – (28) obtained for large times can also be used for small times. Although for small times, the series in these expressions are slowly convergent, the accurate results can be obtained keeping suKcient number of terms in the series.
6.1. Solutions for small times The series solution of velocity, !ux and skin friction obtained for large times are slowly convergent for small times. Therefore, to obtain the accurate values of these quantities using the series expansion for large times, suKcient number of terms must be calculated. In order to 4nd the number of terms to be calculated the Laplace transform method is very useful [8]. The Laplace transform of u(y; t) is de4ned by the equation ∞ u= ue−st dt: 0
Then Eq. (18) and the boundary conditions take the following forms 1 dp s ; u − u =
s d x u(0) = 0;
u(h) =
U ; s
where primes denote di=erentiation with respect to y. The solution subject to boundary conditions is
sinh qy 2 u sinh qy = + 2 2 1− U sinh qh h s sinh qh sinh q(h − y) − ; sinh qh where q = (s=)1=2 . This equation is exact and the inverse of it gives Eq. (26) which is obtained for large times. For small times the method given in [8] is used. Since small t corresponds to large s, then, uO can be written as u e−q[(2n+1)h−y] e−q[(2n+1)h+y] = − U n=0 s s n=0 2 1 e−q[(2n+1)h−y] − + 2 h s2 n=0 s2 +
e−q[(2n+1)h+y] n=0
+
s2
e−q[(2n+2)h−y] n=0
s2
−
e−q[2nh+y] n=0
:
s2
(29)
M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
(2n + 1) − (2n + 1) + u √ √ = erfc − erfc U n=0 2 2 n=0 (2n + 1) − √ i2 erfc + 2 1 − 4 2 n=0
(2n + 1) + 2n + √ + i erfc − i2 erfc √ 2 2 n=0 n=0 (2n + 2) − 2 √ : (30) + i erfc 2 n=0 2
This series expansion in Eq. (30) is rapidly convergent for 1, but it is slowly convergent for 1. However, it can be easily shown that it can also be used for 1. For small times, the volume !ux Q across a plane normal to the !ow and per unit width of this plane can be obtained by Eq. (29). QO is the Laplace transform of Q and it is given as h O u dy: Q= 0
Inserting Eq. (29) into this equation, one 4nds e−q(2n+1)h QO e−2nqh e−(2n+2)qh = + −2 U n=0 qs qs qs n=0 n=0 e−2nqh e−(2n+2)qh 2 1 − 2 + + 2 h s2 qs2 qs2 n=0 n=0 e−(2n+1)qh −2 : qs2 n=0
The inverse of QO can be obtained in the following form [8] n n+1 Q = 41=2 i erfc √ + i erfc √ Uh=2 n=0 n=0 2n + 1 −2 i erfc √ + 4 n=0
The inverse of uO can be found in the following form [8]
× 1 − 16
1=2
n i3 erfc √ n=0
n+1 2n + 1 i erfc √ − 2 i3 erfc √ + 2 n=0 n=0
where n
i erfc x =
x
∞
1105
3
; (31)
in−1 erfc d ;
i0 erfc x = erfc x are integrals of the complementary error function [8] which are obtained from tables. It is easily shown that Eq. (31) which is obtained for small times can also be used for large times without any diKculty. For = 0:5; Q = 0 gives = − 3:011983 accurate to six decimal places which can be compared with that for large times. The frictional force per unit area exerted by the !uid on the stationary plate at y = 0 can be obtained by Eq. (29). The Laplace transform of O 0 = (@u|@y)y=0 is given as O0 = (@u=@y) y=0 and inserting Eq. (29) into the expression for O0 , one 4nds @u O0 = @y y=0 q 2 q −2nqh −(2n+1)qh e = U 2 + 2 e s h s2 n=0 n=0 q q −(2n+2)qh −(2n+1)qh + : e −2 e s2 s2 n=0 n=0 The inverse of O0 is obtained in the following form [8] 2 −(2n+1)2 =4 0 = e + 4 1=2
U=h ( )1=2 n=0 n n+1 × i erfc √ + i erfc √ n=0 n=0 2n + 1 −2 i erfc √ : 2 n=0 The expression of 0 is obtained for small times, however it can also be used for large times. It
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M. Emin Erdo#gan / International Journal of Non-Linear Mechanics 37 (2002) 1091–1106
