LAMINAR MHD FLOW IN THE ENTRANCE REGION OF AN … · 2020. 2. 21. · development time, rather than a development length. ... of annular tubes. Laminar MHD Flow in the Entrance Region

International Journal of Advanced Research in

Engineering and Applied Sciences ISSN: 2278-6252

Vol. 1 | No. 5 | November 2012 www.garph.co.uk IJAREAS | 29

LAMINAR MHD FLOW IN THE ENTRANCE REGION OF AN ANNULAR CHANNEL

WITH DISSIPATION

D. Venkata Rami Reddy*

M. Rama Chandra Reddy**

B. Rama Bhupal Reddy***

Abstract: In this paper we analyze the Laminar MHD flow in the entrance region of an

annular channel under the influence of transverse magnetic field. The motion of fluid is

between the two insulating cylinders which are concentric. The origin of the coordinate

system is located at the extreme left of the channel along the central line of the cylinders, z is

the coordinate which increases in the down stream direction, r is the radial coordinate and

is the angular coordinate and is perpendicular to the (r, z) plane. The flow problem is

described by means of partial differential equations and the solutions are obtained by using

an implicit finite difference technique. The velocity, temperature and pressure profiles are

obtained and their behaviour is discussed computationally for different values of governing

parameters like magnetic field parameter M, Eckert number Ek and Prandtl number Pr.

Keywords: Laminar flow, MHD, Annual Channel

*Lecturer in Mathematics, Government Polytechnic College, Anantapur, A.P., India.

**Reader in Mathematics, S.K.S.C. Degree College, Proddatur, A.P., India.

***Associate Professor in Mathematics, K.S.R.M. College of Engineering, Kadapa, A.P., India.




1. INTRODUCTION

Fluid transport systems composed of pipes or ducts have many practical applications in

chemical plants and oil refineries, city water supply system and Bio fluid mechanism which

one seem to be maze of pipes. Power plants contain many pipes and ducts for transporting

fluid involved in the energy-conservation process. Hartmann [14] determined the steady-

state profile for laminar magneto-hydrodynamic flow in a plane channel and the analog to

that problem in cylindrical co-ordinates was solved by Globe [11]. Uflyand [43] and

Chekmarer [8] analyzed the time transient case of this problem, which implies a

development time, rather than a development length. The entry problem for this case is

solved by a technique similar to that of Shohet, et.al. [38] which was used in the solution of

the entry problem for a plane channel.

Numerical techniques are applied to the wide variety of engineering fields due to the

complex nature and characteristics of the problems in these areas where it is impossible to

find or not practical to compute an analytical solution. For the last thirty years the finite

difference method, the finite element method, the finite volume method and the Boundary

Element Method (BEM) are mostly applied to many problems and become dominant tools

for solving partial differential equations. Newly generated methods can be grouped in two

main areas as, domain discretization methods and meshless methods. Some of the most

important meshless methods are the smooth particle hydrodynamics [10], the reproducing

kernel particle method [26], the method of fundamental solutions [12], the meshless local

Petrov-Galerkin method [6] the local boundary integral method [39] etc. Among these

methods, the Method of Fundamental Solutions (MFS) has emerged as an effective

boundary-only meshless method [12], which is known as the indirect BEM, will be applied to

magnetohydrodynamic flow problem. In MFS neither domain nor boundary discretization

by elements is required. This feature makes it easy to use and implement in different areas.

Magnetohydrodynamic (MHD) flows in a non-uniform magnetic field, or in ducts of a varying

cross-sectional area, have been extensively studied since the seminal paper of Hartmann

[13], where the formation of the axial current loops and the origin of the pressure losses

associated with the fringing magnetic field either at the entry or exist of a magnet are

qualitatively described.




Later, Shercliff, in his two books [35, 37] and educational movie [36], showed that the

presence of the axial currents promotes the formation of the so-called “M-shaped” velocity

profile and associated vorticity. A few important experiments were made in Riga [7, 9, 25,

40] and in Saint Petersburg [42], which revealed the spectacular behavior of such non-

uniform flows. Incidentally, new phenomena of turbulence suppression in a fringing

magnetic field were recently discovered experimentally in Ilmenau, Germany [4, 5].

Attempts to build a theory of those complex flows started with the papers by Kulikovskii [21,

22], where the concept of characteristic surfaces was first introduced. Although this concept

is limited to the asymptotic case of a perfectly conducting fluid, it allows for qualitative

predictions of the main features of many MHD flows, including those in a fringing field.

