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A note on integration of the Navier–Stokes equations
Nonlinear Analysis: Real World Applications 9 (2008) 1823 – 1826 www.elsevier.com/locate/na
Short communication
A note on integration of the Navier–Stokes equations Arieh Pistiner Ministry of Environment Protection, Haifa, Israel Received 14 March 2007; accepted 28 March 2007
Abstract In this study the 2D Navier–Stokes equations are used to obtain a new self-similar equation. The latter equation, subject to appropriate boundary conditions and volume discharge, describes the pressure distribution and velocity field of a plane free jet. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Navier–Stokes equations; Integration; Self-similar equation; Free jet
1. Introduction In this study, the 2D steady-state Navier–Stokes equations for an incompressible fluid and in the absence of body forces [1] 2 ju ju 1 jp j u j2 u (1a) u +v =− + + 2 , jx jy jx jx 2 jy 2 jv 1 jp j v j2 v jv , (1b) +v =− + + u jx jy jy jx 2 jy 2 together with the mass balance equation ju jv + =0 jx jy
(2)
are analyzed. In the latter case, u(x, y) and v(x, y) are, respectively, the steady-state velocity profiles in the x- and the y-directions, p(x, y) is the pressure distribution, is the fluid density and its kinematic viscosity. Substituting (2) in (1a), (1b) the set (1a), (1b) can be rewritten in the following form: 2 ju2 juv 1 jp j u j2 u + =− + , (3a) + jx jy jx jx 2 jy 2 2 j2 v jv 2 1 jp j v juv (3b) + 2 . + =− + jx jy jy jx 2 jy E-mail address: [email protected]. 1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.03.017
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A. Pistiner / Nonlinear Analysis: Real World Applications 9 (2008) 1823 – 1826
We will refer to a case at which a fluid jet is injected from a line source into a fluid at rest. The line source is located parallel to the x-axis at distance y = +y+ from the origin. Accordingly, the volume flux q can be determined by integrating the fluid velocity v(x, +y+ ) over the length of the line source as follows: +D/2 q= (4) v(x, +y+ ) dx, −D/2
where D is the line source length. Now we will define dimensionless variables in the form u uˆ = , U v vˆ = , U xˆ =
Ux ,
(5a) (5b) (5c)
Uy , p pˆ = 2 , U
yˆ =
(5d) (5e)
where U is the characteristic velocity. Substituting (5a)–(5e) in (2) and (3a), (3b), we obtain dimensionless equations of the type juˆ 2 j2 uˆ juˆ vˆ jpˆ j2 uˆ + =− + 2 + 2, jxˆ jyˆ jxˆ jxˆ jyˆ
(6a)
jvˆ 2 jpˆ j2 vˆ j2 vˆ juˆ vˆ + =− + 2 + 2, jxˆ jyˆ jyˆ jxˆ jyˆ
(6b)
juˆ jvˆ + = 0, *xˆ *yˆ
(6c)
where xˆ and yˆ appearing in (5c), (5d) lead to representation, based on Reynolds number coordinates i.e. xˆ ≡ Rex ,
(7a)
yˆ ≡ Rey .
(7b)
In accordance with the above, the velocity components must vanish at an infinite distance from the line source uˆ → 0,
vˆ → 0,
pˆ → 0,
xˆ → ±∞ (yˆ > yˆ+ ),
(8a.c)
uˆ → 0,
vˆ → 0,
pˆ → 0,
yˆ → ±∞.
(8d.f)
Using the symmetry property of the jet in the domain −∞ < xˆ < ∞, +yˆ+ < yˆ < ∞, we can impose an additional boundary condition on xˆ = 0 jvˆ = 0, jxˆ
uˆ = 0,
p(0, ˆ y) ˆ = pˆ 0 (y) ˆ on xˆ = 0 (yˆ > + yˆ+ ).
(9a.c)
The volume flux defined in (4) is a conservative feature of the problem hence it must be independent of the yˆ coordinate. Accordingly, we may define a dimensionless volume flux qˆ in the following form: +∞ qˆ = q/Q = v( ˆ x, ˆ y) ˆ dxˆ = const, (10) −∞
where Q ≡ v.