is well known that separation occurs when the shear stress at the wall is zero. For = 0:1, separation occurs at = 0:4189 accurate to four decimal places and = 0:5 separation occurs at = 0:9836 accurate to four decimal places which can be compared with those of the values for large times. 7. Conclusions Some exact solutions of the Navier–Stokes equations are discussed. These are the Stokes’s 4rst problem, unsteady Couette !ow, unsteady !ow between two parallel plates which are suddenly moved in the same or in the opposite direction, unsteady Poiseuille !ow and unsteady generalized Couette !ow which is a superposition of the unsteady Couette !ow and the unsteady Poiseuille !ow. The solutions of these !ows are in the form of series in terms of time. These series may be rapidly convergent for small times or slowly convergent for large times or vice versa. The series is slowly convergent for large times, if necessary condition is established, this series can also be used for large times. The opposite can also be true. The series solution of velocity for the unsteady Couette !ow is rapidly convergent for large times but slowly convergent for small times. However, it can be used for small times. For example, for t=h2 = 0:001, to attain the value of u=U accurate to fourteen decimal places, the required number of terms is about 54. The series solution of velocity for the unsteady Couette !ow is rapidly convergent for small times but slowly convergent for large times. However, it can also be used for large times. For example, for t=h2 = 10, in order to obtain accurate value of u=U to fourteen decimal places, the required number of terms in the series is about 17. The series solution of velocity for the unsteady Poiseuille !ow is rapidly convergent for large times, but slowly convergent for small times. However, it can be used for small times. For example for t=b2 = 0:001 to obtain the value of the velocity accurate to fourteen decimal
places, the required number of terms in the series is about 44. The series solution of the velocity for the unsteady Poiseuille !ow is rapidly convergent for small times, but slowly convergent for large times. However it can be used for large times. For example, for t=b2 = 10, in order to obtain the accurate value of the velocity to fourteen decimal places, the required number of terms in the series is about 17. These examples show that the expression obtained for small times can also be used to obtain the values of the quantities such as velocity, !ux and skin friction for large times. The opposite can also be true. This is very important, because, sometimes it can be diKcult to obtain the solution for small times but it can be easy to obtain it for large times, then the solution obtained for large times can be used for small times. The opposite can also be true. References [1] R. Berker, IntQegration des equations du mouvement d’un !uid visqueux incompressible, in: Handbook of Fluid Dynamics, VIII=3, Springer, Berlin, 1963. [2] C.Y. Wang, Exact solutions of the unsteady Navier– Stokes equations, Appl. Mech. Rev. 42 (1969) s270–s282. [3] C.Y. Wang, Exact solutions of the steady-state Navier– Stokes equations, Annu. Rev. Fluid Mech. 23 (1991) 159–177. [4] M.E. Erdo*gan, A note on an unsteady !ow of a viscous !uid due to an oscillating plane wall, Int. J. Non-Linear Mech. 35 (2000) 1–6. [5] Y. Zeng, S. Weinbaum, Stokes problem for moving half-planes, J. Fluid Mech. 287 (1995) 59–74. [6] M. Abramovitz, I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. [7] N. Tokuda, On the impulsive motion of a !at plate in a viscous !uid, J. Fluid Mech. 33 (1968) 657– 672, 35 (1968) 828. [8] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, 2nd Edition, Oxford University Press, Oxford, 1973. [9] H. Schlichting, Boundary Layer Theory, 6th Edition, McGraw-Hill, New York, 1968. [10] M.E. Erdo*gan, I.T. GUurgUoze, Generalized Coutte !ow of a dipolar !uid between two parallel plates, J. Appl. Math. Phys. (ZAMP) 20 (1969) 785–790. [11] M.E. Erdo*gan, E=ect of the side walls in generalized Couette !ow, Appl. Mech. Eng. 3 (1998) 271–286.