Subsequently, different research groups, around Walker [15, 17, 18, 30, 31, 33] and more

recently Molokov (24, 27, 28, 29], investigated such flows, using simplifying assumptions,

e.g. limiting the analysis to inertialess flows. These authors obtained velocity and electric

current distributions, and suggested first explanations of how these flows are organized.

In particular, numerical results based on the inertialess flow model [31] have demonstrated

a fair agreement with the experimental data for both pipe and rectangular duct flows in a

strong magnetic field. However, other experiments for flows in a slotted duct at high

Reynolds (Re≈105) and moderate Hartmann numbers (Ha ≈ 102) have demonstrated strong

inertial effects [42], which cannot be explained using inertialess models. As a matter of fact,

except for Holroyd [16] and Ilmenau experiment [4, 5] most of the theoretical efforts are

lacking direct comparisons with the experimental data, which often do not comply with the

theoretical predictions. Quite recently, Alboussière reexamined the theory, paying a special

attention to the full role of the Hartmann layers [2]. His considerations are, however, still

based on a few striking assumptions that limit the model applicability.

Abdul Maleque and Abdul Sattar [1] were studied the effects of variable properties and hall

current on study MHD laminar convective fluid flow due to a porous rotating disk. MHD

flow of a power-law fluid over a rotating disk studied Anderson and Korte [3]. Jain, et.al.,

[20] were studied the forced convective sub-cooled boiling in heated annular channels.

Diffusion of Aerosols in the entrance region of a smooth cylindrical pipe studied Ingham

[19]. Kumari and Nath [23] were studied Transient MHD rotating flow over a rotating

sphere in the vicinity of the equator. Shah and Farnia [34] were studied flow in the entrance




of annular tubes. Laminar MHD Flow in the Entrance Region of an Annular Channel studied

Numerically Venkateswarlu et al [44].

In this paper determines the velocity and temperature of laminar magneto-hydrodynamic

flow in the entrance region of an annular channel. Both the velocity and temperature

profiles are initially flat upon entering channel. The equations of this system are place into a

finite different form and solved numerically for various magnetic field parameter M, Eckert

number Ek and fixed Prandtl number Pr. Velocity, temperature and pressure curves are

presented from the results of computational work

2 FORMULATION AND SOLUTION OF THE PROBLEM

The annular device is shown in figure 1. This representation is best described in cylindrical

coordinates. The motion of the fluid is between the two insulating cylinders, which are

concentric. The origin of the co-ordinate system is located at the extreme left of the

channel along the center line of the cylinders, Z is the co-ordinate which increases in the

down stream direction, r is the radial co-ordinate and is the angular co-ordinate and is

perpendicular to the (r, z) plane. Note that the origin of this co-ordinate system is not

placed in the center of the region in which the fluid passes. The uniform magnetic field will

be in the radial direction. The inner radius of the channel is ri and the outer radius is r0. Te

channel is of length ‘L’, which must be long compared with the entry length.

The governing equations are:

q

Dt

D (1)

2e

Dqp q J H

Dt (2)

2pC q T T

(3)

Where = Dissipation function

= 2 2 2

2r z r zv v v v

z r r z

The Ohm’s law is eJ E q H

Where q = (vz, vr, 0) and H = (0, H0, 0)




Figure 1: Annual channel configuration

The following assumptions are made:

(i) Flow is steady, laminar, viscous, incompressible and developed.

(ii) There are no applied (external) magnetic fields other than in the r-direction.

(iii) Electric field E and induced magnetic field are neglected [32, 41]

(iv) All the physical properties of the fluid are assumed to be constant

The resultant equations in cylindrical coordinates from the above assumptions are listed

below:

z

u

r

ur

r

zr

1 =0 (4)

2 22

0

2 2

1 ez z z zr z z

Hu u u upu u u

r z z r r r r

(5)

2 2

2

1P r z

T T T TC u u

r z r r r

2

r

u z (6)

The boundary conditions are

inrz TTuuu ;0;0 for z = 0 and ri ≤ r ≤ r0

;0ru for z > 0 and r = ri (7)

wrz TTuu ;0;0 for z > 0 and r = r0

A set of dimensionless variables may now be introduced as follows:

0

2

ir

zZ

0u

uU z

ir ruV ir

rR

2

0

pP

u Pr

PC

inW

in

TT

TT

202

2 HM e

2

0

P w in

uEk

C T T

(8)




The determining equations are then:

01

Z

U

R

VR

R (9)

2

2

2

2 1

R

UM

R

U

RR

U

Z

P

Z

UU

R

UV

(10)

2

2

2 1

Pr

1

R

UEk

RRRZU

RV

(11)

The boundary conditions are

U=1, V=0, =0, for Z = 0 and 1 ≤ R ≤ 3

V = 0 for Z > 0 and R = 1 (12)

U=0, V=0, =1, for Z > 0 and R = 3

A finite difference techniques is adopted for solving above differential equation (9)

to (11) together with boundary conditions (12)

3 RESULTS AND DISCUSSION

The following initial values are taken at entrance:

Z =0, U=1.0, V=0, P=0.1357 and Pr = 0.1. For fully developed flow calculations have been

carried. Figures 2 to 11 show the velocity profile, Figures 12 to 17 show the temperature

profile and Figures 18 and 19 shows the pressure profile are obtained for different magnetic

field parameter M and Eckert Number Ek in the annular channel.

For fixed M as Z increases, the velocity decreases for R=1.1, 1.3 and then starts increasing,

for R=1.5 and after. The effect of magnetic field in an annular channel is not produce much

flattening of the velocity profile. Its primary effect is to shift the position of the point

towards the outer at which the maximum velocity occurs and as the magnetic field

increased this point. For R=1.3, the velocity values are decreases and for R=1.9 the velocity

values are increases as magnetic field parameter M increases. The velocity profiles are not

effect for increasing Eckert number Ek and for fixed magnetic field parameter M.

We also notice that the temperature profiles shows that, it reduces initially and then

gradually with increasing R while fixing the other parameters for M=0. This similar

behaviour of the temperature component is observed for the magnetic parameter M=2 and

6. Next we observe that the temperature profiles for different values of the magnetic




parameter. For fixed M the temperature profile follows parabolic path and attains

minimum at R=1.5 for Z=0.001. As Z increases the temperature profiles shows the linear

path for Z=0.1. These similar observations were obtained for the different values of R and

fixing the other parameters. The temperature profiles are effect for increasing Eckert

number Ek and for fixed magnetic field parameter. Temperature increases for increasing

Eckert number Ek and for fixed magnetic field parameter M.

The magnitude of pressure component decreases for increase of the values of Z and the

magnetic parameter M. Therefore in general the magnitude of pressure enhances with

increasing the magnitude field in the radial direction throughout the annular channel. The

pressure profiles are not effect for increasing Eckert number Ek and for fixed magnetic field

parameter M.

Figure 2.3: Velocity profile for fixed M=0

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.002 0.004 0.006 0.008 0.01Z

UR=1.1 R=1.3

R=1.5 R=1.7

R=1.9


0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 0.002 0.004 0.006 0.008 0.01

Z

UR=1.1 R=1.3

R=1.5 R=1.7

R=1.9

Figure 2: Velocity profile for fixed M=0 Figure 3: Velocity profile for fixed M=2


0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.002 0.004 0.006 0.008 0.01

Z

U R=1.1 R=1.3 R=1.5

R=1.7 R=1.9


0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.002 0.004 0.006 0.008 0.01

Z

U R=1.1 R=1.3

R=1.5 R=1.7

R=1.9





Figure 2.7: Velocity profile for fixed R=1.2

0.8

0.85

0.9

0.95

1

1.05

1.1

0 0.02 0.04 0.06 0.08 0.1

Z

U

M=0 M=2

M=4 M=6

Figure 2.8: Velocity profile for fixed R=1.6

0

0.2

0.4

0.6

0.8

1

1.2

0 0.02 0.04 0.06 0.08 0.1

Z

U

M=0 M=2

M=4 M=6

Figure 6: Velocity profile for fixed R=1.2 Figure 7: Velocity profile for fixed R=1.6

Figure 2.9: Velocity Profile for fixed R=1.8

0

0.2

0.4

0.6

0.8

1

1.2

0 0.02 0.04 0.06 0.08 0.1

Z

U

M=0 M=2

M=4 M=6

Figure 2.10: Velocity Profile for fixed M=0

0.1

0.3

0.5

0.7

0.9

1.1

1.3

1.5

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

R

U

Z=0.01 Z=0.05

Z=0.1

Figure 8: Velocity profile for fixed R=1.8 Figure 9: Velocity profile for fixed M=0


