On Two-Dimensional Variable Viscosity Fluid Motion with Body Forcefor Intermediate Peclet Number Via von-Mises Coordinates

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International Journal of Fluid Mechanics & Thermal Sciences

2019; 5(3): 75-81

http://www.sciencepublishinggroup.com/j/ijfmts doi: 10.11648/j.ijfmts.20190503.13

ISSN: 2469-8105 (Print); ISSN: 2469-8113 (Online)

On Two-Dimensional Variable Viscosity Fluid Motion with

Body Forcefor Intermediate Peclet Number Via von-Mises

Coordinates

Mushtaq Ahmed

Department of Mathematics, University of Karachi, Karachi, Pakistan

Email address:

To cite this article:

Mushtaq Ahmed. On Two-Dimensional Variable Viscosity Fluid Motion with Body Forcefor Intermediate Peclet Number Via von-Mises Coordinates. International Journal of Fluid Mechanics & Thermal Sciences. Vol. 5, No. 3, 2019, pp. 75-81.

doi: 10.11648/j.ijfmts.20190503.13

Received: March 13, 2019; Accepted: July 26, 2019; Published: August 26, 2019

Abstract:

This article uses von-Mises coordinates to present a class of new exact solutions of the system of partial differential equations for the plane steady motion of incompressible fluid of variable viscosity in presence of body forcefor moderate Peclet number. This communication applies successive transformation technique and characterizes streamlines through an equation relating a differentiable function f(x) and a function of stream function. Considering the function of stream function satisfies a specific relation, the exact solutions for moderate Peclet number with body force are determined for given one component of the body force when f(x) takes a specific value and when it is not. In both the cases, it shows an infinite set of streamlines, the velocity components, viscosity function, generalized energy function and temperature distribution for intermediate Peclet number in presence of body force. When f(x) takes a specific value, a relation between viscosity and temperature function is observed.

Keywords:

Variable Viscosity Fluids, Navier-Stokes Equations with Body Force, Martin’s System, von-Mises Coordinates, Moderate Peclet Number

1. Introduction

For the motion of a variable viscosity fluidthe equation of conservation of mass, momentum and energy are known as a system of partial differential equations (PDE). The momentum equationsare Navier-Stokes equations (NSE). In Navier-Stokes equations, the product of mass and acceleration of fluid

element are in left-hand side and body forces term in addition to surface forcein right-hand side. The non-dimensional equations for the steady motion of constant density and variable viscosity fluid in tensor notation are following:

0

v x

α α ∂ ₌

∂ (1)

1

e

v p

v F

x x

v v

R x x x

α

β α

β µ β α

∂ ₌ ₋ ∂

∂ ∂

 _∂ _∂ 

∂  

+  _ + _

∂  ∂ ∂ 

(2)

1

e r

c

e

T T

v

x R P x x

v

E v v

R x x x

β

β α α

β

α α

β β α

µ

 

∂ ₌ ∂ ∂

 

∂ ∂ ∂ 

 _∂ 

∂ ∂

+ _ + _

∂ _∂ ∂ _

(2)

whereF_α(x_α)is the body force per unit mass, v_{α α}(x ) the fluid velocity, p= p x( _α) is pressure, the coefficients of viscosity µ>0 , the space coordinates x_α and

, {1, 2, 3}

α β

∈ . The non-dimensionl quantitiesE_c, R_e and

r

P are the Eckert number, the Reynolds number, the Prandtl number respectively.

For the two-dimensional Cartesian space case taking

, {1, 2}

α β

∈ , x₁=x , x₂= y , v₁=u x y( , ) , v₂ =v x y( , ) ,

3 0

v = ,F₁=F x y₁( , ),F₂ =F x y₂( , ), F₃=0in equations (1-3) one finds

x

u + v_y= 0 (4)

u u_x+ vuy= F1−px

+

e

1

R [(2

µ

ux x) +{ (

µ

uy+vx)} ]y (5)

u v_x+ v vy= F2 −py

+

e

1

R [(2

µ

vy y) +{ (

µ

uy+vx)} ]x (6)

u 2 a

   

 + vTy=

1

e

P_′ (Txx+Tyy)

+ c

e

E R

2 2 2

[2 (µ u_x +v_y )+µ(u_y+v_x) ] (7)

whereR P_e _r is the Peclet numberP_e_′.

