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Creeping flow of a second grade fluid in a corner
Applied Mathematical Modelling 35 (2011) 621–636
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Creeping flow of a second grade fluid in a corner A.M. Siddiqui a, Akhlaq Ahmad a,b,*, Naseer Ahmed b,1 a b
Department of Mathematics, The Pennsylvania State University, York Campus, USA Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
a r t i c l e
i n f o
Article history: Received 15 July 2008 Received in revised form 16 June 2010 Accepted 7 July 2010 Available online 14 July 2010 Keywords: Creeping flow Corner Rotating boundary Domain perturbation
a b s t r a c t A variant of Taylor’s (1962) [23] scraper problem, in which, the lower plate rotates is considered. The non-linear partial differential equations governing the flow of a second grade fluid are modeled and solved by using the domain perturbation technique considering the angular velocity of the rotating plate as a small parameter. Also the rheology of the second grade fluid is examined by depicting the profiles of the velocity, stream function, pressure and stress fields. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction The creeping corner flow induced by a steady in-plane motion of one of the walls has been examined by Moffat [1] and Batchelor [2]. Hancock and Lewis [3] have investigated the effects of inertia forces, by constructing a regular perturbation series for the stream function, of which the leading term is the known similarity solution, but their works are restricted to Newtonian Fluids. Stokes flow in corners has also been studied by other authors. Anderson and Davis [4] considered viscous flow in a wedge made up by one rigid and one stress free boundary. Moreover, local solutions and partially local solutions, the latter ones do not satisfy all boundary conditions, were calculated at a rigid wedge where two immiscible viscous fluids meet at the vertex. In a later paper the same authors investigated the flow near a tri-junction including heat transfer and solidification [5]. Other boundary conditions have been considered by Betelu et al. [6]. Neglecting surface tension they calculated the viscous flow and its temporal evolution in a corner bounded by two stress free surfaces initially forming a sharp wedge. Neither of these studies, however, was devoted to the case when one wall is subjected to a given non-zero tangential stress. Despite the above studies of viscous fluids, the flow of the Newtonian fluids between intersecting planes has been extended within the context of various different non-Newtonian fluid models because of the wide range of engineering applications. For example, it will help to provide valuable information on the design of extrusion dies of industrial importance. Theoretical research on steady flow of this type was initiated by Langlois and Rivlin [7] who carried out perturbation analyses about slow flow of a non-Newtonian fluid through a wedge and a cone. They found that the stresses developed by the viscoelastic properties of the fluid were incompatible with radial flow and vortices were predicted in the perturbation solution. Strauss [8] was the first to present the solution of the steady, two-dimensional, and inertial flow of an incompressible Maxwell fluid between intersecting planes by using a series expansion in terms of decreasing powers of r. In his subsequent
* Corresponding author at: Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan. Tel.: +92 3009870978. E-mail addresses: [email protected] (A.M. Siddiqui), [email protected] (A. Ahmad), [email protected] (N. Ahmed). 1 Deceased author. 0307-904X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2010.07.011
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study [9], he considered the stability of the same flow problem. Han and Drexler [10] have studied non-inertial converging flow, using a modified second-order fluid model They observed that all three material functions depend on the second invariant of the rate of deformation. They also made a comparison between the theoretically predicted velocity profiles and stress distributions with the experimentally determined ones. Yoo and Han [11] carried out experiments on the converging slow flow of a polymer between intersecting planes and tried to explain the data in terms of the second grade fluid model. They found that theoretical analysis corroborates qualitatively with experimentally determined stress distributions. In the case of most non-Newtonian fluids a purely radial flow is not possible if inertial terms are to be retained in the equations of motion. Kaloni and Kamel [12] have shown that there cannot be a purely radial flow of Cosserat fluids in convergent channels. Later, Hull [13] studied the non-inertial flow of a general linear viscoelastic fluid in this geometry. He showed that radial flow is obtained for a wedge of 90° and no others. The similar results are valid for the Rivlin-Ericksen fluids. Mansutti and Rajagopal [14] studied the non-inertial flow of a shear thinning fluid between intersecting planes. They showed that sharp and pronounced boundary layers develop adjacent to the solid boundaries, even at zero Reynolds number. Bhatnagar et al. [15] have extended the analysis of Strauss [8] to an Oldroyd-B fluid which is characterized by a viscosity and two material constants with units of time. In their work, the effects of much higher values of the Reynolds number than Strauss’ work and the elastic parameter on the streamline patterns have been discussed. Serdar Baris [16] has investigated the flow of second grade viscoelastic fluid in a porous converging channel, he obtained the expressions for velocity field and temperature distribution by employing a numerical technique. Marco et al. [17] considered general Oldroyd model and have shown that the corner singularity is a sum of Newtonian singularity plus a lower order non-Newtonian singularity. Riedler and Schneider [18] studied the non-inertial flow of a power law fluid in a corner region with a moving wall and showed the streamline patterns near the wall were considerably less affected by the the power law exponent than near the wall at rest. Strauss’ work on the Maxwell fluid has been extended by Huang et al. [19] for the exact same geometry and boundary conditions as Strauss’ study, to the case of an Oldroyd-B fluid. They find that an increase in the elastic parameter reduces cellular structure. The question of true behaviour of a non-Newtonian fluid flow around a mathematically sharp edge is obviously a combined mathematical and physical problem. Whether the mathematical solutions constructed are regular or singular, their agreement with the physical reality must still be examined. It is, however, as difficult to establish a rigorous mathematical proof as it is complicated to obtain experimental evidence of a quantitative nature. Hills and Moffat [20] included experimental evidence for the three-dimensional scraper honing problem, but in general visualization of complicated three-dimensional flows is difficult and little experimental evidence is available. However, the two main papers on stokes flow in three-dimensional corners, Mustakis and Kim [21] and Shankar [22] both included major computational work to support their theories. The computations in Shankar gave a check on self consistency by testing the satisfaction of the boundary conditions. Mustakis and Kim [21] confirmed the traction singularity exponents which they found from two independent numerical calculations. Also they speculated that their work could be compared with experimentally recorded data. Many questions remain open for the three-dimensional problem. That the general form of the Stokes flow in a cone is not yet known. It is surprising, but perhaps the problem is amenable to a treatment similar to Shankar [22]. There is still very little known about the flow at polyhedral vertices, except for the particular cases described by Mustakis and Kim [21] and Shankar [22]. In their paper Hills and Moffat [20] have studied the three dimensional flow induced by the rotation of lower boundary in its plane, the upper plane being fixed at a corner angle h0. A local similarity solution valid in a neighbourhood of centre of rotation is obtained and streamlines are shown to be closed curves. A similar conclusion was reached by Shankar [22] who studied the Stokes problem in a trihedral corner with a non-zero boundary velocity assigned within some ring at a corner’s wall. He established the existence of corner eddies in the antisymmetric flow and found non-closed spirally shaped streamlines. This complicated behaviour was caused by the three dimensional nature of the flow. We consider in this paper a variant of Taylor’s [23] paint scraper problem, in which the lower plate rotates in its plane with constant angular velocity X about a point, taken to be origin, on the line of intersection of two planes. We investigate the Stokes flow of a second grade fluid flow induced by in-plane rotation of lower boundary with constant angular velocity X. This problem has potential relevance to situations in which fluids are mixed or kept in motion by the rotation of a paddle or rotor, or when fluid is scraped from a smooth surface. Such a situation has been investigated experimentally by Takahashi et al. [24], who considered the mixing of two highly viscous Newtonian fluids by the action of a helical blade in scraping contact with the base of a circular cylinder. The importance of understanding rotationally driven flows has been emphasize by Tatterson, Brodkey and Calabrese [25]. We emphasize here that for the configuration under consideration, all three velocity components u, v and w are functions of all three cylindrical polar coordinates (r, hz), where z is the axial coordinate parallel to the line of intersection of the planes. Even within the Stokes approximation, these velocity components are coupled through the pressure field. 2. Basic constitutive equations The second-order approximation to the general constitutive equation of a simple incompressible fluid is given by
T ¼ pI þ S;
ð2:1Þ
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623
where
S¼
2 X
Si ;
ð2:2Þ
i¼1
with
S1 ¼ lA1 ;
ð2:3Þ 2 2 A1 ;
S2 ¼ a1 A2 þ a
ð2:4Þ
where the coefficient l is the viscosity and a1, a2, are the normal stress moduli. For steady motion the Rivlin-Ericksen tensors satisfy the recursion relation T
ACþ1 ¼ ðgradAC Þu þ AC gradu þ ðAC graduÞ ;
ð2:5Þ
A0 ¼ I:
ð2:6Þ
with
For steady motion the field equations are given as
divu ¼ 0;
ð2:7Þ
b 1 p ðgraduÞu ¼ grad þ divS;
q
ð2:8Þ
q
where q is the constant density, b p is the pressure and the effects of body forces are negligible. We now formulate our problem. 3. Formulation of the problem We now consider a variant of Taylor’s [23] scraper problem, in which, the lower plate rotates in its plane with constant angular velocity X about a fixed point, taken to be the origin, on the line of intersection of the two planes. As discussed in [20], we shall find that for the configuration indicated by Fig. 3.1, all three velocity components (u, v, w) are functions of all three cylindrical polar coordinates (r, h, z), where z is now the axial coordinate parallel to the line of intersection of the planes. With the geometry and the coordinate system indicated in Fig. 3.1, the flow in the region 0 < h < h0 is given in the Stokes approximation by the equations
divu ¼ 0; 1 divS ¼ gradp;
ð3:1Þ ð3:2Þ
q
where p is the pressure per unit density and the velocity field relative to the coordinates (r, h, z) with boundary conditions is given by
Fig. 3.1. Geometry of the problem.
