ISSN 2286-4822
www.euacademic.org DRJI Value: 5.9 (B+)
On Two-Dimensional Motion of Incompressible
Variable Viscosity Fluids with Moderate Peclet
Number Via von-Mises Coordinates
MUSHTAQ AHMED
Department of Mathematics University of Karachi, Pakistan
Abstract
The problem of plane steady motion with moderate Peclet number of incompressible and variable viscosity fluid is addressed to announce a class of exact solutions in von-Mises coordinates. The class
is characterized by an equation relating a differentiable function
f
(
x
)
and a stream function
. The successive transformation technique isapplied for exact solutions. The study finds
f
(
x
)
arbitrary in one case and a specific value in other case. In both the cases, this discourse shows an infinite set of new exact solutions for moderate Peclet number.Key words: Variable viscosity fluids, Navier-Stokes equations, Exact solutions to incompressible fluids, Martin’s system, von-Mises coordinates, Moderate Peclet number.
1 Introduction
incompressible fluids of variable viscosity with body force. The successive transformation technique used in [10-14] was to first transform the basic equations in curvilinear coordinates
(
,
)
. The resulting equations were then transformed defining the curvilinear coordinate
as streamline, selecting the curvilinear coordinate
in some specific way and assuming angle
between these coordinate curves. In fact the idea of this transformation is a loan from Martin [15] where the curvilinear coordinate
is defined as streamline and the curvilinear coordinate
is left arbitrary therefore the coordinates
,
will be called here as “Martin’s coordinates”.The arbitrary coordinate lines
constant
of Martin’s coordinates system
,
can be set in many ways. For example, the technique of [10-14] was to retransform the basic equations taking angle
between Martin’s coordinates taking the coordinate
along the radial direction. Another example is the von-Mises coordinates(
x
,
)
here the coordinate lines
constant
of Martin’s coordinates is taken alongx
axis
thus the function
x
and stream function
of Martin’s coordinates as independent variables instead ofy
andx
[16].This communication first transforms the basic non-dimensional flows equations from Cartesian space
(
x
,
y
)
to Martin’s coordinates then to von-Mises coordinates(
x
,
)
. It further characterizes the streamlines of the class of flows under consideration byconst.
)
(
a
b
x
f
y
(1)
Where
f
(
x
)
is a differentiable function anda
0
,b
are constants. The equation (1) implies)
(
)
(
f
x
y
(2)where
(
)
a
b
.system
(
,
)
. Section (3) retransforms the basic equations to von-Mises coordinates. The exact solutions to the problem are given in section 4. Conclusions are given at the end.2. Basic non-dimensional equations in Martin’s coordinates
Consider the following non-dimensional form of the fluid flow model for steady plane motion of incompressible fluid of variable viscosity with constant thermal conductivity with no heat addition [7]
x
u
+v
y= 0 (3)u
u
x+v
u
y=
p
x+e
R
1
]
)}
(
{
)
2
[(
u
x x
u
y
v
x y (4)u
v
x+v
v
y=
p
y+e
R
1
]
)}
(
{
)
2
[(
v
y y
u
y
v
x x (5)u
T
x+v
T
y=r e
P
R
1
(
)
yy xx
T
T
+e c
R
E
]
)
(
)
(
2
[
u
x2
v
y2
u
y
v
x 2 (6)Where
u
,v
are the components of velocity vectorq
(
u
,
v
)
,T
is temperature,p
pressure,
the viscosity. These five quantities are function ofx
andy
. The Reynolds number, the Ecart number and the Prandtl numbers are denoted byR
e,E
c andP
r are respectively. The product ofR
e andP
r is Peclet numberP
e. The solution of these equations for very large and very smallP
e can be found where as finding solutions for moderateP
e is challenging [17-22].A stream function
(
x
,
y
)
satisfies equation (3) such thaty
u
v
x (7)The equations (4-6) in manageable form by introducing the vorticity function
and the total energy functionL
defined by
=v
xu
y (8)L
=p
+2
1
)
(
u
2
v
2 e
R
1
)
2
(
u
x (9)are
–
v
=
L
x+e
R
1
y
A
(10)u
= ye
y
B
R
L
1
+e
R
1
x
A
(11)u
T
x+v
T
y=e
P
1
)
(
T
xx
T
yy +e c
R
E
2 2
4
4
1
A
B
(12)where
A
=
(
u
y
v
x)
andB
=4
u
x (13)Consider a curvilinear coordinate system
(
,
)
in theplane
y
x
,
)
(
through transformation)
,
(
x
x
,y
y
(
,
)
(14)such that the Jacobian
J
=)
,
(
)
,
(
x
y
of the transformation isnon-zero and finite. Follow Martin [15], let
be the angle between the tangent to the streamlines
const
.
