A Class of Exact Solutions of Equations for
Plane Steady Motion of Incompressible Fluids
of Variable Viscosity for finite Peclet Number
through Von-Mises Coordinates
Mushtaq Ahmed
Department of mathematics, University of Karachi, Pakistan
Abstract
This paper determines a class of exact solutions for plane steady motion of incompressible fluids of variable viscosity for finite Peclet number through von-Mises coordinates. The class is characterized by an equation involving a stream function and two differentiable functions f(x) and g(x) . Successive transformations technique is used on non-dimensional form of basic equations. The exact solutions are determined basing on two velocity profile cases. The first velocity profile case fixes the functionsg(x) and demands f (x) to satisfy a second order variable coefficients differential equation whose trivial solution is opted. The second velocity profile case fixes only the function g(x)and leaves f(x) arbitrary. In both the cases, a large set of expressions for streamlines, viscosity function, generalized energy function and temperature distribution for finite Peclet number can be found.
Keywords - Exact solutions for incompressible fluids, Variable viscosity fluids, Navier-Stokes equations with body force, Martin’s coordinates, von-Mises coordinates
I. INTRODUCTION
The basic equations for motion of a fluid element comprises of the equation of momentum, the equation of energy and equation of continuity. The basic equations for plane steady motion of incompressible variable viscosity fluid in Cartesian space(x,y) in non-dimensional form are following
0
y
x v
u (1)
2
{ ( )} ][ x x y x y
x y
x vu p u u v
u
u (2)
] )} (
{ ) 2 [( R
1
e
x x y y
y y
y
x vv p v u v
v
u (3)
[2 ( ) ( ) ]R E
P R
1 2 2 2
e c r
e
x y y
x yy
xx y
x vT T T u v u v
T
u (4)
Where the coefficient of viscosity is 0 , the velocity vector field q
u(x,y),v(x,y)
and) , (x y p
p is pressure. The dimensionless quantitiesRe, Pr and Ec are respectively the Reynolds number,
the Prandtl number and the Eckert number. The product of Re and Pr is Peclet numberPe .
The equation of contunity (1) indicates
v
u x
y
, (5)
where (x,y) is a stream function such that yx xy .
Dimension analysis method, coordinates transformation techniques and successive coordinate’s transformation techniques are available for exact solutions in references [1-9] and references therein. The solution of equation (4) for very large and very smallPe can be found where as finding solutions for finite Pe is fascinating
defines the curves const . as streamlines and leaves the curves const . arbitrary [13]. With this definition the curvilinear coordinates system (, ) is called the Martin’s coordinates system. In Martin’s system the curves const . are arbitrary therefore von-Mises coordinates system (x, ) takes it alongxaxis [14].
Let us characterization the streamlines const . by
. )
( ) (
const x
g x f y
(6)
The equation (7), without loss of generality, implies
) ( ) ( )
(x g x f
y (7)
where f(x) and g(x) are differentiable functions. In this communication, the function g(x) is non-zero and ( ) is zero.
The paper is organized as follow: Section (2) applies successive transformation technique converts the basic to Martin’s coordinates(, ) then to the von-Mises coordinates(x, ). Section (3) finds exact solutions of fundamental equations. The last section presents conclusion.
II. BASIC EQUATIONS IN VON-MISES COORDINATES
In equations (2-4) let us define the vorticity function w and the total energy functionL , the functionsA and B as follow
y
x u
v
w (8)
e 2
2
R 2 ) (
2
1 ux
v u p
L (9)
and
x x
y v B u
u
A ( ), 4 (10)
Consider the allowable change of Martin’s coordinates (, ) through
) , ( ),
,
( y y
x
x (11)
such that the Jacobian of the transformation 0 ) , (
) , (
y x
J is finite.