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u ¼ uer þ v eh þ wez ; u ¼ 0;
ð3:3Þ
on h ¼ 0;
ð3:4Þ
u ¼ ðXz; 0; Xr Þ on h ¼ h0 :
ð3:5Þ
We shall refer to the fixed plane h = h0 as the blade and to the rotating plane as the plate or base. We let the velocity and pressure fields of the form
u ¼ uðr; h; z; XÞ and p ¼ pðr; h; z; XÞ:
ð3:6Þ
and introduce the non-dimensional variables [26]
r r ¼ ; a
z z ¼ ; a
h ¼
h ; h0
u ¼
u
Xa
;
where a is the length scale measured from the corner parallel to the plane of each plate to an arbitrary arc up to which the viscous forces dominate over the inertia forces. That is, a is the length scale within the neighbourhood of the intersection of boundaries, where the present solution is valid. It should be noted that after introducing the new starred variables, we, then drop the asterisks and hence the forms of the Eqs. (3.1)–(3.6) remain the same. Now, following Joseph [27], we assume the series expansion of velocity and pressure fields ass
u¼
n X 1 ðkÞ k u X þ OðU n Þ k! k¼1
ð3:7Þ
p¼
n X 1 ðkÞ k p X þ OðU n Þ; k! k¼1
ð3:8Þ
and
respectively. Here, the following notation
ðÞðkÞ ¼
@k @ Xk
ðÞjX¼0 ;
ð3:9Þ
has been employed. Accordingly, the Eqs. (3.1)–(3.3) take the form
div uðkÞ ¼ 0; 1 divSðkÞ ¼ gradpðkÞ ;
ð3:11Þ
uðkÞ ¼ uðkÞ ðr; h; zÞer þ v ðkÞ ðr; h; zÞeh þ wðkÞ ðr; h; zÞez
ð3:12Þ
pðkÞ ¼ pðkÞ ðr; h; zÞ;
ð3:13Þ
q
ð3:10Þ
where
and
are functions of position only. The boundary conditions (3.4) and (3.5) can now be written in the form
uð1Þ ¼ ð0; 0; 0Þ at h ¼ h0 ; u
ðkÞ
¼ ð0; 0; 0Þk > 1 at h ¼ h0
ð3:14Þ ð3:15Þ
and
uð1Þ ¼ ðz; 0; r Þ at h ¼ 0;
ð3:16Þ
uðkÞ ¼ ð0; 0; 0Þk > 1 at h ¼ 0:
ð3:17Þ
4. First order problem For k = 1, the field equations become
divuð1Þ ¼ 0; 1 divSð1Þ ¼ gradpð1Þ :
q
ð4:1Þ ð4:2Þ
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625
Here ð1Þ
ð1Þ
Sð1Þ ¼ S1 þ S2 ;
ð4:3Þ
with ð1Þ
ð1Þ
S1 ¼ lA1 ;
ð4:4Þ
T ð1Þ A1 ¼ graduð1Þ þ graduð1Þ
ð4:5Þ
where
and ð1Þ
Ai
ð1Þ
¼ 0; Si
¼ 0 for i P 2:
ð4:6Þ
From Eqs. (3.12) and (3.13), the first order velocity and pressure fields are
uð1Þ ¼ uð1Þ ðr; h; zÞer þ v ð1Þ ðr; h; zÞeh þ wð1Þ ðr; h; zÞez
ð4:7Þ
pð1Þ ¼ pð1Þ ðr; h; zÞ;
ð4:8Þ
and
respectively. For the problem under consideration, the continuity Eq. (4.1) in terms of cylindrical polar coordinates, assumes the form
uð1Þ @uð1Þ 1 @ v ð1Þ @wð1Þ þ þ þ ¼ 0: r @h r @r @z
ð4:9Þ
Now substituting the expression for grad u(1) into the Eqs. (4.4) and (4.5) and then taking into account the results so obtained, the components of the momentum Eq. (4.2) are given by
@pð1Þ ¼m @r
2 @uð1Þ r @r
@pð1Þ ¼m @h
3 @uð1Þ r @h
2 ð1Þ
@ 2 uð1Þ @h2
þ 2 @ @ru2 þ r12
þ 1r
2 ð1Þ ð1Þ ð1Þ r32 @ v@h þ @ @zu2 2ur2 2 ð1Þ
2 ð1Þ
u þ @@h@r þ @ v@r þ r @ @rv2
v r þ 2r ð1Þ
ð1Þ
@ 2 v ð1Þ @h2
2 ð1Þ
þ r @ @zv2
@ 2 v ð1Þ @h@r
! ð4:10Þ
;
! ð4:11Þ
;
@pð1Þ ¼ mr2 wð1Þ ; @z 2
@ 1 where r2 ¼ @r 2 þ r
@ @r
þ r12
ð4:12Þ @2 @h2
2
@ þ @z 2 , is the Laplacian operator in cylindrical polar coordinates.
Eliminating the pressure p(1) from the Eqs. (4.10) and (4.11), and then introducing the stream function w(1)(r, h, z) analogous to the stream function w(r, h, z) according to [20,28,29], the derived velocity field have the following expressions:
uð1Þ ¼
1 @wð1Þ ð1Þ @wð1Þ ;v ¼ ; r @h @r
ð4:13Þ
By virtue of these results, the continuity equation leads to the fact that w(1) does not depend upon the coordinate z and the first two momentum Eqs. (4.10) and (4.11) reduce to
0
2 ð1Þ 4 @ w @h2
3 m@ r
r22
@ 3 wð1Þ @r@h2
4 ð1Þ
þ 2r
þr @ @rw4 1r
@ 4 wð1Þ @r 2 @h2
@ 2 wð1Þ @r 2
þ r13
@ 4 wð1Þ @h4
3 ð1Þ
þ 1r 4 ð1Þ
@ 4 wð1Þ @z2 @h2
w @ w 1 þ @@r@z 2 þ r @r2 @z2 þ r 2
3 ð1Þ
þ 2 @ @rw3
@wð1Þ @r
1 A ¼ 0:
ð4:14Þ
The boundary conditions for w(1), in view of (4.13), take the forms
@wð1Þ @wð1Þ ¼ ¼ 0; at h ¼ h0 ; @h @r @wð1Þ @wð1Þ ¼ rz; ¼ 0; at h ¼ 0: @h @r
ð4:15Þ ð4:16Þ
In what follows, we find the solution of Eq. (4.14). 4.1. Solution of the first order problem The solution of the system (4.14)–(4.16) is given by
wð1Þ ¼ rzf ðhÞ;
ð4:17Þ
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where f is defined by
f ¼
U h20
2
sin h0
n o 2 h20 sin h þ sin h0 h cos h þ ðh0 sin h0 cos h0 Þh sin h :
Thus, the velocity components u(1) and
uð1Þ ¼ zf
ð4:18Þ
v(1), become
0
ð4:19Þ
and
v ð1Þ ¼ zf ;
ð4:20Þ
respectively. To obtain the third component w(1) of the velocity field, we first find the pressure field p(1). For this purpose, we substitute the last two results into the pressure Eq. (4.10), we find that
@pð1Þ g0 ¼ mz 2 ; @r r
ð4:21Þ
where g is given by
g ¼ f 0 þ f 00 :
ð4:22Þ
Integration of this equation yields that
pð1Þ ¼ mz
g0 þ p0 ðh; zÞ: r
ð4:23Þ
Also the other pressure equation takes the form
@pð1Þ g ¼ mz : r @h
ð4:24Þ
Differentiating Eq. (4.23) partially with respect to h and then comparing the result with Eq. (4.24), we obtain
@p0 ¼ 0; @h
ð4:25Þ
which leads to the fact that p0 is independent of h. Consequently, we let p0 = p0(z) = constant and we obtain the required expression for the pressure distribution p(1) as
pð1Þ ¼ mz
g0 þ p0 : r
ð4:26Þ (1)
In view of the last result, the third component of velocity w ð1Þ
satisfies the equation
0
@p g ¼ m : @z r
mr2 wð1Þ ¼
ð4:27Þ
Following Taylor [23], we analogously may assume the solution in the form
wð1Þ ¼ rFðhÞ;
ð4:28Þ
which reduces (4.27) to the following boundary value problem: 00
F þF ¼ F ¼ 0; F ¼ 1;
2 2 k sin h þ sin h0 cos h 2
h20 sin h0
;
at h ¼ h0 ; at h ¼ 0:
ð4:29Þ ð4:30Þ ð4:31Þ
The solution of this problem can be easily found as
F ¼ f 0 ðhÞ;
ð4:32Þ
so that 0
wð1Þ ¼ rf ðhÞ:
ð4:33Þ
This yields a complete first order solution of the problem under consideration. 5. Second order problem We now turn to find the solution of our problem for k = 2, for which the governing equations are
divuð2Þ ¼ 0;
ð5:1Þ
A.M. Siddiqui et al. / Applied Mathematical Modelling 35 (2011) 621–636
1
q
divSð2Þ ¼ gradpð2Þ ;
627
ð5:2Þ
where ð2Þ
ð2Þ
Sð2Þ ¼ S1 þ S2 :
ð5:3Þ
Here ð2Þ
ð2Þ
S1 ¼ lA1
ð5:4Þ
2 ð2Þ ð2Þ ð1Þ S2 ¼ a1 A2 þ 2a2 A1 ;
ð5:5Þ
and
ð2Þ
ð2Þ
in which A1 and A2 are given by
T ð2Þ A1 ¼ graduð2Þ þ graduð2Þ
ð5:6Þ
and ð2Þ
A2 ¼ 2
ð1Þ
gradA1
ð1Þ T ð1Þ ; uð1Þ þ A1 graduð1Þ þ A1 graduð1Þ
ð5:7Þ
respectively. Also we have ð2Þ
Ai
ð2Þ
¼ 0; Si
¼ 0 for i P 3:
ð5:8Þ
In the present case, the velocity and pressure fields are assumed to be
uð2Þ ¼ uð2Þ ðr; h; zÞer þ v ð2Þ ðr; h; zÞeh þ wð2Þ ðr; h; zÞez
ð5:9Þ
and
pð2Þ ¼ pð2Þ ðr; h; zÞ;
ð5:10Þ
respectively. Computing all the tensors involved in Eqs. (5.4)–(5.7) by making use of Eqs. (5.9) and (5.10), the continuity and the momentum equations for the second-order are
uð2Þ @uð2Þ 1 @ v ð2Þ @wð2Þ þ þ þ ¼0 r @h r @r @z
ð5:11Þ
and
@pð2Þ ¼m @r
2 @uð2Þ r @r
( þ 2a1
2 ð2Þ
þ 2 @ @ru2 þ r12
@ 2 uð2Þ @h2
þ 1r
@ 2 v ð2Þ @h@r
!