The solution of the equation (4) is a stream function

( , )x y

ψ

such that

u y

ψ

∂ =

∂ x v

ψ ∂ = −

∂ (8)

The solutions of momentum and energy equations are there through dimension analysis methods and coordinates transformation techniques [1-6]. For solution of these equations when NSE includes body force some transformations technique are applied [7-10]. Further, solutions are there for very small and very large P_e_′where as the solution with intermediate P_e_′ is challenging[11-14]. This communicationapplies successive transformation scheme to meet the challenge of moderateP_e_′. According to this scheme the basic non-dimensional flows equations with body force in Cartesian space ( , )x y are first transformed into Martin’s coordinates

( )

φ ψ

, then to von-Mises coordinates( , )x

ψ

. In Martin’s coordinates, the curvilinear coordinates

( )

φ ψ

, are such that the coordinate lines

ψ

=const. are streamlines and the coordinate lines φ=constantare arbitrary [15]. Whereas

in the von-Mises coordinates, the arbitrary coordinate lines of Martin’s system is taken along the x axis− . Thus, the function φ=x and stream function ψ of Martin’s coordinates as independent variables instead of y and x

[16]. Further, the characterization of the streamlines is through

( ) const.

y− f x = (9)

where f x( )is a differentiable function and ψ =const. are the streamlines. Therefore, it is reasonable to take

( ) ( )

y= f x +

ν ψ

(10)

with

ν

as a differentiable function of stream function ψ . The paper is organized as follow: Section (2) shows transformation of basic equations into Martin’s coordinates

( , )

φ ψ

. Section (3) retransforms equations from Martin’s systemto von-Mises coordinates ( , )x

ψ

. Section (4), calculates exact solutions in von-Mises coordinates. Last sectionpresents conclusions.

2. Basic Equations to Martin’s System

Let us write the equations (5-7) to a convenient form, before transforming to Martin’s coordinates, in terms of the vorticity function Ω and the total energy function T_x defined by

Ω= v_x−u_y (11)

L = p+ 1 2

2 2

(u +v )−

e

1

R yψ (12)

and find

–vΩ= F₁ −L_x+

e

1

R Ay (13)

u Ω = F₂ _y 1 _y

e

L B R

− − +

e

1

R Ax (14)

uT_x+ vT_y= 1 e

P_′ (Txx+Tyy)

+ c

e

E

R

(

)

2 2

1

4

4µ B + A (15)

where

A =

µ

(u_y+v_x) B= 4

µ

u_x (16)

Let us consider the following allowable change of curvilinear coordinates ( , )

φ ψ

through

( , )

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76 Mushtaq Ahmed: On Two-Dimensional Variable Viscosity Fluid Motion with Body Forcefor Intermediate Peclet Number Via von-Mises Coordinates

such that the Jacobian ( , ) ( , )

x y J

φ ψ

∂ =

∂ of the transformation is non-zero and finite. Let ξ be the angle between the tangents to the streamlines

ψ

=const. and the curvesφ=const. at a common pointP x y( , ), basic equations in Martin’s system are following [17]

e

R J E

− Ω =

(

)

(

11 22

)

cos sin

sin cos e

F F F

R J E

J F F

ξ ξ

₋ ₊ 

 

+ −

 

 

+R J E L_e _ψ +

(

₍ 2 2_{) cos 2} ₂ _{sin 2}

)

A_φ F −J ξ− FJ ξ

+ E A_ψ

(

Jsin 2

ξ

−Fcos 2 )

ξ

)

– 1₍ 2 2_{) sin 2} _{cos 2}

2

B_φ_ F −J ξ+FJ ξ_

 

+E B_ψ sin 2 cos2 2

F

J

ξ

 

+

 

 , (18)

0 = R J E F_e

[

₁cosξ+F₂sinξ

]

–R J L_e _φ+E A_ψ cos 2

ξ

–A F_φ

[

cos 2

ξ

−Jsin 2

ξ

]

+ 1 sin 2 sin2 2

B_φ_ F

ξ

−J

ξ

_

  – 2 sin 2

E B_ψ

ξ , (19)

and

1 JP_e_′

G T F T E T F T

J J

φ ψ ψ φ

φ ψ

_ ₋ _ _ ₋ _ 

_ _ ₊_ _ 

_ _ _ _ 

 

= – c

e

E

R

(

)

2 2

1

4 4µ B + A +

T

J φ

(20)

where

E = x_φ2+y_φ2,

F = x_φ x_ψ+y_φ y_ψ,

G = (x_ψ )2 + (y_ψ)2 (21)

are the coefficients of first fundamental form and

J = ± E G−F2 (22)

( , )

B

φ ψ

=

3

4

EJ

µ

[E_φ( sinF ξ+Jcos )ξ 2

–2E F( sin

ξ

+Jcos )

