The Plane Viscous Incompressible MHD Flows

The Plane Viscous Incompressible MHD Flows

Anirban Roy

Department of Sciences & Humanities, Faculty of Engineering, Christ University Bengaluru, India

Hari Baskar R Department of Mathematics Christ University Bengaluru, India

Abstract The fundamental coefficients for a plane surface with a curvilinear coordinate system is used to study the two dimensional plane Magnetohydrodynamic flows of viscous incompressible fluids with mutually orthogonal magnetic and velocity fields. Geometry of streamlines are studied for some specific flows by expressing the governing equations in terms of the fundamental coefficients.

div H 0

where v v1, v2

1 2

with v v v 2 v 2 ,

(4)

denotes the velocity vector

H H1, H2 the magnetic

Keywords Magnetohydrodynamic flows; streamlines; fundamental coefficients.

1. INTRODUCTION

   O. P. Chandna and M. R. Garg [1] investigated the geometries of steady plane magnetohydrodynamic flows of a

   field vector, the density, p the pressure, the coefficient of viscosity, the magnetic permeability of flow.

   Also the vorticity function (x, y), current density function

   (x, y) and the energy function h(x, y) are given by

   v v

   steady viscous incompressible fluid when the streamlines and magnetic lines form an isometric net and when magnetic force vanishes. They [2] also investigated these flows with mutually orthogonal magnetic and velocity fields using Hodograph transformation method. Kingston and Talbot [3] classified the

   corresponding flows of an inviscid incompressible fluid. Nath

   2 1

   x y

   H2 H1

   x y

   1 2

   2

   1 2 2

   (5)

   (6)

   and Chandna [4] determined the flow geometries for these type of flows when the streamlines are straight lines or

   h v

   2

   • p or h

     v1

   • v2

     p

     (7)

     involutes of a curve. M. H. Martin [5] introduced the curvilinear coordinates , in the plane of flow in which the coordinate lines = constant are the stream lines of the flow and the coordinate lines = constant are left arbitrary.

     In the present work, we obtain a system of partial differential equations for the fundamental coefficients E, F, G

     Now (1) becomes

     v1 v2 0

     x y

     Using (5) and (7), (2) becomes

     h

     (8)

     for a plane surface, as functions of , , to study steady plane Magnetohydrodynamic flows of a viscous incompressible fluid, having infinite electrical conductivity, when the magnetic fields and velocity fields are mutually orthogonal. This approach is illustrated by considering two examples in which the curves = constant and = constant form an orthogonal curvilinear

     y

     x

     v2 H2 x

     h

     v1 H1 y

     (9)

     (10)

     coordinate system.

2. BASIC EQUATIONS

   From (3) we get

   v1H2 v2 H1 k

   where k is an arbitrary non zero constant.

   (11)

   The steady plane Magnetohydrodynamic flow of a viscous incompressible fluid of infinite electrical conductivity is governed by the following system of equations

   Finally (4) gives

   H1 H2

   0.

   (12)

   x y

   div v 0

   (1)

   We shall determine the seven unknown functions v1, v2, H1,

   v.grad v grad p curl H H 2 v

   H2, , and h.

   1. III. CONCEPTS OF DIFFERENTIAL GEOMETRY

      curl v H 0

   2. Let x x , and y y , define a system of

curvilinear coordinates in (x, y) plane.

then we can obtain x, y and x, ysuch that

line and = constant, directed in the sense of increasing

If	the	Jacobian		J	is	given	by	lower parameter values of according as J is positive or
		x	x					negative. He has also proved that
J x, y , y y				x y x y , WV E and v iv E ei (17) 1 2 J where is is the angle between the tangent to the co-ordinate

and the new form of the vorticity is

x J ,

y J ,

x J ,

y J

1

F

E

(18)

y

x

y x

W W W

(13)

provided 0 J

Nath and Chandna [4] have shown that the magnetic field acts along the magnetic lines towards the higher or lower

Then the first fundamental form for the xy plane is

ds2 Ed 2 2Fdd Gd 2

where

parameter values of according as J is positive or negative.

Also they have shown that

x 2

E

y 2

,

WH

G AND H

• iH

G ei

(19)

1 2 J

F x x y y ,

where is is the angle between the tangent to the c(o1-4o)rdinate

x 2 y 2

line and = constant, directed in the sense of increasing

with x axis. The new form of the current density function is

G

1 G F

W W W

(20)

The incompressibility constraint equation (8) implies the

existence of a stream function (x, y) and the solenoidal

Using (14), (15) and (16), we get from (9)

equation (12) implies the existence of a magnetic flux

h 1 G F

(21)

function (x, y) such that

2 J

v ,

v ,

H

, H

Using (14), (15) and (16), we get from (10)

x 2 y

1 x

2 y 1

h 1 F E

(22)

(15)

We assume that the curves = constant and the curves

2 J

Differentiating (21) w. r. t. and (22) w.r.t. , and then using

= constant form the curvilinear coordinate system. Then using (15) we get from (11),

the integrability condition

2h

2h

we obtain

1 G F

1 F E

, 1

2 J J

y x x

y x, y J k 0

i.e. J 1 0

(16)

0 (23)

k

And

Thus we conclude that when streamlines, = constant and the magnetic lines = constant of steady plane viscous