2 ð2Þ ð2Þ ð2Þ r32 @ v@h þ @ @zu2 2ur2 2
2
0
2zr3 gðf þ gÞ zr3 ðfgÞ00 1r ð2f 0 g Þ 2
)
2
2gr þ 2zr3 fg 2 fgr 2 z g2 4a2 3 g 2 þ ; r r 2 ð2Þ 2 ð2Þ ! ð2Þ 3 @uð2Þ u þ @@h@r þ @ v@r þ r @ @rv2 r @h
@pð2Þ ¼m @h
2 ð2Þ
2 ð2Þ
v r þ 2r @ @hv2 þ r @ @zv2 2z2 0 þ 2a1 4f 0 g 2 ðfgÞ0 þ 2ðfgÞ0 þ fg f 0 g r 2 z þ 4a2 2 þ 1 gg 0 ; r nz 0 o @pð2Þ 0 ¼ mr2 wð2Þ þ 2a1 2 fg f 0 g : r @z ð2Þ
ð5:12Þ
ð5:13Þ ð5:14Þ
Now, we eliminate the pressure p(2) by cross differentiation and then introduce the stream function w(2)(r, h, z) such that
uð2Þ ¼
1 @wð2Þ ð2Þ @wð2Þ ;v ¼ ; r @h @r
ð5:15Þ
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by virtue of which, the continuity equation leads to the fact that w(2) is independent of z. Similarly, the Eqs. (5.12), (5.13) reduce to the following equation:
0 0¼m
2 ð2Þ 4 @ w @ r3 @h2
r22
@ 3 wð2Þ @r@h2
þ 2r
4 ð2Þ
þr @ @rw4 1r
@ 4 wð2Þ @r 2 @h2
2 ð2Þ
@ w @r 2
þ r13
@ 4 wð2Þ @h4
3 ð2Þ
þ 1r
@ 4 wð2Þ @z2 @h2
4 ð2Þ
w @ w 1 þ @@r@z 2 þ r @r2 @z2 þ r 2
3 ð2Þ
þ 2 @ @rw3 ð2Þ
@w @r
1
0
A 8ð2a1 þ a2 Þ gg : r
ð5:16Þ
Also, the corresponding boundary conditions are
@wð2Þ @wð2Þ ¼ ¼ 0; at h ¼ h0 ; @h @r ð2Þ ð2Þ @w @w ¼ 0; ¼ 0; at h ¼ 0: @h @r
ð5:17Þ ð5:18Þ
We now solve this second-order boundary value problem. 5.1. Solution of the second-order problem As discussed in [23], we may propose the following solution
wð2Þ ¼ r 2 f1 ðhÞ;
ð5:19Þ
so that the Eq. (5.16) becomes the following fourth order ordinary differential equation
f1iv þ 4f100 ¼
8
m
ð2a1 þ a2 Þgg 0 :
ð5:20Þ
The corresponding boundary conditions on f1 are
f1 ¼ f10 ¼ 0;
at h ¼ 0;
ð5:21Þ
f1
at h ¼ h0 :
ð5:22Þ
¼
f10
¼ 0;
The solution of the system (5.20)–(5.22) is
f1 ¼ R1 þ R2 h þ R3 cos 2h þ R4 sin 2h þ
1 ð2a1 þ a2 Þðd1 h cos 2h d2 h sin 2hÞ; 2m
ð5:23Þ
where the constants R’s and d’s are given by
R1 ¼ R3 ; R2 ¼
cos 2h0 1 sin 2h0 1 R4 R3 ð2a1 þ a2 Þðd1 h0 cos 2h0 d2 h0 sin 2h0 Þ; h0 2m h0
ð5:24Þ ð5:25Þ
where R3 and R4 are given by
sin 2h0 2h0 4ðcos 2h0 1Þð1 cos 2h0 h0 sin 2h0 Þ 1 ð2a1 þ a2 Þðcos 2h0 1Þðd1 cos 2h0 2d1 h0 sin 2h0 d2 sin 2h0 2d2 h0 cos 2h0 Þ 2m 1 1 h0 sin 2h0 ð2a1 þ a2 Þd1 þ ð2a1 þ a2 Þd1 ðcos 2h0 1Þ ð2a1 þ a2 Þ sin 2h0 ðd1 h0 cos 2h0 d2 h0 sin 2h0 Þ þ 2m m m ð2a1 þ a2 Þ h0 ð2a1 þ a2 Þd1 ðd1 h0 cos 2h0 d2 h0 sin 2h0 Þ þ ð5:26Þ 2mðcos 2h0 1Þ 2mðcos 2h0 1Þ
R3 ¼
and
R4 ¼
1 ð1 cos 2h0 h0 sin 2h0 Þ 1 ð2a1 þ a2 Þðcos 2h0 1Þðd1 cos 2h0 2d1 h0 sin 2h0 d2 sin 2h0 2d2 h0 cos 2h0 Þ 2m 1 1 h0 sin 2h0 þ ð2a1 þ a2 Þd1 ðcos 2h0 1Þ ð2a1 þ a2 Þ sin 2h0 ðd1 h0 cos 2h0 d2 h0 sin 2h0 Þ þ ð2a1 þ a2 Þd1 ; 2m m m ð5:27Þ
A.M. Siddiqui et al. / Applied Mathematical Modelling 35 (2011) 621–636
629
respectively. We also have
2 4 2 k sin h0 d1 ¼ 2 ; 2 h20 sin h0
ð5:28Þ
2
4k sin h0 d2 ¼ 2 : 2 2 h0 sin h0
ð5:29Þ
Substituting the expression of stream function from (5.19) into the expressions of velocity components u(2) and v(2), one can readily find 0
uð2Þ ¼ rf 1
ð5:30Þ
and
v ð2Þ ¼ 2rf 1 :
ð5:31Þ
Substituting the values of u
(2)
and v
pð2Þ ¼ m 4f10 þ f1000 ln r þ 2a1
(2)
"
in the Eqs. (5.12), (5.13), we get the expression for the pressure distribution p(2) as zfg r2 0
þ zðfgÞ 2r 2
00
ð2f 0 g Þ þ 2g 2 þ 2fg ln r
#
þ 2a2
2 zg 2 2g ln r : r2
ð5:32Þ
Differentiating Eq. (5.32) partially with respect to z and then comparing the result with Eq. (5.14), we find that 2
ð2Þ
mr w
( 0 0 ) fg f 0 g 2fg ðfgÞ00 g2 þ 4a2 2 : ¼ 2a1 þ 2 2 2 r r r r
ð5:33Þ
Let us suppose that the solution of Eq. (5.33) is
wð2Þ ¼ f2 ðhÞ;
ð5:34Þ
then partial differential Eq. (5.33) reduces to the following ordinary differential equation:
f200 ¼ 2a1 2g 2 2f 0 g 0 þ 4a2 g 2 ;
ð5:35Þ
where the boundary conditions on f2 are
f2 ¼ 0;
at h ¼ 0;
ð5:36Þ
f2 ¼ 0;
at h ¼ h0 :
ð5:37Þ
The system (5.35)–(5.37) has the solution 2
b1 þ C b 2 h þ A1 h A2 cos 2h A3 sin 2h A4 ðh cos 2h sin 2hÞ A5 ðh sin 2h þ cos 2hÞ: f2 ¼ C 4 4 2 4 4
ð5:38Þ
The constants involved in Eq. (5.38) have the following values:
b 1 ¼ A1 þ A5 ; C 4 A1 þ A5 A1 h0 A2 cos 2h0 A3 sin 2h0 b þ þ C2 ¼ 4h0 2 4h0 4h0 A4 A5 þ ðh0 cos 2h0 sin 2h0 Þ þ ðh0 sin 2h0 þ cos 2h0 Þ; 4h0 4h0 16ða1 þ a2 Þ 8a1 b1 A1 ¼ 2 a3 ; 2 2 2 h0 sin h0 h20 sin h0
ð5:39Þ
ð5:40Þ ð5:41Þ
16ða1 þ a2 Þ 8a1 A2 ¼ 2 b2 2 a4 ; 2 2 2 2 h0 sin h0 h0 sin h0
ð5:42Þ
16ða1 þ a2 Þ 8a1 A3 ¼ 2 a5 2 a1 ; 2 2 2 2 h0 sin h0 h0 sin h0
ð5:43Þ
8a1 A4 ¼ 2 a5 ; 2 2 h0 sin h0
ð5:44Þ
8a1 A5 ¼ 2 a2 ; 2 2 h0 sin h0
ð5:45Þ
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k 2 2 h 2 sin h0 ; 2 0 1 4 2 sin h0 k ; a2 ¼ 2 1 2 2 2 4 a3 ¼ h0 sin h0 sin h0 k ; 2
a1 ¼
ð5:46Þ ð5:47Þ ð5:48Þ
Fig. 6.1. Variation of velocity component u with X for viscous fluid.
Fig. 6.2. Variation of velocity component u with X for second grade fluid.
Fig. 6.3. Variation of velocity component
v with X for viscous fluid.
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1 2 2 2 4 h0 sin h0 sin h0 þ k ; 2 2 a5 ¼ k sin h0 ; 1 2 4 k þ sin h0 ; b1 ¼ 2 1 2 4 b1 ¼ k sin h0 : 2
a4 ¼
Fig. 6.4. Variation of velocity component
631
ð5:49Þ ð5:50Þ ð5:51Þ ð5:52Þ
v with X for second grade fluid.
Fig. 6.5. Variation of velocity component w with X for viscous fluid.
Fig. 6.6. Variation of velocity component w with X for second grade fluid.
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Summarizing the results of the perturbation series up to second-order, we have
uðr; h; zÞ ¼ Xuð1Þ þ
uð2Þ þ O X3 ;
X2 2 2
v ðr; h; zÞ ¼ Xv ð1Þ þ
X
2
v ð2Þ þ O
X3 ;
X ð2Þ wðr; h; zÞ ¼ Xwð1Þ þ w þ O X3 ; 2 X2 ð2Þ ð1Þ pðr; h; zÞ ¼ Xp þ p þ O X3 : 2
ð5:53Þ ð5:54Þ
2
Fig. 6.7. Variation of pressure p with X for viscous fluid.