ξ

(F_φsin

ξ

+J_φcos )

α

+ E2(J_ψ sin 2ξ+G_φsin2ξ)], (23)

( , )

A

φ ψ

=µ[−

2 5

( cos sin ) 4

F J E J

ξ

−

ξ

{E_φ(2E J3cosξ+F Esin )ξ

– 4E J J2 2 _φcosξ−2E E F_φsinξ

+ E E E_ψ sinξ }

+

3

cos 2J

α

{E_ψ( sinF

ξ

+Jcos )

ξ

– 2EJ_ψ cos

ξ

−E G_φsin

ξ

}

+

3

( sin cos ) 2

F J EJ

ξ+ ξ

{(J E_φ−2EJ_φ) sin

ξ

+ cosξ [−F E_φ+2E F_φ−E E_ψ]}

−sin₃ 2J

ξ

{(E_ψ( sinJ

ξ

−Fcos )

ξ

–2E J_ψ sin

ξ

+E G_φcos

ξ

}], (24)

and

Ω=

3

( sin cos ) 2

F J EJ

ξ+ ξ

{(J E_φ−2EJ_φ) sin

ξ

+ cosξ [−F E_φ+2E F_φ−E E_ψ]}

−sin₃ 2J

ξ

{E_ψ( sinJ

ξ

−Fcos )

ξ

–2EJ_ψ sin

ξ

+EG_φcos

ξ

}]

+

2 5

( cos sin ) 4

F J E J

ξ

−

ξ

{E_φ(2E J3cosξ+F Esin )ξ

– 4E J J2 2 _φcosξ−2E E F_φsinξ

+ E E E_ψ sinξ }

– [

3

cos 2J

ξ

{E_ψ( sinF

ξ

+Jcos )

ξ

– 2EJ_ψ cos

ξ

−E G_φsin

ξ

}], (25)

3. Retransformation to von-Mises

Coordinates

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coordinates, thereforesetting

x

φ= (26)

in equations (21-22), one have 2 1 [ ( )] E= + x f x′ (27)

F= J E−1 (28)

G = x2

ν ψ

′( )2 (29)

J= x

ν ψ

′( ) (30)

and the angleξ between the coordinate curvesis 1 cos E ξ = (31)

Thus equations (18-20), equation (25) and equations (23-24) are e R − Ω = −R J F_e ₂ +R L_e _ψ– JA_x + E−1A_ψ+ B_ψ (32)

0 = R_e

(

F₁+F₂ E−1

)

–R L_e _x+A (2 E) J ψ − +A_x E−1 – E 1B J ψ − (33)

x x J T −2 E−1 T_ν_x

ν

′+ E T ( )2 J νν ν ′ + 2 1 x e x E J P T E ψ ′   − −   −   + 2 2 1 x E E EJ E J E J J ψ ψ ν ν   _′′ − − +   _′ −    Tν

ν

′ = –

(

2 4 2

)

4 c r J E P B A µ + (34)

2 2 2 ( ) ( ) 1 ( ) 1 ( ) [ ( )] { ( )} f x f x x f x x ν ψ ν ψ ν ψ ′  _′′   +         Ω = _ _ __ ′ _   ′′ _ ′ + + _ _ ′   _ _   (35)

where 2 2 1 2 1 ( , ) (2 ) x x J E E J E A x J E J J ψ µ ψ ₋ ₋  +   −   = _ _ − ₋      (36)

( , ) B x

ψ

= 4µ 3 1 J [− J Jx+ E−1 Jψ ] (37)

andqthe magnitude ofq=( , )u v is E q J = (38)

4. Exact Solutions in von-Mises

Coordinates

Follow [1-3], the conditionL_x_ψ =L_ψ_x on equations (29-30) provides x x x

ν

′A – 2x f A′ _x_ψ – 2 2 1 x (f ) A x

ν

ψψ  ₋ _′    ′ +

ν

′A_x–A_ψ

(

f′+x f′′

)

– B_x f Bψ ψ ν ′     −   ′     =R_eΩ_x +R_e

(

F₁+F x f₂ ′

)

_ψ _e

(

₂

)

x R xν′F − (39)

The solution of equation (39) is expected to lead to the exact solution therefore let us simplifyΩ involved in the right-hand side of it through 2 d

ν

′′=

ν

′ (40)

where d is constant. The case for d=0 is considered separately. For d≠0 equation (40)gives 1 2 1 1 ln ( ) d d k k ν ψ  ₋  =   +   (41)

wherek₁ and k₂ are constants. Equations (35-37) using (40) implies Ω= 1

ν

   _′   2 2 (1 ) N c N x x  _′ ₊  +      , (42)