A. Pistiner / Nonlinear Analysis: Real World Applications 9 (2008) 1823 – 1826
1825
2. The self-similar formulation We will further refer to the circumstances in which the steady-state velocities and the pressure distributions (defined in the domain −∞ < xˆ < ∞, +yˆ+ < yˆ < ∞) achieve a certain asymptotic deriving from a certain universal behavior, such that, they can be described by a single independent variable in the form u( ˆ x, ˆ y) ˆ = f ()yˆ −1 ,
(11a)
v( ˆ x, ˆ y) ˆ = F ()yˆ −1 ,
(11b)
Pˆ (x, ˆ y) ˆ = g()yˆ −2
(11c)
and =
xˆ . yˆ
Substituting (11b), (11d) in (10), the conservative, dimensionless volume flux qˆ takes the form +∞ qˆ = F () d = const. −∞
(11d)
(12)
It should be noted that the self-similar functional form (11a)–(11d) exists irrespective of the actual velocity v(x, ˆ +yˆ+ ) and an actual pressure p(x, ˆ +yˆ+ ). Eventually, it tends to the similarity solution derived above for an asymptotically large distance of yˆ from the line source. Both solutions, in the physical x. ˆ yˆ and the self-similar domain are, however, characterized by the same amount of volume flux given by (10) and (12). Substituting (11a)–(11d) in (6a), one obtains (f 2 ) − f F − (f F ) = −g + (f (1 + 2 )) ,
(13)
which can also be rewritten in the following form: f 2 − f F + g − (f (1 + 2 )) = G(),
(14a)
G = f F .
(14b)
The set (14a), (14b) combines to yield f 2 + g = (f (1 + 2 ) + G) .
(15)
Introduction of (11a)–(11d) into (6b) one obtains (f F ) − F 2 − (F 2 ) = g + (g) + (F (1 + 2 )) .
(16)
Expression (16) can also be presented in the form −F 2 + f F − g − (F (1 + 2 )) = H (),
(17a)
H = F 2 + g.
(17b)
Combination of (17a), (17b) yields the following expression: f F = (F (1 + 2 ) + H ) ,
(18)
and substituting of (14b) in (18) and integration yield H + F (1 + 2 ) − G = , where is a constant of integration.
(19)
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A. Pistiner / Nonlinear Analysis: Real World Applications 9 (2008) 1823 – 1826
Substituting (11a), (11b), (11d) in (6c) we obtain F=
f +c ,
(20)
where c is a constant to be determined. Introduction of (9a,b), into (11a), (11b), and (11d) results in F (0) = 0,
(21a)
f (0) = 0.
(21b)
According to (21b), the constant c in (20) must vanish and the equation reduces to F=
f .
(22)
Combination of (14a), (17a), (19) and (22) leads to g = 2F −
1 + 2
,
(23)
and substituting (23) in (13) yields (F (1 + 2 )) − 2F + = −F 2 . 2 1+
(24)
Differentiation of (24) yields the following differential equation: F (1 + 2 ) + 6F + 6F −
2 (1 + 2 )2
+ F 2 = 0.
(25)
In view of the boundary condition (9c) together with (11c) and (11d), we may impose g(0) = g0 . This, together with (23) immediately yields F (0) =
g0 + . 2
(26)
Eq. (25) can be solved with the aid of a standard numerical integration procedure, subjected to (12) and the boundary conditions (21a) and (26). In the limit, one obtains from (25) f ∼ F ∼
C1 C1 2
as → ±∞,
(27a)
as → ∞,
(27b)
where C1 is an integration constant. System (27a), (27b) shows that the boundary conditions (8a–c) are automatically satisfied. In addition to the above, the boundary conditions (8d–f) are automatically satisfied by (11a–c). 3. Concluding remark In principle, the similarity representation does not generally reproduce the actual boundary conditions on the line source for the pressure distribution and velocity profiles but rather exhibits the following dependence for any finite x: ˆ v( ˆ x, ˆ +yˆ+ ) → ∞,
p( ˆ x, ˆ +yˆ+ ) → ∞
as + yˆ+ → 0.