Fig. 6.8. Variation of pressure p with X for second grade fluid.
Fig. 6.9. Variation of tangential stress Tp with X for viscous fluid.
ð5:55Þ ð5:56Þ
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633
Substituting the values of u(i), v(i), w(i) and p(i) (i = 1, 2) in (5.53)–(5.56), we finally obtain the required velocity and pressure fields up to the second-order as 0
uðr; h; zÞ ¼ Xzf ðhÞ þ
X2
0
rf ðhÞ; 2 1 v ðr; h; zÞ ¼ Xzf ðhÞ þ X2 rf1 ðhÞ;
ð5:57Þ ð5:58Þ
2
wðr; hÞ ¼ Xrf ðhÞ þ
X
2
f2 ;
ð5:59Þ
Fig. 6.10. Variation of tangential stress Tb with X for second grade fluid.
Fig. 6.11. Variation of tangential stress Tb with X for second grade fluid.
Fig. 6.12. Variation of normal stress Trr with X for rotating plate.
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" # 00 2 zfg þ zðfgÞ g 0 X2 0 zg 2 000 2r 2 r2 þ X2 a2 2 2g 2 ln r : pðr; h; zÞ ¼ Xmz þ m 4f1 þ f1 ln r þ X a1 0 0 r 2 r ð2f g Þ þ 2g 2 þ 2fg ln r
ð5:60Þ
The tangential stresses Trz and Thz are given by
zg 2 T rz ¼ X2 ð2a1 þ a2 Þ ; r X2 2mf20 2a1 z 0 fg f 0 g : þ T hz ¼ mg X þ r 2 r
ð5:61Þ ð5:62Þ
Let us denote the tangential stress at lower rotating plate by Tp and on upper fixed plate (i.e. blade) by Tb, then their expressions can readily be obtained by setting h = 0 and h = h0 in (5.61) and (5.62) respectively. In what follows, we analyze our results by sketching the graphs. 6. Graphs and discussion In order to observe the geometrical structure in various orientations of the scraper, we draw the following necessary graphs in Figs. 6.1–6.11 and to examine the rheology of the second grade fluid by depicting velocity profiles, the stream function, the pressure and the stress fields. In Figs. 6.1–6.11, we have tried to expose the characteristics of velocity, pressure and stress fields for flow of both viscous and second grade fluids. Fig. 6.1 shows the behavior of velocity component u for viscous fluid in the domain 0 < h < p against the angular velocity X of the rotating plate. Here we note that for small values of angle h, u starts with low positive values, then it become negative for h 2 (0.5, 1) (h in radian measurements). After this reverse flow, the component u increases rapidly for increasing values of angular velocity X. Fig. 6.2 is plotted for the u component of the velocity in the case of non-Newtonian fluid, but now we observe that every curve of the velocity u remain almost constant in the domain h 2 (0, 2.5) (h in radian measurements), when the angular velocity X is fixed for every curve. At h = 2.4 radians, all the velocity curves change sign and then velocity increases in the reverse direction with an increase in angular velocity X.
Fig. 6.13. Variation of normal stress Thh with X for fixed plate.
Fig. 6.14. Variation of radial velocity u with material parameter a1.
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635
Fig. 6.15. Variation of radial velocity u with material parameter a2.
Fig. 6.3 depicts the v component of the velocity for viscous fluid, in this figure we note that the velocity increases in the negative direction with X, it vanishes at h = 1.5 radians and then again increasing behavior is shown by v component in the reverse direction. In Fig. 6.4, curves are plotted for v component, when the second grade fluid flows in the flow domain, again we observe that as the angular velocity X is increased v increases in the reverse direction and this trend repeats itself because the sinusoidal nature of the flow. The third component w of the velocity field for viscous fluid flow is pictured in Fig. 6.5, where we see that for small values of angle h the response against the angular velocity X is not appreciable but as h increases the velocity also increases in the negative direction with an increase in X. In Fig. 6.6, the profiles of velocity component w are plotted against the angular velocity X for second grade fluid keeping the other parameters fixed. In this case it is very surprising that no reverse flow occurs in the whole flow domain and velocity goes on increasing with X in the specified area of flow. The next two figures are sketched to discuss the pressure distribution in the flow field for viscous and second grade fluids. Fig. 6.7 shows the pressure distribution for the viscous fluid flow, here we note that for small h pressure curves start with almost zero value and then go on increasing with X. Then again pressure becomes zero at h = 2.5 radians, afterwards it starts increasing in the negative direction. Note that the pressure curves change sign at h = 2.5 radians in the flow field under consideration. Fig. 6.8 pictures the pressure field for non-Newtonian fluid flow, it demonstrates that for small h pressure is also small and as h increases, there appear considerable variations in the pressure field and it jumps high and remain bounded as we increase the angular velocity X. It is worth noting that pressure remain always positive for second grade fluid flow. Figs. 6.9 and 6.10 depict the behavior of tangential stress on the fixed plane (blade). In Fig. 6.9 we note that the stress is in opposite direction to the fluid flow and goes on increasing in this direction as the angular velocity increases. In the mathematical expression for tangential stress Tb, if we put h0 = 0, an unbounded stress is obtained, this behavior can also be seen in the profiles shown in Fig. 6.9 i.e. there exist a flow singularity in the stress field for h0 = 0. For the non-Newtonian case (Fig. 6.10) the behavior of the tangential stress is opposite to that for viscous fluid, for present case stress remain positive, increases with X and it shoots up as h0 ? 0. As it is discussed earlier that the viscous contribution in the tangential stress for the rotating plate vanish, so in Fig. 6.11, we tried to explain the contribution due to non-Newtonian parameters in the stress field on the lower rotating plate. We observe that the parameters a1 and a2 do contribute in the stress filed Tp and stress increases with second grade parameters. Further as we increase angular velocity X the shear stress on the plate increases in the reverse direction. Normal stress at lower and upper plates is shown in the Figs. 6.12 and 6.13 respectively. The behavior of the normal stress is identical on both the plates, however, the strength of the normal stress at upper plate is very much stronger as compared to the lower rotating plate. In this flow velocity behaves like ra near the corner. Very near the corner, one hopes that length scales other than the distance from the corner become unimportant. Thus the radial flow dominates over the the flow in the angular direction and the assumption of dominant viscous forces imply that the velocity field along the z- direction is considerably less effected by the second grade parameters a1 and a2. Validating these considerations, we have plotted the radial component of the velocity in the Figs. 6.14 and 6.15 against the parameters a1 and a2 respectively. Against a1, the radial velocity is initially negative but it decreases with this parameter as the angle between the plates is increased. The radial velocity shows an opposite behavior against the parameter a2 provided that the other circumstances remain the same. 7. Conclusion The problem under consideration is a variant of Taylor’s paint scraper, here the lower plate (base) rotates with constant angular velocity X about a fixed point in its own plane. This problem has potential relevance to situations in which fluids are mixed, kept in motion by the rotation of a paddle or rotor and when fluid is scraped from a plane surface.
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It is to be noted that by setting a1 = a2 = 0, we recover the results obtained by Hills et al. The solutions (5.57)–(5.62) show that the flow fields are strongly dependent upon the angular velocity X, but they are valid only for small values of angular velocity. By setting X = 0, we see that velocity field vanish and no flow occurs, this matches with the physical situation because flow is generated by the rotation of the lower plate. The solutions (5.57)–(5.62) of the non-linear partial differential equations governing the flow reveal that there is an appreciable contribution of the second grade parameters in the flow regime, but the viscous effects on the tangential stress at the plate (base) do not contribute. It is also noted that as for as the radial distance is concerned there appears no singularity in the velocity field with respect to r (radial distance) but stress field vary as r1 and singularity appears as r ? 0. Acknowledgements We are grateful to the referee of previous version of this paper, whose penetrating comments led to significant improvements. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
H.K. Moffat, Viscous and resistive eddies near a sharp corner, J. Fluid Mech. 18 (1964) 1. G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967. C. Hancock, Lewis, Effects of inertia in forced corner flows, J. Fluid Mech. 112 (1981) 315. D.M. Anderson, S.H. Davis, Two-fluid viscous flow in a corner, J. Fluid Mech. 257 (1993) 1. D.M. Anderson, S.H. Davis, Fluid flow, heat transfer and solidification near tri-junctions, J. Cryst. Growth 142 (1994) 245. S. Betelû, J. Diez, R. Gratton, L. Thomas, Instantaneous viscous flow in a corner bounded by free surfaces, Phys. Fluids 8 (1996) 2269. W.E. Langlois, R.S. Rivlin, Steady flow of slightly viscous fluids, Technical Report no. DA-457/3, Division of Applied Mathematics, Brown University, 1959. K. Strauss, Die strömung einer einfachen viscoelastichen fl üssigkeit in einem konvergenten kanal. Tiel I: Die stationare strö mung, Acta Mech. 20 (1974) 233. K. Strauss, Die strömung einer einfachen viscoelastichen fl üssigkeit in einem konvergenten kanal. Tiel II: Die stabiltät der strömung, Acta Mech. 21 (1975) 141. C.D. Han, L.H. Drexler, Studies of converging flows of viscoelastic polymeric melts. III. Stress and velocity distribution in the entrance region of a tapered slit die, J. Appl. Polym. Sci. 17 (1973) 2369. H.J. Yoo, C.D. Han, Stress distribution of polymers in extrusion through a converging die, J. Rheol. 25 (1981) 115. P.N. Kaloni, M.T. Kamel, A note on the Hamel flow of Cosserat fluids, ZAMP 31 (1980) 293. A.M. Hull, An exact solution for the slow flow of a general linear viscoelastic fluid through a slit, J. Non-Newtonian Fluid Mech. 8 (1981) 327. D. Mansutti, K.R. Rajagopal, Flow of shear thinning fluid between intersecting planes, Int. Non-Linear Mech. 26 (1991) 769. R.K. Bhatnagar, K.R. Rajagopal, G. Gupta, Flow of an Oldroyd-B fluid between intersecting planes, J. Non-Newtonian Fluid Mech. 46 (1993) 498. S. Baris, Flow of second grade viscoelastic fluid in a porous converging channel, Turkish J. Eng. Env. Sci. 27 (2003) 73. M.A. Fontelos, A. Friedman, The flow of a class of Oldroyd fluids around a re-entrant corner, J. Non-Newtonian Fluid Mech. 95 (2000) 185. J. Riedler, W. Schneider, Viscous flow in corner regions with a moving wall and leakage of fluid, Acta Mech. 48 (1983) 95. Y.N. Huang, Flow of a non-Newtonian fluid between intersecting planes of which one is moving, Rheol. Acta. 32 (1993) 490. C.P. Hills, H.K. Moffat, Rotary honing: a variant of the Taylor paint-scraper problem, J. Fluid Mech. 418 (2000) 119. I. Mustakis, S. Kim, Microhydrodynamics of sharp corners and edges, traction singularities, J. AICHE 44 (1998) 1469. P.N. Shankar, On stokes flow in a semi-infinite wedge, J. Fluid Mech. 422 (2000) 69. G.I. Taylor, On scraping viscous fluid from a plane surface, in: M. Schäfer (Ed.), Miszellangen der Angewandten Mechanik, vol. 313, Akademie, 1962. K. Takahashi, M. Sasaki, K. Arai, S. Saito, Effects of geometrical variables of helical ribbon impellers on mixing of highly viscous Newtonian liquids, J. Chem. Eng. Jpn. 15 (1982) 217. G.B. Tatterson, R.S. Brodkey, R.V. Calabrese, More mixing technology into the 21st century, Chem. Eng. Prog. 87 (1991) 45. C.P. Hills, Eddies induced in cylinderical containers by a rotating end wall, Phys. Fluids 13 (8) (2008). D.D. Joseph, R.L. Fosdick, The free surface on a liquid between cylinders rotating at different speeds, Arch. Ration. Mech. Anal. 49 (1973) 321. H.K. Moffat, V. Mak, Corner singularities in three-dimensional Stokes flow, IUTAM Symposium on Non-Linear Singularities in Deformation and Flow, Kluwer, 1999, pp. 21–26. K.M. Jansons, A new angle on G.I. Taylor’s scraper, Phys. Fluids 30 (1987) 4.