Suppose be the angle between the tangents to the streamlines lines const . and the curves const . at a common pointP(x, y), then
) tan( =
x y
(12)
Applying the differential geometric technique [15], the fundamental equations (2-4) in Martin’s coordinates system( , ) are following
E J w Re
= Re J E L +
( )cos 2 2 sin 2
2 2
FJ J
F
A
+ EA
J sin 2 F cos 2)
–
( )sin 2 cos 2
2
1 2 2
FJ J
F B
+E B
2
cos 2
sin 2 1
J
F , (13)
0 = –ReJ L +E A cos 2 –A
F cos 2 J sin 2
+
sin 2 sin 2
2 1
J F
B – sin 2
2
B E
, (14)
e P
J 1
J T F T E
J T F T G
= – e c
R E
2 2
4 4
1
A
B
+ J
T
(15)
where the coefficients of first fundamental form are
E = x2 y2 , F = x x + y y , G = (x )2 + (y )2 , (16)
J = E G F 2 , (17)
and
) , (
A = [
5 2
4
) sin cos
(
J E
J
F
{E (2E J3cos F E sin )
– 4E2J2J cos 2E E F sin + E E E sin }
+
3
2 cos
J
{E (F sin J cos )– 2EJ cos EG sin }
+
3
2
) cos sin
(
EJ J
F
{(J E 2EJ ) sin
+ cos [FE 2 E F E E ]}
3
2 sin
J
{(E (J sin F cos )–2EJ sin +EG cos }], (18)
) , (
B =
3
4
EJ
[E (F sin J cos )2–2E(F sin J cos )
(F sin J cos ) + E2(J sin 2 G sin 2)] , (19) and
w =
3
2
) cos sin
(
EJ J
F
{(J E 2EJ ) sin + cos [FE 2E F E E ]}
3
2 sin
J
{E (J sin F cos )–2EJ sin +EG cos }]
+
5 2
4
) sin cos
(
J E
J
F
{E(2E J3 cos F E sin )
– 4E2J2J cos 2E E F sin + E E E sin } – [
3
2 cos
J
{E (F sin J cos )– 2EJ cos E G sin } ], (20) Since the von-Mises coordinates system(x, ), takes the curves const . alongxaxis , therefore
x
(21)
Applying equation (21) and the streamlines (7) in equation (12) and equations (16-17) and using the trigonometric identities, we have
E
1
cos (22)
2
) (
1 M N
E (23)
F = J E 1 (24)
G = J 2 (25)
) (xg a g x
J (26)
) ( ) ( ) ( )
(x xg x M x x f x
N , (27)
and
b
a
(28)
with constantsa 0, b . The equations (13-15) and equations (18-20), in von-Mises coordinates (x, ) are following
J A E A B
L R w
Re e x
1 (29)
J B E E A J E A L
Re x x
1 1 ) 2 ( 0
(30)
4
(31)4 1 2 1 2 1 2 2 2 2 2 A B P E J T J EJ E E J E a T P E E J T J E a T E a T J r c x x e x x x x
( ) 2( )( )
) ( 2 g N N M N M g x g x a
A (32)
2 2 ) ( 4 g x a g N
B (33)
a g x N g x N a g x N M g x M w 2 2 2 ) ( 2 1 ) ( 2 (34)
and the magnitude of velocity vector q (u,v) is
J N M q 2 ) (
1
(35)
Eliminating the functionL from equations (28-29) assuming Lx L x provides the following compatibility equation
) 36 ( ) ( ) ( ) ( ) ( ) ( ) ( 1 ) ( 2 2 2 x e x x x x x w R g a B g f B g f x g f A g f x g f A A g a A g x a g f x A g f x A g x a Finding solution of equation (36) L and T are obtained from equations (29-30) and (31) respectively. The viscosity is obtained either from equation (32) or (33). The pressure and velocity components are obtained from equation (9) and equation (5) respectively.
The number of unknown in equation (36) can be reduced by eliminating through a relation between A and
B but it is not impossible here. Therefore, following two cases are considered
Case I: A 0
III. EXACT SOLUTIONS The case
0
A (37)
and equations (27) in (32) provides
0 ) ( ) ( 2
g x g
g xg
(38)
0 ) ( ) ( 2
f x g
f g x
(39)
It is easy to see that equation (38) is satisfied by
1 2 0
1 )
(
C x C x g
(40)
Equation (39) on utilizing equation (40) provides
0 ) 3
( )
(C0 x2 C1 f C0 x2 C1 f
x (41)
The equation (41) possesses trivial and non-trivial solution. A trivial solution of equation (41) is
0 ) (x
f (42)
Equation (36) on utilizing equation (42), becomes
x
B
g g
B
g g
B = g
a C Re
2 0
4
(43)
This suggests seeking a solution of the form