Reference [1] F.M. White, Fluid Mechanics, McGraw-Hill Kogakusha Ltd, 1979.
(28a,b)
Short communication
A note on integration of the Navier–Stokes equations Arieh Pistiner Ministry of Environment Protection, Haifa, Israel Received 14 March 2007; accepted 28 March 2007
Abstract In this study the 2D Navier–Stokes equations are used to obtain a new self-similar equation. The latter equation, subject to appropriate boundary conditions and volume discharge, describes the pressure distribution and velocity field of a plane free jet. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Navier–Stokes equations; Integration; Self-similar equation; Free jet
1. Introduction In this study, the 2D steady-state Navier–Stokes equations for an incompressible fluid and in the absence of body forces [1] 2 ju ju 1 jp j u j2 u (1a) u +v =− + + 2 , jx jy jx jx 2 jy 2 jv 1 jp j v j2 v jv , (1b) +v =− + + u jx jy jy jx 2 jy 2 together with the mass balance equation ju jv + =0 jx jy
(2)
are analyzed. In the latter case, u(x, y) and v(x, y) are, respectively, the steady-state velocity profiles in the x- and the y-directions, p(x, y) is the pressure distribution, is the fluid density and its kinematic viscosity. Substituting (2) in (1a), (1b) the set (1a), (1b) can be rewritten in the following form: 2 ju2 juv 1 jp j u j2 u + =− + , (3a) + jx jy jx jx 2 jy 2 2 j2 v jv 2 1 jp j v juv (3b) + 2 . + =− + jx jy jy jx 2 jy E-mail address: [email protected]. 1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.03.017
1824
A. Pistiner / Nonlinear Analysis: Real World Applications 9 (2008) 1823 – 1826
We will refer to a case at which a fluid jet is injected from a line source into a fluid at rest. The line source is located parallel to the x-axis at distance y = +y+ from the origin. Accordingly, the volume flux q can be determined by integrating the fluid velocity v(x, +y+ ) over the length of the line source as follows: +D/2 q= (4) v(x, +y+ ) dx, −D/2
where D is the line source length. Now we will define dimensionless variables in the form u uˆ = , U v vˆ = , U xˆ =
Ux ,
(5a) (5b) (5c)
Uy , p pˆ = 2 , U
yˆ =
(5d) (5e)
where U is the characteristic velocity. Substituting (5a)–(5e) in (2) and (3a), (3b), we obtain dimensionless equations of the type juˆ 2 j2 uˆ juˆ vˆ jpˆ j2 uˆ + =− + 2 + 2, jxˆ jyˆ jxˆ jxˆ jyˆ
(6a)
jvˆ 2 jpˆ j2 vˆ j2 vˆ juˆ vˆ + =− + 2 + 2, jxˆ jyˆ jyˆ jxˆ jyˆ
(6b)
juˆ jvˆ + = 0, *xˆ *yˆ
(6c)
where xˆ and yˆ appearing in (5c), (5d) lead to representation, based on Reynolds number coordinates i.e. xˆ ≡ Rex ,
(7a)
yˆ ≡ Rey .
(7b)
In accordance with the above, the velocity components must vanish at an infinite distance from the line source uˆ → 0,
vˆ → 0,
pˆ → 0,
xˆ → ±∞ (yˆ > yˆ+ ),
(8a.c)
uˆ → 0,
vˆ → 0,
pˆ → 0,
yˆ → ±∞.
(8d.f)
Using the symmetry property of the jet in the domain −∞ < xˆ < ∞, +yˆ+ < yˆ < ∞, we can impose an additional boundary condition on xˆ = 0 jvˆ = 0, jxˆ
uˆ = 0,
p(0, ˆ y) ˆ = pˆ 0 (y) ˆ on xˆ = 0 (yˆ > + yˆ+ ).
(9a.c)
The volume flux defined in (4) is a conservative feature of the problem hence it must be independent of the yˆ coordinate. Accordingly, we may define a dimensionless volume flux qˆ in the following form: +∞ qˆ = q/Q = v( ˆ x, ˆ y) ˆ dxˆ = const, (10) −∞
where Q ≡ v.