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Creeping flow of a second grade fluid in a corner A.M. Siddiqui a, Akhlaq Ahmad a,b,*, Naseer Ahmed b,1 a b
Department of Mathematics, The Pennsylvania State University, York Campus, USA Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
a r t i c l e
i n f o
Article history: Received 15 July 2008 Received in revised form 16 June 2010 Accepted 7 July 2010 Available online 14 July 2010 Keywords: Creeping flow Corner Rotating boundary Domain perturbation
a b s t r a c t A variant of Taylor’s (1962) [23] scraper problem, in which, the lower plate rotates is considered. The non-linear partial differential equations governing the flow of a second grade fluid are modeled and solved by using the domain perturbation technique considering the angular velocity of the rotating plate as a small parameter. Also the rheology of the second grade fluid is examined by depicting the profiles of the velocity, stream function, pressure and stress fields. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction The creeping corner flow induced by a steady in-plane motion of one of the walls has been examined by Moffat [1] and Batchelor [2]. Hancock and Lewis [3] have investigated the effects of inertia forces, by constructing a regular perturbation series for the stream function, of which the leading term is the known similarity solution, but their works are restricted to Newtonian Fluids. Stokes flow in corners has also been studied by other authors. Anderson and Davis [4] considered viscous flow in a wedge made up by one rigid and one stress free boundary. Moreover, local solutions and partially local solutions, the latter ones do not satisfy all boundary conditions, were calculated at a rigid wedge where two immiscible viscous fluids meet at the vertex. In a later paper the same authors investigated the flow near a tri-junction including heat transfer and solidification [5]. Other boundary conditions have been considered by Betelu et al. [6]. Neglecting surface tension they calculated the viscous flow and its temporal evolution in a corner bounded by two stress free surfaces initially forming a sharp wedge. Neither of these studies, however, was devoted to the case when one wall is subjected to a given non-zero tangential stress. Despite the above studies of viscous fluids, the flow of the Newtonian fluids between intersecting planes has been extended within the context of various different non-Newtonian fluid models because of the wide range of engineering applications. For example, it will help to provide valuable information on the design of extrusion dies of industrial importance. Theoretical research on steady flow of this type was initiated by Langlois and Rivlin [7] who carried out perturbation analyses about slow flow of a non-Newtonian fluid through a wedge and a cone. They found that the stresses developed by the viscoelastic properties of the fluid were incompatible with radial flow and vortices were predicted in the perturbation solution. Strauss [8] was the first to present the solution of the steady, two-dimensional, and inertial flow of an incompressible Maxwell fluid between intersecting planes by using a series expansion in terms of decreasing powers of r. In his subsequent
* Corresponding author at: Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan. Tel.: +92 3009870978. E-mail addresses: [email protected] (A.M. Siddiqui), [email protected] (A. Ahmad), [email protected] (N. Ahmed). 1 Deceased author. 0307-904X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2010.07.011
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study [9], he considered the stability of the same flow problem. Han and Drexler [10] have studied non-inertial converging flow, using a modified second-order fluid model They observed that all three material functions depend on the second invariant of the rate of deformation. They also made a comparison between the theoretically predicted velocity profiles and stress distributions with the experimentally determined ones. Yoo and Han [11] carried out experiments on the converging slow flow of a polymer between intersecting planes and tried to explain the data in terms of the second grade fluid model. They found that theoretical analysis corroborates qualitatively with experimentally determined stress distributions. In the case of most non-Newtonian fluids a purely radial flow is not possible if inertial terms are to be retained in the equations of motion. Kaloni and Kamel [12] have shown that there cannot be a purely radial flow of Cosserat fluids in convergent channels. Later, Hull [13] studied the non-inertial flow of a general linear viscoelastic fluid in this geometry. He showed that radial flow is obtained for a wedge of 90° and no others. The similar results are valid for the Rivlin-Ericksen fluids. Mansutti and Rajagopal [14] studied the non-inertial flow of a shear thinning fluid between intersecting planes. They showed that sharp and pronounced boundary layers develop adjacent to the solid boundaries, even at zero Reynolds number. Bhatnagar et al. [15] have extended the analysis of Strauss [8] to an Oldroyd-B fluid which is characterized by a viscosity and two material constants with units of time. In their work, the effects of much higher values of the Reynolds number than Strauss’ work and the elastic parameter on the streamline patterns have been discussed. Serdar Baris [16] has investigated the flow of second grade viscoelastic fluid in a porous converging channel, he obtained the expressions for velocity field and temperature distribution by employing a numerical technique. Marco et al. [17] considered general Oldroyd model and have shown that the corner singularity is a sum of Newtonian singularity plus a lower order non-Newtonian singularity. Riedler and Schneider [18] studied the non-inertial flow of a power law fluid in a corner region with a moving wall and showed the streamline patterns near the wall were considerably less affected by the the power law exponent than near the wall at rest. Strauss’ work on the Maxwell fluid has been extended by Huang et al. [19] for the exact same geometry and boundary conditions as Strauss’ study, to the case of an Oldroyd-B fluid. They find that an increase in the elastic parameter reduces cellular structure. The question of true behaviour of a non-Newtonian fluid flow around a mathematically sharp edge is obviously a combined mathematical and physical problem. Whether the mathematical solutions constructed are regular or singular, their agreement with the physical reality must still be examined. It is, however, as difficult to establish a rigorous mathematical proof as it is complicated to obtain experimental evidence of a quantitative nature. Hills and Moffat [20] included experimental evidence for the three-dimensional scraper honing problem, but in general visualization of complicated three-dimensional flows is difficult and little experimental evidence is available. However, the two main papers on stokes flow in three-dimensional corners, Mustakis and Kim [21] and Shankar [22] both included major computational work to support their theories. The computations in Shankar gave a check on self consistency by testing the satisfaction of the boundary conditions. Mustakis and Kim [21] confirmed the traction singularity exponents which they found from two independent numerical calculations. Also they speculated that their work could be compared with experimentally recorded data. Many questions remain open for the three-dimensional problem. That the general form of the Stokes flow in a cone is not yet known. It is surprising, but perhaps the problem is amenable to a treatment similar to Shankar [22]. There is still very little known about the flow at polyhedral vertices, except for the particular cases described by Mustakis and Kim [21] and Shankar [22]. In their paper Hills and Moffat [20] have studied the three dimensional flow induced by the rotation of lower boundary in its plane, the upper plane being fixed at a corner angle h0. A local similarity solution valid in a neighbourhood of centre of rotation is obtained and streamlines are shown to be closed curves. A similar conclusion was reached by Shankar [22] who studied the Stokes problem in a trihedral corner with a non-zero boundary velocity assigned within some ring at a corner’s wall. He established the existence of corner eddies in the antisymmetric flow and found non-closed spirally shaped streamlines. This complicated behaviour was caused by the three dimensional nature of the flow. We consider in this paper a variant of Taylor’s [23] paint scraper problem, in which the lower plate rotates in its plane with constant angular velocity X about a point, taken to be origin, on the line of intersection of two planes. We investigate the Stokes flow of a second grade fluid flow induced by in-plane rotation of lower boundary with constant angular velocity X. This problem has potential relevance to situations in which fluids are mixed or kept in motion by the rotation of a paddle or rotor, or when fluid is scraped from a smooth surface. Such a situation has been investigated experimentally by Takahashi et al. [24], who considered the mixing of two highly viscous Newtonian fluids by the action of a helical blade in scraping contact with the base of a circular cylinder. The importance of understanding rotationally driven flows has been emphasize by Tatterson, Brodkey and Calabrese [25]. We emphasize here that for the configuration under consideration, all three velocity components u, v and w are functions of all three cylindrical polar coordinates (r, hz), where z is the axial coordinate parallel to the line of intersection of the planes. Even within the Stokes approximation, these velocity components are coupled through the pressure field. 2. Basic constitutive equations The second-order approximation to the general constitutive equation of a simple incompressible fluid is given by
T ¼ pI þ S;
ð2:1Þ
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623
where
S¼
2 X
Si ;
ð2:2Þ
i¼1
with
S1 ¼ lA1 ;
ð2:3Þ 2 2 A1 ;
S2 ¼ a1 A2 þ a
ð2:4Þ
where the coefficient l is the viscosity and a1, a2, are the normal stress moduli. For steady motion the Rivlin-Ericksen tensors satisfy the recursion relation T
ACþ1 ¼ ðgradAC Þu þ AC gradu þ ðAC graduÞ ;
ð2:5Þ
A0 ¼ I:
ð2:6Þ
with
For steady motion the field equations are given as
divu ¼ 0;
ð2:7Þ
b 1 p ðgraduÞu ¼ grad þ divS;
q
ð2:8Þ
q
where q is the constant density, b p is the pressure and the effects of body forces are negligible. We now formulate our problem. 3. Formulation of the problem We now consider a variant of Taylor’s [23] scraper problem, in which, the lower plate rotates in its plane with constant angular velocity X about a fixed point, taken to be the origin, on the line of intersection of the two planes. As discussed in [20], we shall find that for the configuration indicated by Fig. 3.1, all three velocity components (u, v, w) are functions of all three cylindrical polar coordinates (r, h, z), where z is now the axial coordinate parallel to the line of intersection of the planes. With the geometry and the coordinate system indicated in Fig. 3.1, the flow in the region 0 < h < h0 is given in the Stokes approximation by the equations
divu ¼ 0; 1 divS ¼ gradp;
ð3:1Þ ð3:2Þ
q
where p is the pressure per unit density and the velocity field relative to the coordinates (r, h, z) with boundary conditions is given by
Fig. 3.1. Geometry of the problem.