A= 2 x µ ν ′[

(

)

2 2 1 x N′ − N−d −N ] (43)

and

[

]

2 4 1 B d N x µ ν = − + ′ (44)

Where ( ) ( ) N x =x f x′ (45)

Let us attempt solutions of equation (39) by eliminating µ

(5)

through

A=Y x B( ) (46)

where

2

2 (1 )

( ) 0

4 ( 1 )

x N N d N

Y x

d N

′ − − −

= ≠

− + (47)

Thus, it becomes

x x

xY B –(1 2+ N Y B) _ν_x+

2

(1 )

N N Y

B x νν  _{− −}       

+B_ν

[

−2N Y′−Y N′

]

+ B_x

(

2xY′+Y

)

+B xY( ′′+Y′)=R_e 1₂

ν

    ′   2 2 (1 )

N d N

x x ′  _′ ₊  +      

+R_e

(

F₁+N F₂

)

_ν _e

(

₂

)

x

R x F

− (48)

The use of (41) and (46) in the equation (34)involves

factor 1 Pe ν′

 

−

 _′

  therefore, this guides to searchthe function B

of the type

( , ) 1 Pe ( )

B xψ R x

ν′

 

= − _′

  (49)

where the functionR x( ) is to be determined. Equation (48) on utilizing equation (49)gives

(

)

1 Pe xY R R 2xY Y R xY( Y )

ν

′   ′′ ′ ′ ′′ ′   − + + + +  _′     +

{

}

2 (1 ) (1 2 )

2

e

d N N Y

d P

N Y R R _x

N Y Y N

ν ′   _{− −}     ′ − + −    ′ _ _ __ ′ ′ + +    

=R_e 1₂

ν

    ′   2 2 (1 )

N d N

x x ′  _′ ₊  +    

  +Re

( )

F1 ν

+R N F_e

( )

₂ _ν _e

(

₂

)

x

R x F

− (50)

The search for the appropriate form of F₁ andF₂providing the solution of equations (32-34) leads toF x₂( , )

ψ

=F x₂( )as a solution of the following differential equation

(

2

)

[

( )

]

e x

R a x F = −x Y R ′ ′ (51)

or

2

e

R F = –(Y R)′+ h1

x (52)

whereh₁ is constant.

The equation (50) and (52)implies

1

e

R F=

[

]

1

( ) e

P _d

x Y R e c K ν ′   _{′ −} ′     + 1 e d P e K ν

′ −

{

2

}

(1 2 )

(1 )

2

N Y R

d N N Y

R _x

N Y Y N

′ − +      _{− −}      ₋         _₊ _′₊ _′ _   2 2 2 1 2 (1 ) 2 e d

R e N d N

x

d K x

ν ′ −  _{ } _′ _ +   +_  +      

+ H x( ) (53)

Substitution of equations (52–53) in equations (32–33) provides the functionL

e

R L=

2 2

1 2 2

1

(1 )

2 d e

R e N d N

h

x

d K x

ν ν +_ −    ′+ +       + 1 ( ) ( 1) d e

P e x Y R

N Y R

K d ν − ′  ′₊ ₊     

+ N h2 2N Y R( ) H x( ) dx x   ′ + +    

∫

+ h₂ (54)

whereh₂ is constant and thus equation(43) or equation (44) provides

µ =

2

4 ( 1 )

x d N

ν′ −

− + 1 ( )

e P R x

ν

′   −  _′

  (55)

The equation (34) on using (27), (40), (46), (49) and (55) becomes

(x

ν

′)T_{x x} −2N T_ν_x

ν

′+

2

(1 )

( ) N

T

x νν ν

+ _′

+

(

ν

′ −P T_e_′

)

_r −N′T_ν

ν

′

= – E Pc r( 1 d N)

(

1 4Y2

)

1 Pe R x( )

x

ν

′

− +  

+  − _′

  (56)

The right-hand side of equation (56) suggestssearching for its solution of the type

( ) ( , ) K x T xν

ν =

′ (57)

(6)

x K′′ +2d N K′+

2 2

(1 )

d N K

x

+

+ 1 Pe K

ν

′

 

′ −

 _′

  +d K N′

= – E Pc r( 1 d N)

(

1 4Y2

)

1 Pe R x( )

x

ν

′

− +  

+  − _′

  (58)

Comparing the coefficient of 1 Pe

ν

′

 