and the curves
const
.
at a pointP
(
x
,
y
)
, then)
tan(
=
x
y
(15)
2
ds
=E
(
,
)
d
2 + 2F
(
,
)
d
d
+G
(
,
)
2d
(16) andE
=x
2
y
2 ,F
=x
x
+y
y
,G
= (x
)2 + ( Differentiating equation (14) with respect to
x
andy
, and solving the resulting equations
y
= J
x,y
=J
x,x
=J
y,x
= J
y (18) andJ
= E
G
F
2 = (x
y
y
x
) = W
(19) Trigonometric identities on equation (15) and equation (18) provides
x
=E
Cos
(
)
,x
=E
1
[
F
Cos
(
)
J
Sin
(
)
]
y
=E
Sin
(
)
,y
=E
1
[
F
Sin
(
)
+J
Cos
(
)
] (20)The integrability conditions:
x
=x
y
=y
(21) forx
andy
, yield:
=E
J
112,
=E
J
122(22)
wherein
2 11
=[
2
]
2
1
2
FE
E
F
E
E
W
,
122 = [ ]2 1
2 EG FE
W
(23)
Equation (21), applying the integrability condition
for)
,
(
, yields
E
W E
W W K
2 12 2
11
1 (24)
where K is called the Gaussian curvature.
Now equations (10-11), on substituting equation (15), equation (18), equation (20) , equations (22-23) simplifies to following
E
J
R
e
=R
eJ
E
L
+A
(
F
2
J
2)
cos
2
2
FJ
sin
2
+
EA
J
sin
2
F
cos
2
)
–
( )sin2 cos2
2
1 2 2
FJ J
+
E
B
2
cos 2 sin 2 1 J F (25) and
0 =–
R
eJ
L
+E
A
cos
2
–A
F
cos
2
J
sin
2
+
sin
2
sin
22
1
J
F
B
– sin
2
2
B
E
(26)
Differential geometry [23], says that
)
(
T
xx
T
yy =J
1
J
T
F
T
E
J
T
F
T
G
(27)The expression
u
T
x
v
T
y becomesJ
T
, thus equation (12) on using
(27) becomes 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
(28) and J E q (29) where 2 2v
u
q
.Equation (13) on substitute equations (18-23), provides
)
,
(
B
= 34
EJ
[ 2)
cos
sin
(
F
J
E
–2E
(
F
sin
J
cos
)
)
cos
sin
(
F
J
+E
2(
J
sin
2
G
sin
2
)
] , (30))
,
(
A
=
[5 2
4
)
sin
cos
(
J
E
J
F
{
E
(
2
E
J
3cos
F
E
sin
)
+
3
2
cos
J
{
E
(
F
sin
J
cos
)
–2
EJ
cos
E
G
sin
}+ 3
2
)
cos
sin
(
EJ
J
F
{
(
J
E
2
EJ
)
sin
+
cos
[
FE
2
E
F
E
E
]
}
3
2
sin
J
{
(
E
(
J
sin
F
cos
)
–2
EJ
sin
+EG
cos
}] (31)In Martin’s system, equation (8) becomes
y y
x
x
v
u
u
v
(32)Equation (32) on substituting equation (15), equation (18) and equation (20) provides
= 32
)
cos
sin
(
EJ
J
F
{
(
J
E
2
EJ
)
sin
+cos
]
2
[
FE
E
F
E
E
}
3
2
sin
J
{
E
(
J
sin
F
cos
)
–2
EJ
sin
+EG
cos
}]+
5 2
4
)
sin
cos
(
J
E
J
F
{
E
(
2
E
J
3cos
F
E
sin
)
–
4
E
2J
2J
cos
2
E
E
F
sin
+E
E
E
sin
}– [ 3
2
cos
J
{
E
(
F
sin
J
cos
)
–2
EJ
cos
sin
G
E
} ] (33)The basic system of non-dimensional partial differential equations for the problem concerned in Martin’s system comprises of equations (25-26), equations (28), with equations (30-31) and equation (33).