1 1 Magnetohydrodynamic flows are taken as curvilinear co-

W 2 EG F 2 J 2 J W , EG F 2

k 2 k 2

By inverse theorem of differential calculus, if we know x

and y as functions of and , then we can find and as functions of x and y.

ordinates system (, ) in the physical plane then the set of seven of differential equations (5), (6), (8), (9), (10), (13),

(14) for v1, v2, H1, H2, , and h as functions of x and y may be replaced by the system of six equations (16), (17), (18), (19), (20) and (23) in five dependent variables E, F, G, and . If the solutions to these equations are given we can find x and y as functions of and and hence E, F, G, , as

IV. CONTINUITY EQUATION AND VORTICITY

functions of x and y, since 0 J

. Once we obtain E, F,

Martin [5] has shown that the equation of continuity implies the fluid flows along the streamlines towards the higher or

G, , as functions of x and y then v1, v2, H1, H2 as functions of x and y are given by (17) and (19). Finally, the energy and

pressure functions h and p of x & y can be obtained from momentum equation and energy expression (3).

We now study an example in which streamlines are straight lines and not parallel but envelop a curve C. Taking the tangent lines to the curve C and their orthogonal

Thus we conclude that when the streamlines in a two dimensional flow of a plane viscous MHD fluid are straight lines, then they must be concurrent or parallel.

Radial Flow: The square of the element of arc length in polar coordinate system is given by

trajectories, the involutes of C as the system of orthogonal curvilinear coordinates, we find the square of the element of arc length ds in this orthogonal curvilinear coordinate system

ds2 dr2 r2d 2

Since the flows are radial, we have

(29)

is given by

ds2 d 2 2 d 2

(24)

= (r) and = () (30)

where = () denotes the arc length of the curve C and the angle of elevation of the tangent line to the curve C. In this coordinate system, = constant are the involutes of the curve

We find the values of E, F, G, J, , and hence the values of

V, H, h and p as

C and = constant are the tangent lines to the curve C.

Investigating the flows for which

= () and = () (25) We find the values of E, F, G, J, and as

E 2 ,

1

J r 1 ,

k

0,

F 0,

2

r

G 2 ,

,

1

E 2

F 0,

2

G 2 ,

2k

A

where

kr A

(say)

V E

k k

A ,

J

,

3 ,

(26)

W

kr A r

H G

kr kr

W

A

Then the Gauss equation

K 1 W FE 2EF EE

2

W EG FE 0

2

(27)

h

k 2 r 2

A2

k 2r 2

f (r) c1

A2

W E 2W

E 2W

p

A2

2r 2

f (r) c1 ,

is satisfied by the E, F and G.

where c1

is an arbitrary constant .

Substituting all these values in (23) we get

22 4 iv 4 4 3

6 4 9 6 2 2

15 2 10 15 3 0

(28)

Next we study the example where magnetic lines are the tangent lines to the curve C and streamlines are the involutes, the orthogonal trajectories of C.

In this coordinate system, = constant are the involutes of the curve C and the curves = constant its tangent lines. The square of the element of arc length ds in this orthogonal curvilinear coordinate system is given by

Since and are independent variables, for the above

ds2 d 2 2 d 2

(31)

relation to hold identically, it must hold on the curve C, =

() and consequently15 3 0 .

Since cannot vanish identically, 0 . Therefore, the radius of curvature of C vanishes identically and hence the streamlines are concurrent lines.

where = () denotes the arc length. In this coordinate system, = constant are the involutes of the curve C and the curves = constant its tangent lines.

Investigating the flows for which

= () and = () (32) We find the values of E, F, G, J, and as

2

E 2 ,

F 0,

G 2 ,

iv 3 2 2 4 0 (35)

1

J ,

This is a differential equation in () where is involute in

this case and hence the distance along the radius from the common centre of the circular stream line.

If q gives the velocity of flow on circular stream lines, for

q the above differential equation (35) becomes

,

3

(33)

3q 2 2q 4 q q 0

Also we find that the Gauss equation (27) is satisfied by E, F

and G as calculated above.

Substituting all these values in (23) we get

REFERENCES

1. Garg M. R. and Chandna O.P. (1976) Viscous orthogonal MHD flows SIAM J.Appl. Math., Vol. 30(3)577 585.

   iv 5 2

   4

2. Chandna O.P. and Garg M. R. (1971) On steady plane magnetohydrodynamic flows with orthogonal magnetic and

   3 2

   4

   3 2 62 2 2 0

   Moreover putting 0 in (34) we get

   Since and are independent variables, for the above relation to hold identically, it must hold on the curve C, = () and consequently

   3 62 2 2 0 .

   (34)

   velocity fields. Int. J. of Eng. Sci., Vol. 17, 3, 251 257.

3. Kingston J. G. and Talbot Z. (1969) Z. Angew, Math. Phys. 20 956

4. Nath V. I. and Chandna O.P. (1973) On Plane viscous Magnetohydrodynamic Flows, Quarterly of Applied Mathematics, Vol. 31, No. 3, pp. 351-362

5. Martin M.H. (1971) The flow of a viscous fluid. I, Arch. Rational Mech. Anal. 41 266 – 286.

Since and cannot vanish identically, 0

i.e. constant

i.e. radius of curvature is same

in all directions i.e. bending is same in all directions. Hence the streamlines are concentric circles.

Thus we conclude that if the streamlines in a two dimensional flow of a plane viscous MHD fluids are involutes of a curve C then C reduces to a point and the streamlines are circles concentric at this point.