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u ¼ uer þ v eh þ wez ; u ¼ 0;
ð3:3Þ
on h ¼ 0;
ð3:4Þ
u ¼ ðXz; 0; Xr Þ on h ¼ h0 :
ð3:5Þ
We shall refer to the fixed plane h = h0 as the blade and to the rotating plane as the plate or base. We let the velocity and pressure fields of the form
u ¼ uðr; h; z; XÞ and p ¼ pðr; h; z; XÞ:
ð3:6Þ
and introduce the non-dimensional variables [26]
r r ¼ ; a
z z ¼ ; a
h ¼
h ; h0
u ¼
u
Xa
;
where a is the length scale measured from the corner parallel to the plane of each plate to an arbitrary arc up to which the viscous forces dominate over the inertia forces. That is, a is the length scale within the neighbourhood of the intersection of boundaries, where the present solution is valid. It should be noted that after introducing the new starred variables, we, then drop the asterisks and hence the forms of the Eqs. (3.1)–(3.6) remain the same. Now, following Joseph [27], we assume the series expansion of velocity and pressure fields ass
u¼
n X 1 ðkÞ k u X þ OðU n Þ k! k¼1
ð3:7Þ
p¼
n X 1 ðkÞ k p X þ OðU n Þ; k! k¼1
ð3:8Þ
and
respectively. Here, the following notation
ðÞðkÞ ¼
@k @ Xk
ðÞjX¼0 ;
ð3:9Þ
has been employed. Accordingly, the Eqs. (3.1)–(3.3) take the form
div uðkÞ ¼ 0; 1 divSðkÞ ¼ gradpðkÞ ;
ð3:11Þ
uðkÞ ¼ uðkÞ ðr; h; zÞer þ v ðkÞ ðr; h; zÞeh þ wðkÞ ðr; h; zÞez
ð3:12Þ
pðkÞ ¼ pðkÞ ðr; h; zÞ;
ð3:13Þ
q
ð3:10Þ
where
and
are functions of position only. The boundary conditions (3.4) and (3.5) can now be written in the form
uð1Þ ¼ ð0; 0; 0Þ at h ¼ h0 ; u
ðkÞ
¼ ð0; 0; 0Þk > 1 at h ¼ h0
ð3:14Þ ð3:15Þ
and
uð1Þ ¼ ðz; 0; r Þ at h ¼ 0;
ð3:16Þ
uðkÞ ¼ ð0; 0; 0Þk > 1 at h ¼ 0:
ð3:17Þ
4. First order problem For k = 1, the field equations become
divuð1Þ ¼ 0; 1 divSð1Þ ¼ gradpð1Þ :
q
ð4:1Þ ð4:2Þ
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Here ð1Þ
ð1Þ
Sð1Þ ¼ S1 þ S2 ;
ð4:3Þ
with ð1Þ
ð1Þ
S1 ¼ lA1 ;
ð4:4Þ
T ð1Þ A1 ¼ graduð1Þ þ graduð1Þ
ð4:5Þ
where
and ð1Þ
Ai
ð1Þ
¼ 0; Si
¼ 0 for i P 2:
ð4:6Þ
From Eqs. (3.12) and (3.13), the first order velocity and pressure fields are
uð1Þ ¼ uð1Þ ðr; h; zÞer þ v ð1Þ ðr; h; zÞeh þ wð1Þ ðr; h; zÞez
ð4:7Þ
pð1Þ ¼ pð1Þ ðr; h; zÞ;
ð4:8Þ
and
respectively. For the problem under consideration, the continuity Eq. (4.1) in terms of cylindrical polar coordinates, assumes the form
uð1Þ @uð1Þ 1 @ v ð1Þ @wð1Þ þ þ þ ¼ 0: r @h r @r @z
ð4:9Þ
Now substituting the expression for grad u(1) into the Eqs. (4.4) and (4.5) and then taking into account the results so obtained, the components of the momentum Eq. (4.2) are given by
@pð1Þ ¼m @r
2 @uð1Þ r @r
@pð1Þ ¼m @h
3 @uð1Þ r @h
2 ð1Þ
@ 2 uð1Þ @h2
þ 2 @ @ru2 þ r12
þ 1r
2 ð1Þ ð1Þ ð1Þ r32 @ v@h þ @ @zu2 2ur2 2 ð1Þ
2 ð1Þ
u þ @@h@r þ @ v@r þ r @ @rv2
v r þ 2r ð1Þ
ð1Þ
@ 2 v ð1Þ @h2
2 ð1Þ
þ r @ @zv2
@ 2 v ð1Þ @h@r
! ð4:10Þ
;
! ð4:11Þ
;
@pð1Þ ¼ mr2 wð1Þ ; @z 2
@ 1 where r2 ¼ @r 2 þ r
@ @r
þ r12
ð4:12Þ @2 @h2
2
@ þ @z 2 , is the Laplacian operator in cylindrical polar coordinates.
Eliminating the pressure p(1) from the Eqs. (4.10) and (4.11), and then introducing the stream function w(1)(r, h, z) analogous to the stream function w(r, h, z) according to [20,28,29], the derived velocity field have the following expressions:
uð1Þ ¼
1 @wð1Þ ð1Þ @wð1Þ ;v ¼ ; r @h @r
ð4:13Þ
By virtue of these results, the continuity equation leads to the fact that w(1) does not depend upon the coordinate z and the first two momentum Eqs. (4.10) and (4.11) reduce to
0
2 ð1Þ 4 @ w @h2
3 m@ r
r22
@ 3 wð1Þ @r@h2
4 ð1Þ
þ 2r
þr @ @rw4 1r
@ 4 wð1Þ @r 2 @h2
@ 2 wð1Þ @r 2
þ r13
@ 4 wð1Þ @h4
3 ð1Þ
þ 1r 4 ð1Þ
@ 4 wð1Þ @z2 @h2
w @ w 1 þ @@r@z 2 þ r @r2 @z2 þ r 2
3 ð1Þ
þ 2 @ @rw3
@wð1Þ @r
1 A ¼ 0:
ð4:14Þ
The boundary conditions for w(1), in view of (4.13), take the forms
@wð1Þ @wð1Þ ¼ ¼ 0; at h ¼ h0 ; @h @r @wð1Þ @wð1Þ ¼ rz; ¼ 0; at h ¼ 0: @h @r
ð4:15Þ ð4:16Þ
In what follows, we find the solution of Eq. (4.14). 4.1. Solution of the first order problem The solution of the system (4.14)–(4.16) is given by
wð1Þ ¼ rzf ðhÞ;
ð4:17Þ
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where f is defined by
f ¼
U h20
2
sin h0
n o 2 h20 sin h þ sin h0 h cos h þ ðh0 sin h0 cos h0 Þh sin h :
Thus, the velocity components u(1) and
uð1Þ ¼ zf
ð4:18Þ
v(1), become
0
ð4:19Þ
and
v ð1Þ ¼ zf ;
ð4:20Þ
respectively. To obtain the third component w(1) of the velocity field, we first find the pressure field p(1). For this purpose, we substitute the last two results into the pressure Eq. (4.10), we find that
@pð1Þ g0 ¼ mz 2 ; @r r
ð4:21Þ
where g is given by
g ¼ f 0 þ f 00 :
ð4:22Þ
Integration of this equation yields that
pð1Þ ¼ mz
g0 þ p0 ðh; zÞ: r
ð4:23Þ
Also the other pressure equation takes the form
@pð1Þ g ¼ mz : r @h
ð4:24Þ
Differentiating Eq. (4.23) partially with respect to h and then comparing the result with Eq. (4.24), we obtain
@p0 ¼ 0; @h
ð4:25Þ
which leads to the fact that p0 is independent of h. Consequently, we let p0 = p0(z) = constant and we obtain the required expression for the pressure distribution p(1) as
pð1Þ ¼ mz
g0 þ p0 : r
ð4:26Þ (1)
In view of the last result, the third component of velocity w ð1Þ
satisfies the equation
0
@p g ¼ m : @z r
mr2 wð1Þ ¼
ð4:27Þ
Following Taylor [23], we analogously may assume the solution in the form
wð1Þ ¼ rFðhÞ;
ð4:28Þ
which reduces (4.27) to the following boundary value problem: 00
F þF ¼ F ¼ 0; F ¼ 1;
2 2 k sin h þ sin h0 cos h 2
h20 sin h0
;
at h ¼ h0 ; at h ¼ 0:
ð4:29Þ ð4:30Þ ð4:31Þ
The solution of this problem can be easily found as
F ¼ f 0 ðhÞ;
ð4:32Þ
so that 0
wð1Þ ¼ rf ðhÞ:
ð4:33Þ
This yields a complete first order solution of the problem under consideration. 5. Second order problem We now turn to find the solution of our problem for k = 2, for which the governing equations are
divuð2Þ ¼ 0;
ð5:1Þ
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1
q
divSð2Þ ¼ gradpð2Þ ;
627
ð5:2Þ
where ð2Þ
ð2Þ
Sð2Þ ¼ S1 þ S2 :
ð5:3Þ
Here ð2Þ
ð2Þ
S1 ¼ lA1
ð5:4Þ
2 ð2Þ ð2Þ ð1Þ S2 ¼ a1 A2 þ 2a2 A1 ;
ð5:5Þ
and
ð2Þ
ð2Þ
in which A1 and A2 are given by
T ð2Þ A1 ¼ graduð2Þ þ graduð2Þ
ð5:6Þ
and ð2Þ
A2 ¼ 2
ð1Þ
gradA1
ð1Þ T ð1Þ ; uð1Þ þ A1 graduð1Þ þ A1 graduð1Þ
ð5:7Þ
respectively. Also we have ð2Þ
Ai
ð2Þ
¼ 0; Si
¼ 0 for i P 3:
ð5:8Þ
In the present case, the velocity and pressure fields are assumed to be
uð2Þ ¼ uð2Þ ðr; h; zÞer þ v ð2Þ ðr; h; zÞeh þ wð2Þ ðr; h; zÞez
ð5:9Þ
and
pð2Þ ¼ pð2Þ ðr; h; zÞ;
ð5:10Þ
respectively. Computing all the tensors involved in Eqs. (5.4)–(5.7) by making use of Eqs. (5.9) and (5.10), the continuity and the momentum equations for the second-order are
uð2Þ @uð2Þ 1 @ v ð2Þ @wð2Þ þ þ þ ¼0 r @h r @r @z
ð5:11Þ
and
@pð2Þ ¼m @r
2 @uð2Þ r @r
( þ 2a1
2 ð2Þ
þ 2 @ @ru2 þ r12
@ 2 uð2Þ @h2
þ 1r
@ 2 v ð2Þ @h@r
!