3. Basic equations in von-Mises coordinates
In order to achieve the objective of this discourse set
x
The equation (15), equation (17), equations (19), equations (30-31) and equation (33) reduces to
E
1
cos
(35)
2)
(
1
x
f
x
E
(36)x
a
J
(37)2
4
x
a
B
(38)
(
)
2
(
)
2
x
x
f
x
x
f
x
x
a
A
(39)
x
a
x
f
x
(
)
(40)The equations (25-26) and equation (28) on utilizing equations (34-40), give
R
e =R
eL
–a
x
A
x+M
A
+B
(41)0 = –
R
eL
x+x
a
M
A
(
1
2)
+
M
A
x –x
a
B
M
(42)
x x
T
x
a
2
M
T
x+T
x
a
M
)
1
(
2+
a
P
e
T
x +M
T
= –
24
2
4
B
A
P
E
x
a
c r
(43)and
)
(
x
f
x
M
(44)Considering
L
x
L
x on equations (41-42) yieldsx x
A
x
a
2
M
A
x–
Ax a
M2
1 +
x
A
a
–A
M
–
a B f
Bx =
R
ew
x (45))
,
(
u
v
q
from equation (7),p
from equation (9), and streamlines from equation (2) are determined.4. Exact solutionsThe compatibility equation (45) involves the functions
A
,B
and derivative off
(
x
)
. For its solution this discourse eliminates
from the functionA
andB
. The equation (38) and equation (39), on eliminate
, providesA
=X
(
x
)
B
(46)where
x
M
M
x
X
2
4
1
)
(
(47)provided
x
M
2
M
0
.Use of equation (46) in equation (45), gives
x x
B
X
x
a
–(
1
2
M
X
)
B
x +
M
X
(
1
M
2)
x
a
B
+
x
X
X
B
a
x2
–
B
2
M
X
M
X
+a
B
x
X
=
x
a
M
R
e (48)Since the coefficients of
B
xx,B
x ,B
,B
x,B
andB
are all functions ofx
only therefore the solution of equation (48) is of the formB
=G
(
x
)
+K
(
x
)
(49)where
G
(
x
)
andK
(
x
)
are to be determined. Equation (48) on substituting equation (49) gives
[x X
K
(
x
)
+(X
+ 2x X
)K
(
x
)
+ (X
+x X
)K
(
x
)
]=
x
a
M
R
e +a
(
1
2
M
X
)
K
(
x
)
+a
2
M
X
M
X
K
(
x
)
(50)Since
x
and
are von-Mises coordinates therefore equation (50) providesx X
K
(
x
)
+(X
+ 2x X
)K
(
x
)
+ (X
+x X
)K
(
x
)
=0 (51) andX
x
a
G
(
x
)
+a
X
2
r
X
G
(
x
)
+a
x
X
G
(
x
)
=Z
1(
r
)
(52) where)
(
1
x
Z
= x a M
Re +
a
(
1
2
M
X
)
K
(
x
)
+a
2MXMX
K
(
x
)
(53)Equation (51) in exact differential form is
0
)
(
dx
K
X
d
x
dx
d
(54)Equation (54) implies
)
(
)
(
ln
)
(
1 2x
X
k
x
X
x
k
x
K
(55)where
k
1 andk
2 are constants.Equation (52) in exact differential form is
dx
G
X
d
x
dx
d
(
)
=
Z
1(
x
)
(56)Equation (56) implies
)
(
x
G
=X
1
[x
1
dx
x
Z
1(
)
]dx
+X
k
x
k
3ln
4(57)
where
k
3 andk
4 are constants.On substituting equation (55) and equation (57) in equation (49), one can have
B
=X
1
[x
1
dx
x
Z
1(
)
]dx
+X k x
k3ln 4 +
X k x
k1ln 2 (58)