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

R

U

Z=0.01 Z=0.05

Z=0.1


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

R

U

Z=0.01 Z=0.05

Z=0.1





Figure 2.13(a): Temperature profile for fixed M=0 and Ek=0

0

0.2

0.4

0.6

0.8

0 0.002 0.004 0.006 0.008 0.01

Z

Te

mp

era

ture

R=1.1 R=1.3 R=1.5

R=1.7 R=1.9

Figure 2.13(b): Temperature profile for fixed M=0 and Ek=5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Z

Tem

pe

ratu

re

R=1.1 R=1.3

R=1.5 R=1.7

R=1.9

Figure 12(a): Temperature profile for fixed M=0 and Ek=0

Figure 12(b): Temperature profile for fixed M=0 and Ek=5

Figure 2.13(c): Temperature profile for fixed M=0 and Ek=10

0

0.5

1

1.5

2

2.5

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Z

Tem

pe

ratu

re

R=1.1 R=1.3

R=1.5 R=1.7

R=1.9

Figure 2.14(a):Temperature profile for fixed M=2 and Ek=0

0

0.2

0.4

0.6

0.8

0 0.002 0.004 0.006 0.008 0.01

Z

Tem

pera

ture

R=1.1 R=1.3

R=1.5 R=1.7

R=1.9

Figure 12(c): Temperature profile for fixed M=0 and Ek=10



0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Z

Te

mp

era

ture

R=1.1 R=1.3

R=1.5 R=1.7

R=1.9


0

0.4

0.8

1.2

1.6

2

2.4

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Z

Tem

pe

ratu

re

R=1.1 R=1.3

R=1.5 R=1.7

R=1.9







0

0.2

0.4

0.6

0.8

0 0.002 0.004 0.006 0.008 0.01

Z

Tem

peratu

re

R=1.1 R=1.3

R=1.5 R=1.7

R=1.9

Figure 2.15(b): Temperature profile for Fixed M=6 and Ek=5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Z

Tem

pe

ra

ture

R=1.1 R=1.3

R=1.5 R=1.7

R=1.9




0

0.5

1

1.5

2

2.5

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Z

Te

mp

era

ture

R=1.1 R=1.3

R=1.5 R=1.7

R=1.9


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

R

Te

mp

era

ture

Z=0.001 Z=0.005

Z=0.010 Z=0.10




0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

R

Tem

pe

ra

ture

Z=0.001 Z=0.005

Z=0.010 Z=0.10


0

0.5

1

1.5

2

2.5

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

R

Tem

pe

ratu

re

Z=0.001 Z=0.005

Z=0.010 Z=0.10







0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

R

Tem

pe

ra

ture

Z=0.001 Z=0.005

Z=0.010 Z=0.10


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

R

Tem

pe

ra

ture

Z=0.001 Z=0.005

Z=0.010 Z=0.10


Figure 16(b): Temperature profile for fixed

M=2 and Ek=5


0

0.5

1

1.5

2

2.5

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

R

Te

mp

era

ture

Z=0.001 Z=0.005

Z=0.010 Z=0.10

Figure 2.18(a): Temperature profile for fixed M=6 and Ek = 0

0

0.2

0.4

0.6

0.8

1

1.2

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

R

Te

mp

era

ture

Z=0.001 Z=0.005

Z=0.010 Z=0.10



Figure 2.18(b): Temperature profile for fixed M=6 and Ek = 5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

R

Tem

peratu

re

Z=0.001 Z=0.005

Z=0.010 Z=0.10

Figure 2.18(c): Temperature profile for fixed M=6 and Ek = 10

0

0.5

1

1.5

2

2.5

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2R

Te

mp

era

ture

Z=0.001 Z=0.005

Z=0.010 Z=0.10

Figure 17(b): Temperature profile for fixed Figure 17(c): Temperature profile for fixed




M=6 and Ek=5 M=6 and Ek=10

Figure 2.19: Pressure profile for fixed Z=0.001

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Z

P

M=0 M=2

M=4 M=6

Figure 2.20: Pressure profile for fixed Z=0.01

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Z

P

M=0 M=2

M=4 M=6

Figure 18: Pressure profile Figure 19: Pressure profile

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