) (
2 x Q
B (44)
Equation (43) on substituting equation (44) provides
Q– 2
g g
Q = g
a C Re
2 0
2
(45)
whose solution is
) (x
Q =
g
a
C
R
e2 0
2
+ C2 g2 (46)
Equation (44) on substituting equation (46) gives
B = 2 { g
a C Re
2 0
2
+ C2 g2} (47)
and equation (33) on using equation (47) provides
=
) ( 4
)
( 2
xg g x a
{ g
a C Re
2 0
2
+ C2 g2} 2 (48)
Solution of equation (29) and (30), utilizing equation (47) and (37) is
L R
a e = {
a Re
2 2
) (
2
g x
N g x N
– 2a Q(x)}
2
2
+p6 (49)
Equation (31), using (42) and (47) provides
x x T g x
a )
( 2a(xg)Tx+
T gx g x a 1( )2
+ x
Pe Txg x
g x a g x a
) ( 2
) ( 1 )
(
2
+
) ( )
(
) ( 2
g x g
x g x g x
a T =Ec Pr a2(xg)(xg g) 2 Q(x) (50)
Let us seek solution of equation (50) of the form
) ( ) ( )
,
(x 2 R x S x
T (51)
R g x
a 2 2 +x g
ag Pe 4a xg
R+2a
3(x g)2 xg (x g)
R= Ec Pr (xg g){ g a
C Re
2 0
2
+ C2 g2} (52)
S g x
a 2 2 +xg
a g Pe
S= 2a R(x) (53)On substituting the value of g(x) from equation (40) in equation (52) we get
R C x C x
a 2( 0 2 1)2 +x(C0 x2 C1)
a(9C0 x2 C1) Pe(C0 x2 C1)2
R+8aC0 x2(2C0 x2 C1)R = –Ec Pr(C0 x2 C1){ 2 2
1 0 2
2 2
0 2
2
C a
C C R x
a C
Re e
} (54)
where C0, C1and C2 are arbitrary constants. Equation (54) suggests searching a solution of (54) setting
1
C =0 (55)
Equation (54) on substituting C1=0, becomes
R x
a 2 +x
9a PeC0 x2
R–16 a R = –Ec Pr {2 0
2
2 2 0
2
x C
C
a C Re
} (56)
Consider the function R(x) satisfying equation (56) takes the form 2
2 1
)
(x B B x
R (57)
implies
28 1
8
2 2
1
C P
a a
R P E
B c r e r (58)
and
0 2 2
28 aC C P E
B c r (59)
Inserting equations (58-59) in equation (57), we get
) (r
R =
28 1
8
2
2
C P a a
P
Ec e r
0 2
28
a
C
C
P
E
c r
2
1
x
(60)
Employing equation (60) in equation (53), we get
) (x
S = – 2 C02 x e R x dx dx
x
e x
a P C x
a P C
e e
) ( 2 0 2
0
2 3 2
+ C3 dx
x e
x a
P
C e
0 2
2
+C4 (61)
On substituting equations (60-61)) in equation (51), we get the temperature distribution.
Similarly, for non-trivial solution of equation (41), the exact the solutions to the basic equations can be obtained.
The case
0
B (62)
in equation (33) implies
x c
g (63)
where c 0 is constant.
Equation (36) on supplying B 0 gives
x x A c
a1 – 2a (M g)Ax –
Ac g M
a 1 ( )2
+ c g M g g M
aA ( ) 2( )( ) =
x M M c a
Re 2
3
2
x a
Re
(64)
Consider the solution of equation (64) of the form
) ( ) (x P x R
A (65)
Equation (64) employing equation (65) become
R
ac –2aMP+ agR+
c gM M P a 2
+ {ac P +3a g P+ c g g P a 2 2 }= x M M c a
Re 2
3
2
x a
Re
(66)
Since and x are independent variables therefore the equation (66) yields
P
x2 +3x P+P=
x c a Re 2 2 (67) and
xR
= x M M c a R
x e 2
2
2 + c P
xM 2 + P x M M c x
2 (68)
The solutions of equations (67-68) are
x c a x R x x s x s x P e 2 2 2
1 ln (ln )
)
( (69)
and ) (x R = x 1 dx dx x Z x ( ) 1
1 +C1 ln x+C2 (70)
where ) ( 1 x Z = x M M c a R
x e 2
2
2 + P
c xM 2 + P x M M c x
2 (71)
On substituting equations (69-71) in equation (32), we get the value of =
)
(M g
c a
R(x) P(x)
(72)The solution of equation (29) and (30) using equation (63), is
L R
a e =
2 ) ( ) ( 2 2 x gP a x P c a x a Re +{ x x M x M c a
Re 2 ( )
) (
+acR(x)–aM (x)P(x)} +
M
P x dx ca
1 2 ( ) + M (x)R(x)dx +p5 (73)
The equation (31) on utilizing equation (72) becomes
x x T c
a 2a(M g ) T x+
Tc g M
a 1 ( )2
+ Pe Tx
x ac
2( )(M g )
x N M T a
= –Ec Pr
RM (PM gR) P g 2
(74)The structure of equation (74) suggests 2 3 2
1(x) R (x) R (x) R
and
2
[acR3+4ag R3
c g aR3 2 2
2 3 ( 2 )
x g g
aR
P R3
x ac
e