−

 _′

  on both side of

equation (58), one finds

(

2

)

( )

( 1 ) 1 4

c r

x K R x

E P d N Y

′ = −

− + + (59)

and

2

x K′′ +2d x N K′+K d_ 2(1+N2)+d x N′_ =0 (60)

Equation (59) and (60) are coupled equations, the function

( )

K x from equation (60) will lead to R x( ) from equation (59) and the solution of equation (60) can be found for a choice off x( ). For example considering f x( )=m₁lnx m+ ₂

it reduces to Cauchy-Euler equation, when

3 4

1

( ) ln cos ( ln )

f x m d x m d

= − + , reduction of order method

is applicable and when

2

( ) ln

2

x

f x d x

d

= − , the equation is

solvable by transforming to normal form. For other f x( ), the solution of variable coefficient differential equation (60) is easy to find from computer algebra system (CAS) software. This leads to T from equation (57), µ from equation (55), p from (12) using equation (54) and

q=( , )u v from equation (8) for F₁ and F₂ from equations (52-53) for intermediate Peclet number.

WhenY x( )=0, the equation (47) implies

2

2 (1 ) 0

x N′ − N−d −N = (61)

Since d≠0therefore solution of equation (61) for d=1

and d= −1 is presented as an example and likewise one can find for d>0ord<0. When d=1equation (61) provides

(

)

2 1

( ) ln ln cosh 2 ( ln )

f x = +c x+ _ c − x _ (62)

and for d= −1 it provides

(

)

2 1

( ) ln ln cosh 2 ( ln )

f x = −c x− _ c + x _ (63)

For both the equations(62-63), the equation (43) gives

0

A= (64) Using (64) in (39), we get

– B_x f Bψ

ψ ν

′

 

 

−

 

′

 

  =ReΩx

+ R_e

(

F₁+F x f₂ ′

)

_ψ −Re

(

xν′F2

)

_x (65)

Our search for the appropriate form of F₁ and F₂

providing the solution of equations (32-34) leads to

2( , ) 2( )

F x

ψ

=F x as a solution of the following differential equation

(

2

)

e _x e x

R xν′F =R Ω (66)

or

(

2

)

( )

e e

R x

ν

′F =R Ω +G

ψ

(67)

where the function of integration is G( )

ψ

. Utilizing (67) in (65), one finds

1 2 1( )

e e x

f B

R F R F x f B ψ H x

ν ′

 

 

′

= − − − +

′

 

  (68)

where H x₁( )is a function of integration. Solving (32-33), using equations (67-68), we have

1 3

( ) ( )

e

R L= − −B

∫

Gψ ψd +

∫

H x dx+h (69)

whereh₃ is constant.

In light of equation (64), the equation (34)becomes

2

(1 )

2 1

4

e

x x x x

c r

P N

x T N T T T

x x E P B N T

ν νν

ν

µ

′

+  

− + + − _′ 

 

′

− = −

(70)

The function B from equation (70) in equation (44) gives a relation between viscosity andtemperature function

(

)

2

3 2

2

(1 ) ( )

4 1

1

x x x

e

c r

x

x T N T

N T

x x

P E P d N

T

N T

ν

νν

ν ν

µ

ν′

−

 

 

+

 

+

 

′

− _ _

=

 

−  

+ −

   

′

 

 

₋ _′ 

 

(71)

The streamline patterns can be drawn using CAS software to observe the effect of various parameters for d either +ve

(7)

5. Conclusion

This communication finds a class of new exact solutions of the equations governing the two-dimensional steady motion with moderate Peclet number of incompressible fluid of variable viscosity in presence of body force in von-Mises coordinates. The characteristic equation for the streamlinesis

the equation

1 2

1 1

( ) ln

( )

y f x

d d kψ k

 ₋ 

= +  

+

  where the

differentiable function f x( ) is either

(

)

2 1

( ) ln cosh 2 ( ln )

f x = +c x c − x 

  or f x( )=c2−ln_xcosh

(

2 (c1+ln )x

)

_ ,

ψ is the stream function and the constantsdis either +veor

ve

− . The pressure p and velocity components for given the component of body force F₁ orF₂are found when Peclet number is moderate and a relation between viscosity µ and temperature function T is observed. For other f x( ) in streamlines equation, temperature, viscosity, pressure and velocity components for given component of body force are found for moderate Peclet number. It shows that in both the cases an infinite set of velocity components, viscosity function, generalized energy function and temperature distribution for intermediatePeclet number in presence of body force can be constructed and graph of streamlines using CAS software can be drawn to observe the streamline patterns.

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