Viscosity from either from equation (40) or equation (41) is
) , (
x =
4 2 x a [ X
1
[x
1
dx x Z
1( ) ]
dx
+X k x
k3ln 4+
X k x
The solution of equations (41-42) implies the following expression of function
L
L R
a e =–
x a M
Re
+ax
GX
–aK(MX1)
2 2 X K x a
+
a
M
G
X
dx
+
aX M aM
dx xK
(1 2) +
p
1 (60)The temperature distribution
T
is determined from equation (38). Equation (43) on utilizing equations (58-59), becomex x
T
x
a
2MTx+ T x a M ) 1
( 2 +
aPe
Tx +MT= x X ) 4 1 ( P
Ec r 2 {
G
(
x
)
+K
(
x
)
} (61)Equation (61) suggests seeking solution of the form
)
(
)
(
21
x
T
x
T
T
(62)The use of equation (62) in equation (61), gives
}
2
,
1
{
for
),
(
)
(
1
T
Z
x
x
a
P
a
T
e , (63)where
)
(
2
x
Z
=E
P
(
1
4
)
(
)
2 2 r
c
G
x
x
a
X
+2
M
T
2
+M
T
2 (64))
(
3
x
Z
=E
P
(
1
4
)
(
)
2 2 r
c
K
x
x
a
X
. (65)
The solution of equations (63) and (64) are
) (x
T =
dx dx x Z x x a P a a P
a e e
) ( 1 ) ( ) (
+
H
x a dxP a e) (
+
H
2 (67)where
H
i,
i
1
,
2
,...
4
are constant.Therefore, temperature distribution moderate Peclet number is obtained by substituting equations (64-67) in equation (62) for
{
1
,
2
}
. Thus, exact solutions for arbitraryf
(
x
)
is found.2 2 1
2
1
)
(
x
c
x
c
f
(68)The compatibility equation (48) on substituting (68) provides
0
a
B
f
B
x (69)The solution of equation (69) is
3 1
2 1
1
(
)
2
a
x
b
I
x
dx
c
b
c
B
(70)Where
c
1,c
2,c
3,b
1are constants andI
(
x
)
is function of intergation.Equation (41) utilization of equation (70) provides
=4
2
x
a
1 32 1
1
(
)
2
a
x
b
I
x
dx
c
b
c
(71)
The solution of equations (41-42) is
Re
L
=
2 1 1
Re
2
a
c
b
– 1 1 22
a
x
b
c
+
M
0 (72)where
M
0 is a real constant.The energy equation (43) on employing equations (69), equations (71-72), becomes
a
x
2T
xx– 21
c
x
3T
r+(
1
2 4)
1x
c
a
x
T
– 21
c
x
2T
+x
(
a
P
e)
T
r=
E
cP
r
1 32 1
1
(
)
2
a
x
b
I
x
dx
c
b
c
(73)
Equation (73) suggests
a
x
2T
5
+x
(
a
P
e)
T
5
=6
a
c
1A
1x
2 (75)a
x
2T
4
+x
(
a
P
e)
T
4
=4
a
c
1x
3T
5
+4
a
c
1x
2T
5–
6
a
A
1x
(
1
c
12x
4)
+b
1E
cP
r (76)a
x
2T
3
+x
(
a
P
e)
T
3
=2
a
c
1x
3T
4
+2
a
c
1x
2T
4–5 4 2 1
)
1
(
2
a
x
c
x
T
+
x
I
x
dx
a
b
c
c
P
E
c r(
)
2
2 1 1
14 (77)
When
(
a
P
e)
0
the equations (75-77) give)
(
5x
T
=)
2
(
3
1 1 2e
P
a
x
c
A
a
+ aP e e
x
P
n
a
1 + 2n
,(
2
a
R
eP
r)
0
(78))
(
4x
T
=)
5
(
5
6
1 12 5e
P
a
x
c
A
a
)
(
6
1 eP
a
x
A
a
+)
2
)(
4
(
9
2 1 12 4e
e
a
P
P
a
x
c
A
a
+)
2
(
2
1 2 2e