2 ð2Þ ð2Þ ð2Þ r32 @ v@h þ @ @zu2 2ur2 2
2
0
2zr3 gðf þ gÞ zr3 ðfgÞ00 1r ð2f 0 g Þ 2
)
2
2gr þ 2zr3 fg 2 fgr 2 z g2 4a2 3 g 2 þ ; r r 2 ð2Þ 2 ð2Þ ! ð2Þ 3 @uð2Þ u þ @@h@r þ @ v@r þ r @ @rv2 r @h
@pð2Þ ¼m @h
2 ð2Þ
2 ð2Þ
v r þ 2r @ @hv2 þ r @ @zv2 2z2 0 þ 2a1 4f 0 g 2 ðfgÞ0 þ 2ðfgÞ0 þ fg f 0 g r 2 z þ 4a2 2 þ 1 gg 0 ; r nz 0 o @pð2Þ 0 ¼ mr2 wð2Þ þ 2a1 2 fg f 0 g : r @z ð2Þ
ð5:12Þ
ð5:13Þ ð5:14Þ
Now, we eliminate the pressure p(2) by cross differentiation and then introduce the stream function w(2)(r, h, z) such that
uð2Þ ¼
1 @wð2Þ ð2Þ @wð2Þ ;v ¼ ; r @h @r
ð5:15Þ
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by virtue of which, the continuity equation leads to the fact that w(2) is independent of z. Similarly, the Eqs. (5.12), (5.13) reduce to the following equation:
0 0¼m
2 ð2Þ 4 @ w @ r3 @h2
r22
@ 3 wð2Þ @r@h2
þ 2r
4 ð2Þ
þr @ @rw4 1r
@ 4 wð2Þ @r 2 @h2
2 ð2Þ
@ w @r 2
þ r13
@ 4 wð2Þ @h4
3 ð2Þ
þ 1r
@ 4 wð2Þ @z2 @h2
4 ð2Þ
w @ w 1 þ @@r@z 2 þ r @r2 @z2 þ r 2
3 ð2Þ
þ 2 @ @rw3 ð2Þ
@w @r
1
0
A 8ð2a1 þ a2 Þ gg : r
ð5:16Þ
Also, the corresponding boundary conditions are
@wð2Þ @wð2Þ ¼ ¼ 0; at h ¼ h0 ; @h @r ð2Þ ð2Þ @w @w ¼ 0; ¼ 0; at h ¼ 0: @h @r
ð5:17Þ ð5:18Þ
We now solve this second-order boundary value problem. 5.1. Solution of the second-order problem As discussed in [23], we may propose the following solution
wð2Þ ¼ r 2 f1 ðhÞ;
ð5:19Þ
so that the Eq. (5.16) becomes the following fourth order ordinary differential equation
f1iv þ 4f100 ¼
8
m
ð2a1 þ a2 Þgg 0 :
ð5:20Þ
The corresponding boundary conditions on f1 are
f1 ¼ f10 ¼ 0;
at h ¼ 0;
ð5:21Þ
f1
at h ¼ h0 :
ð5:22Þ
¼
f10
¼ 0;
The solution of the system (5.20)–(5.22) is
f1 ¼ R1 þ R2 h þ R3 cos 2h þ R4 sin 2h þ
1 ð2a1 þ a2 Þðd1 h cos 2h d2 h sin 2hÞ; 2m
ð5:23Þ
where the constants R’s and d’s are given by
R1 ¼ R3 ; R2 ¼
cos 2h0 1 sin 2h0 1 R4 R3 ð2a1 þ a2 Þðd1 h0 cos 2h0 d2 h0 sin 2h0 Þ; h0 2m h0
ð5:24Þ ð5:25Þ
where R3 and R4 are given by
sin 2h0 2h0 4ðcos 2h0 1Þð1 cos 2h0 h0 sin 2h0 Þ 1 ð2a1 þ a2 Þðcos 2h0 1Þðd1 cos 2h0 2d1 h0 sin 2h0 d2 sin 2h0 2d2 h0 cos 2h0 Þ 2m 1 1 h0 sin 2h0 ð2a1 þ a2 Þd1 þ ð2a1 þ a2 Þd1 ðcos 2h0 1Þ ð2a1 þ a2 Þ sin 2h0 ðd1 h0 cos 2h0 d2 h0 sin 2h0 Þ þ 2m m m ð2a1 þ a2 Þ h0 ð2a1 þ a2 Þd1 ðd1 h0 cos 2h0 d2 h0 sin 2h0 Þ þ ð5:26Þ 2mðcos 2h0 1Þ 2mðcos 2h0 1Þ
R3 ¼
and
R4 ¼
1 ð1 cos 2h0 h0 sin 2h0 Þ 1 ð2a1 þ a2 Þðcos 2h0 1Þðd1 cos 2h0 2d1 h0 sin 2h0 d2 sin 2h0 2d2 h0 cos 2h0 Þ 2m 1 1 h0 sin 2h0 þ ð2a1 þ a2 Þd1 ðcos 2h0 1Þ ð2a1 þ a2 Þ sin 2h0 ðd1 h0 cos 2h0 d2 h0 sin 2h0 Þ þ ð2a1 þ a2 Þd1 ; 2m m m ð5:27Þ
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respectively. We also have
2 4 2 k sin h0 d1 ¼ 2 ; 2 h20 sin h0
ð5:28Þ
2
4k sin h0 d2 ¼ 2 : 2 2 h0 sin h0
ð5:29Þ
Substituting the expression of stream function from (5.19) into the expressions of velocity components u(2) and v(2), one can readily find 0
uð2Þ ¼ rf 1
ð5:30Þ
and
v ð2Þ ¼ 2rf 1 :
ð5:31Þ
Substituting the values of u
(2)
and v
pð2Þ ¼ m 4f10 þ f1000 ln r þ 2a1
(2)
"
in the Eqs. (5.12), (5.13), we get the expression for the pressure distribution p(2) as zfg r2 0
þ zðfgÞ 2r 2
00
ð2f 0 g Þ þ 2g 2 þ 2fg ln r
#
þ 2a2
2 zg 2 2g ln r : r2
ð5:32Þ
Differentiating Eq. (5.32) partially with respect to z and then comparing the result with Eq. (5.14), we find that 2
ð2Þ
mr w
( 0 0 ) fg f 0 g 2fg ðfgÞ00 g2 þ 4a2 2 : ¼ 2a1 þ 2 2 2 r r r r
ð5:33Þ
Let us suppose that the solution of Eq. (5.33) is
wð2Þ ¼ f2 ðhÞ;
ð5:34Þ
then partial differential Eq. (5.33) reduces to the following ordinary differential equation:
f200 ¼ 2a1 2g 2 2f 0 g 0 þ 4a2 g 2 ;
ð5:35Þ
where the boundary conditions on f2 are
f2 ¼ 0;
at h ¼ 0;
ð5:36Þ
f2 ¼ 0;
at h ¼ h0 :
ð5:37Þ
The system (5.35)–(5.37) has the solution 2
b1 þ C b 2 h þ A1 h A2 cos 2h A3 sin 2h A4 ðh cos 2h sin 2hÞ A5 ðh sin 2h þ cos 2hÞ: f2 ¼ C 4 4 2 4 4
ð5:38Þ
The constants involved in Eq. (5.38) have the following values:
b 1 ¼ A1 þ A5 ; C 4 A1 þ A5 A1 h0 A2 cos 2h0 A3 sin 2h0 b þ þ C2 ¼ 4h0 2 4h0 4h0 A4 A5 þ ðh0 cos 2h0 sin 2h0 Þ þ ðh0 sin 2h0 þ cos 2h0 Þ; 4h0 4h0 16ða1 þ a2 Þ 8a1 b1 A1 ¼ 2 a3 ; 2 2 2 h0 sin h0 h20 sin h0
ð5:39Þ
ð5:40Þ ð5:41Þ
16ða1 þ a2 Þ 8a1 A2 ¼ 2 b2 2 a4 ; 2 2 2 2 h0 sin h0 h0 sin h0
ð5:42Þ
16ða1 þ a2 Þ 8a1 A3 ¼ 2 a5 2 a1 ; 2 2 2 2 h0 sin h0 h0 sin h0
ð5:43Þ
8a1 A4 ¼ 2 a5 ; 2 2 h0 sin h0
ð5:44Þ
8a1 A5 ¼ 2 a2 ; 2 2 h0 sin h0
ð5:45Þ
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k 2 2 h 2 sin h0 ; 2 0 1 4 2 sin h0 k ; a2 ¼ 2 1 2 2 2 4 a3 ¼ h0 sin h0 sin h0 k ; 2
a1 ¼
ð5:46Þ ð5:47Þ ð5:48Þ
Fig. 6.1. Variation of velocity component u with X for viscous fluid.