]
+ [acR2–4aM R3 +2agR2
c Mg
aR3
4 2 R P x ac
e
x g g R x M M R
a 2 3 2 2 2 ]
+ acR1 2aM R2+
c M
aR (1 )
2 3 2
+ P R1
x ac
e
+
( 2 )R2
x M M
a
= –
E
cP
r
R
M
(
P
M
g
R
)
P
g
2
(76)Comparison of the coefficients of like terms on both sides gives
3 2
R
x 5 x2 R3
c a P
x e
4R3= Z4(x) (77)
2 2
R
x 2
2
3 R
ac x P
x e
2 R
= Z3(x) (78)
and
1
R + 1 R1
ac P x
e
= Z2(x) (79)
where ) ( 2 x Z = c R M 2 2
+1 2 R2
x M M
c
– 3
2 2 ) 1 ( 2 R c M –
R
c
a
M
P
E
c r
(80)
) (
3 x
Z =4a x2M R3
x M M R x a 4
2 2 3 –Ec Pr
x2PM cR
(81)) ( )
(
4 P x
a P E x
Z c r (82)
The solution of equation (79) is
) (
1 x
R = xe Z x dx dx
x
e ac x
P x c a P e e ) (
2 + C5
dx x e x c a Pe
+C6 (83)
The solutions of equations (77) and (78) using Mathematica is
) (
2 x
R =
x A C 1 1 1
3 + C4 MeijerG[{{ },{1}}, {{1,1},{}}, A1 x]
+ x 1 ) 1 1 (
1 A x
A eA1r xMeijerG[{{ },{1}}, {{1,1},{}}, A1 x] Z3(x)dx
] 1 A {}}, }, 1 , 1 {{ {1}}, }, MeijerG[{{ x
x
A1
1 A1 x
eA1x Z3(x)dx
(84)) ( 3 x R =
2 1 1 1 1 2 2 1 x A x A+ 2
1
x
A12 (2 4 A1 x A12 x2)eA1x x3 MeijerG[{{ },{1}}, {{2, 2},{}}, A1 x] Z4(x)dx
] 1 A {}}, 2}, 2, {{ {1}}, }, MeijerG[{{
2
x
x
A x A x
xe Z x dx
A12 2 4 1 12 2 A1x 4( )
(85)
where
c a P
A1 e (86)
Therefore equations (83-85) in equation (75) provides temperature distribution for finite Pe .
IV. CONCLUSION
Applying the successive transformation technique on non-dimensional equations for plane steady motion of incompressible fluids of variable viscosity in von-Mises coordinates a class of exact solutions are determined. In this class of flows the streamlines are characterized by equation y f (x) g(x)() with
b
a
( ) with constants a 0 andb . Based on two velocity field shapes exact solutions are
obtained. The first velocity field requires
1 2 0
1 )
(
C x C x g
and f (x) satisfies a second order variable
coefficients differential equation with a trivial solution f(x) 0 . Therefore the streamlines for this case are
1 2
0 x C
C
b a y
. The second velocity field demands
x c x
g( ) and leaves the function f (x) arbitrary.
Therefore the streamlines for this case are
x b a c x f
y ( ) ( ) . Both the velocity field cases create
infinite set of streamlines, pressure, viscosity, generalized energy function, temperature distribution. The symbolsc 0, C0 0 , C1 , are constants.
The software Mathematica is used to determine the solution of some of the ordinary differential equations. The same software can easily draw the streamlines pattern to find the effect of various parameters on the streamlines and discuss the flow characteristic.
REFERENCES
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[2] Mushtaq A., On Some Thermally Conducting Fluids: Ph. D Thesis, Department of Mathematics, University of Karachi, Pakistan, 2016.
[3] Mushtaq A.; Naeem R.K.; S. Anwer Ali; A class of new exact solutions of Navier-Stokes equations with body force for viscous incompressible fluid,: International Journal of Applied Mathematical Research, 2018, 7(1), 22-26.
www.sciencepubco.com/index.php/IJAMR , doi:10.14419/ijamr.v7i1.8836
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www.sciencepubco.com/index.php/IJAMR, doi:10.14419/ijamr.v7i2.12326
[6] Mushtaq Ahmed, (2018), A Class of New Exact Solution of equations for Motion of Variable Viscosity Fluid In presence of Body Force with Moderate Peclet number , International Journal of Fluid Mexhanics and Thermal Sciences, 4 (4) 429-
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