P
a
n
x
c
a
+ aP e e
x
P
n
a
1 + 4n
– e cR
x
E
b
1ln
+
)
2
(
)
2
(
)
(
2
1 2 1 3 1 2 1 31 e e e a P
P
a
P
n
n
x
c
P
n
n
x
c
a
x
n
a
e
,where
(
2
a
P
e)
0
,(
4
a
P
e)
0
,(
5
a
P
e)
0
(79))
(
3
x
T
=n
6+
M
1(
x
)
dx
(80))
(
1
x
M
= a P a e
x
n
) ( 5 + dx x dx x I x a b c c P E x a P a r c a Pa e e
( ) 2 1 1 14 ) ( ) ( 2 +
dx x T x c x a T x c a T x c a x a P a a Pa e ( e)
Therefore, temperature distribution is obtained by substituting equations (78-81) in equation (74) for moderate Peclet number. Thus, exact solutions for
f
(
x
)
given by equation (68) is found when0
)
(
a
P
e
.When
(
a
P
e)
0
, equations (76–78) yield8 7 2 1 1
5
(
x
)
3
c
A
x
n
x
n
T
(82)10
3
)
(
5 2 1 1 4
x
c
A
x
T
+3
c
12A
1x
4+ 1 7 33
4
x
n
c
+2
c
1n
8x
2–
6
A
1x
ln
x
+(
6
A
1
n
9)
x
–x
a
P
E
b
c rln
1 +
10
n
(83))
(
3
x
T
=
M
2(
x
)
dx
dx
n
11x
n
12(84) where
)
(
2
x
M
2
c
1x
T
4
2
c
1x
–2
(
1
c
12x
4)
T
5(
x
)
x
+
c
b
x
I
x
dx
c
x
a
P
E
c r)
(
2
2 1 1 14
2
(85)
and
n
7,n
8n
9,n
10n
11,n
12are non zero constants.Therefore, temperature distribution is obtained by substituting equations (82-85) in equation (74) for moderate Peclet number. Thus, exact solutions, for
f
(
x
)
given by equation (68), is found when(
a
P
e)
0
.5. Conclusion
The following dimensionless parameters
0 *
L
x
x
0 *
L
y
y
0 *
U
u
u
0 *
U
v
0 *
0 *
p
p
p
0 1 * 1
F
F
F
0 2 * 2
F
F
F
where the thermal conductivity
k
k
0
Const
.
, density.
0
Const
andc
v
c
p
Const
.
is used for writing the basic equations governing the two-dimensional steady motion of incompressible fluid of variable viscosity in non-dimensional form.Successive transformation technique is used in finding a class of new exact solutions of the basic non-dimensional equations with moderate Peclet number in von-Mises coordinates. The exact solutions are determined for arbitrary and non-arbitrary
f
(
x
)
. For arbitrary)
(
x
f
the streamlines
const
.
arey
f
(
x
)
b
and for2 2 1 2 1 )
(x c x c
f the streamlines are
b c x c
y 1 2 2
2 1
where
c
1and
c
2 are constants. In both the cases an infinite set of velocity components, viscosity function, generalized energy function and temperature distribution for moderate Peclet number can be constructed. The graph of streamlines can be drawn using computer algebra system software to observe the streamline patterns.References
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