Fig. 6.2. Variation of velocity component u with X for second grade fluid.
Fig. 6.3. Variation of velocity component
v with X for viscous fluid.
A.M. Siddiqui et al. / Applied Mathematical Modelling 35 (2011) 621–636
1 2 2 2 4 h0 sin h0 sin h0 þ k ; 2 2 a5 ¼ k sin h0 ; 1 2 4 k þ sin h0 ; b1 ¼ 2 1 2 4 b1 ¼ k sin h0 : 2
a4 ¼
Fig. 6.4. Variation of velocity component
631
ð5:49Þ ð5:50Þ ð5:51Þ ð5:52Þ
v with X for second grade fluid.
Fig. 6.5. Variation of velocity component w with X for viscous fluid.
Fig. 6.6. Variation of velocity component w with X for second grade fluid.
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Summarizing the results of the perturbation series up to second-order, we have
uðr; h; zÞ ¼ Xuð1Þ þ
uð2Þ þ O X3 ;
X2 2 2
v ðr; h; zÞ ¼ Xv ð1Þ þ
X
2
v ð2Þ þ O
X3 ;
X ð2Þ wðr; h; zÞ ¼ Xwð1Þ þ w þ O X3 ; 2 X2 ð2Þ ð1Þ pðr; h; zÞ ¼ Xp þ p þ O X3 : 2
ð5:53Þ ð5:54Þ
2
Fig. 6.7. Variation of pressure p with X for viscous fluid.
Fig. 6.8. Variation of pressure p with X for second grade fluid.
Fig. 6.9. Variation of tangential stress Tp with X for viscous fluid.
ð5:55Þ ð5:56Þ
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Substituting the values of u(i), v(i), w(i) and p(i) (i = 1, 2) in (5.53)–(5.56), we finally obtain the required velocity and pressure fields up to the second-order as 0
uðr; h; zÞ ¼ Xzf ðhÞ þ
X2
0
rf ðhÞ; 2 1 v ðr; h; zÞ ¼ Xzf ðhÞ þ X2 rf1 ðhÞ;
ð5:57Þ ð5:58Þ
2
wðr; hÞ ¼ Xrf ðhÞ þ
X
2
f2 ;
ð5:59Þ
Fig. 6.10. Variation of tangential stress Tb with X for second grade fluid.
Fig. 6.11. Variation of tangential stress Tb with X for second grade fluid.
Fig. 6.12. Variation of normal stress Trr with X for rotating plate.
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" # 00 2 zfg þ zðfgÞ g 0 X2 0 zg 2 000 2r 2 r2 þ X2 a2 2 2g 2 ln r : pðr; h; zÞ ¼ Xmz þ m 4f1 þ f1 ln r þ X a1 0 0 r 2 r ð2f g Þ þ 2g 2 þ 2fg ln r
ð5:60Þ
The tangential stresses Trz and Thz are given by
zg 2 T rz ¼ X2 ð2a1 þ a2 Þ ; r X2 2mf20 2a1 z 0 fg f 0 g : þ T hz ¼ mg X þ r 2 r
ð5:61Þ ð5:62Þ
Let us denote the tangential stress at lower rotating plate by Tp and on upper fixed plate (i.e. blade) by Tb, then their expressions can readily be obtained by setting h = 0 and h = h0 in (5.61) and (5.62) respectively. In what follows, we analyze our results by sketching the graphs. 6. Graphs and discussion In order to observe the geometrical structure in various orientations of the scraper, we draw the following necessary graphs in Figs. 6.1–6.11 and to examine the rheology of the second grade fluid by depicting velocity profiles, the stream function, the pressure and the stress fields. In Figs. 6.1–6.11, we have tried to expose the characteristics of velocity, pressure and stress fields for flow of both viscous and second grade fluids. Fig. 6.1 shows the behavior of velocity component u for viscous fluid in the domain 0 < h < p against the angular velocity X of the rotating plate. Here we note that for small values of angle h, u starts with low positive values, then it become negative for h 2 (0.5, 1) (h in radian measurements). After this reverse flow, the component u increases rapidly for increasing values of angular velocity X. Fig. 6.2 is plotted for the u component of the velocity in the case of non-Newtonian fluid, but now we observe that every curve of the velocity u remain almost constant in the domain h 2 (0, 2.5) (h in radian measurements), when the angular velocity X is fixed for every curve. At h = 2.4 radians, all the velocity curves change sign and then velocity increases in the reverse direction with an increase in angular velocity X.
Fig. 6.13. Variation of normal stress Thh with X for fixed plate.
Fig. 6.14. Variation of radial velocity u with material parameter a1.
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635
Fig. 6.15. Variation of radial velocity u with material parameter a2.
Fig. 6.3 depicts the v component of the velocity for viscous fluid, in this figure we note that the velocity increases in the negative direction with X, it vanishes at h = 1.5 radians and then again increasing behavior is shown by v component in the reverse direction. In Fig. 6.4, curves are plotted for v component, when the second grade fluid flows in the flow domain, again we observe that as the angular velocity X is increased v increases in the reverse direction and this trend repeats itself because the sinusoidal nature of the flow. The third component w of the velocity field for viscous fluid flow is pictured in Fig. 6.5, where we see that for small values of angle h the response against the angular velocity X is not appreciable but as h increases the velocity also increases in the negative direction with an increase in X. In Fig. 6.6, the profiles of velocity component w are plotted against the angular velocity X for second grade fluid keeping the other parameters fixed. In this case it is very surprising that no reverse flow occurs in the whole flow domain and velocity goes on increasing with X in the specified area of flow. The next two figures are sketched to discuss the pressure distribution in the flow field for viscous and second grade fluids. Fig. 6.7 shows the pressure distribution for the viscous fluid flow, here we note that for small h pressure curves start with almost zero value and then go on increasing with X. Then again pressure becomes zero at h = 2.5 radians, afterwards it starts increasing in the negative direction. Note that the pressure curves change sign at h = 2.5 radians in the flow field under consideration. Fig. 6.8 pictures the pressure field for non-Newtonian fluid flow, it demonstrates that for small h pressure is also small and as h increases, there appear considerable variations in the pressure field and it jumps high and remain bounded as we increase the angular velocity X. It is worth noting that pressure remain always positive for second grade fluid flow. Figs. 6.9 and 6.10 depict the behavior of tangential stress on the fixed plane (blade). In Fig. 6.9 we note that the stress is in opposite direction to the fluid flow and goes on increasing in this direction as the angular velocity increases. In the mathematical expression for tangential stress Tb, if we put h0 = 0, an unbounded stress is obtained, this behavior can also be seen in the profiles shown in Fig. 6.9 i.e. there exist a flow singularity in the stress field for h0 = 0. For the non-Newtonian case (Fig. 6.10) the behavior of the tangential stress is opposite to that for viscous fluid, for present case stress remain positive, increases with X and it shoots up as h0 ? 0. As it is discussed earlier that the viscous contribution in the tangential stress for the rotating plate vanish, so in Fig. 6.11, we tried to explain the contribution due to non-Newtonian parameters in the stress field on the lower rotating plate. We observe that the parameters a1 and a2 do contribute in the stress filed Tp and stress increases with second grade parameters. Further as we increase angular velocity X the shear stress on the plate increases in the reverse direction. Normal stress at lower and upper plates is shown in the Figs. 6.12 and 6.13 respectively. The behavior of the normal stress is identical on both the plates, however, the strength of the normal stress at upper plate is very much stronger as compared to the lower rotating plate. In this flow velocity behaves like ra near the corner. Very near the corner, one hopes that length scales other than the distance from the corner become unimportant. Thus the radial flow dominates over the the flow in the angular direction and the assumption of dominant viscous forces imply that the velocity field along the z- direction is considerably less effected by the second grade parameters a1 and a2. Validating these considerations, we have plotted the radial component of the velocity in the Figs. 6.14 and 6.15 against the parameters a1 and a2 respectively. Against a1, the radial velocity is initially negative but it decreases with this parameter as the angle between the plates is increased. The radial velocity shows an opposite behavior against the parameter a2 provided that the other circumstances remain the same. 7. Conclusion The problem under consideration is a variant of Taylor’s paint scraper, here the lower plate (base) rotates with constant angular velocity X about a fixed point in its own plane. This problem has potential relevance to situations in which fluids are mixed, kept in motion by the rotation of a paddle or rotor and when fluid is scraped from a plane surface.
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It is to be noted that by setting a1 = a2 = 0, we recover the results obtained by Hills et al. The solutions (5.57)–(5.62) show that the flow fields are strongly dependent upon the angular velocity X, but they are valid only for small values of angular velocity. By setting X = 0, we see that velocity field vanish and no flow occurs, this matches with the physical situation because flow is generated by the rotation of the lower plate. The solutions (5.57)–(5.62) of the non-linear partial differential equations governing the flow reveal that there is an appreciable contribution of the second grade parameters in the flow regime, but the viscous effects on the tangential stress at the plate (base) do not contribute. It is also noted that as for as the radial distance is concerned there appears no singularity in the velocity field with respect to r (radial distance) but stress field vary as r1 and singularity appears as r ? 0. Acknowledgements We are grateful to the referee of previous version of this paper, whose penetrating comments led to significant improvements. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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