A. Pistiner / Nonlinear Analysis: Real World Applications 9 (2008) 1823 – 1826
1825
2. The self-similar formulation We will further refer to the circumstances in which the steady-state velocities and the pressure distributions (defined in the domain −∞ < xˆ < ∞, +yˆ+ < yˆ < ∞) achieve a certain asymptotic deriving from a certain universal behavior, such that, they can be described by a single independent variable in the form u( ˆ x, ˆ y) ˆ = f ()yˆ −1 ,
(11a)
v( ˆ x, ˆ y) ˆ = F ()yˆ −1 ,
(11b)
Pˆ (x, ˆ y) ˆ = g()yˆ −2
(11c)
and =
xˆ . yˆ
Substituting (11b), (11d) in (10), the conservative, dimensionless volume flux qˆ takes the form +∞ qˆ = F () d = const. −∞
(11d)
(12)
It should be noted that the self-similar functional form (11a)–(11d) exists irrespective of the actual velocity v(x, ˆ +yˆ+ ) and an actual pressure p(x, ˆ +yˆ+ ). Eventually, it tends to the similarity solution derived above for an asymptotically large distance of yˆ from the line source. Both solutions, in the physical x. ˆ yˆ and the self-similar domain are, however, characterized by the same amount of volume flux given by (10) and (12). Substituting (11a)–(11d) in (6a), one obtains (f 2 ) − f F − (f F ) = −g + (f (1 + 2 )) ,
(13)
which can also be rewritten in the following form: f 2 − f F + g − (f (1 + 2 )) = G(),
(14a)
G = f F .
(14b)
The set (14a), (14b) combines to yield f 2 + g = (f (1 + 2 ) + G) .
(15)
Introduction of (11a)–(11d) into (6b) one obtains (f F ) − F 2 − (F 2 ) = g + (g) + (F (1 + 2 )) .
(16)
Expression (16) can also be presented in the form −F 2 + f F − g − (F (1 + 2 )) = H (),
(17a)
H = F 2 + g.
(17b)
Combination of (17a), (17b) yields the following expression: f F = (F (1 + 2 ) + H ) ,
(18)
and substituting of (14b) in (18) and integration yield H + F (1 + 2 ) − G = , where is a constant of integration.
(19)
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A. Pistiner / Nonlinear Analysis: Real World Applications 9 (2008) 1823 – 1826
Substituting (11a), (11b), (11d) in (6c) we obtain F=
f +c ,
(20)
where c is a constant to be determined. Introduction of (9a,b), into (11a), (11b), and (11d) results in F (0) = 0,
(21a)
f (0) = 0.
(21b)
According to (21b), the constant c in (20) must vanish and the equation reduces to F=
f .
(22)
Combination of (14a), (17a), (19) and (22) leads to g = 2F −
1 + 2
,
(23)
and substituting (23) in (13) yields (F (1 + 2 )) − 2F + = −F 2 . 2 1+
(24)
Differentiation of (24) yields the following differential equation: F (1 + 2 ) + 6F + 6F −
2 (1 + 2 )2
+ F 2 = 0.
(25)
In view of the boundary condition (9c) together with (11c) and (11d), we may impose g(0) = g0 . This, together with (23) immediately yields F (0) =
g0 + . 2
(26)
Eq. (25) can be solved with the aid of a standard numerical integration procedure, subjected to (12) and the boundary conditions (21a) and (26). In the limit, one obtains from (25) f ∼ F ∼
C1 C1 2
as → ±∞,
(27a)
as → ∞,
(27b)
where C1 is an integration constant. System (27a), (27b) shows that the boundary conditions (8a–c) are automatically satisfied. In addition to the above, the boundary conditions (8d–f) are automatically satisfied by (11a–c). 3. Concluding remark In principle, the similarity representation does not generally reproduce the actual boundary conditions on the line source for the pressure distribution and velocity profiles but rather exhibits the following dependence for any finite x: ˆ v( ˆ x, ˆ +yˆ+ ) → ∞,
p( ˆ x, ˆ +yˆ+ ) → ∞
as + yˆ+ → 0.
Reference [1] F.M. White, Fluid Mechanics, McGraw-Hill Kogakusha Ltd, 1979.
(28a,b)