Unsteady MHD Couette flow between two Parallel Plates with Uniform Suction

Mrs.P.H.Nirmala¹*, A.Saila Kumari¹, C.S.K.Raju²

¹Department of Mathematics, JNTU University, Ananthapur, A.P, India.

²Department of Mathematics, GITAM University, Bangalore, Karnataka, India.

*Corresponding Author E-mail:nirmalaph83@gmail.com

ABSTRACT:

In the present paper analysis is carried out to study on an unsteady magnetohydrodynamic flow between two parallel plates with one in uniform motion and the other plate at rest. We also incorporated the uniform suction at stationary plate. The applications of magnetohydrodynamic cover a wide range of engineering areas as power generators, aerodynamics heating, petroleum industry, cooling system, purification of crude oil, separation of matter from fluid, fluid droplets and sprays. We also discussed the physical characteristics of longitudinal and transverse velocity and pressure distributions. The arising set of partial governing differential equations (PDEs) of the flow is solved by similarity transformation. Analytical expression is given for the velocity field and the effects of the various parameters entering into the problem are discussed with the help of graph.

KEYWORDS:MHD, Parallel Plates, Pressure gradient, Reynolds number, Hartmann number.

INTRODUCTION:

Magneto hydrodynamic study deals with eventually an electrically conducting fluid. The study of flow and heat transfer of electrically conducting fluids in various channels is of extreme theoretical interest because it has been applied to astrophysical phenomena and a variety of geophysical. It has many applications in electrostatic precipitation, aerodynamic heating, petroleum industry, polymer technology, accelerators, Magneto hydrodynamic pumps, fluid droplets, purification of crude oil, power generators, geothermal energy ballistics and nuclear reactors and in many branches of science and engineering. Attia and Kotb[1] they initiated the study of heat transfer in magneto hydrodynamics case using finite difference method.

Hassanien and Mansour [2] a detailed analysis of the effect of magnetic parameters, frequency parameter and permeability of porous medium is studied. Venkateswarlu et al.[3] discussed a problem of an unsteady flow of an incompressible viscous fluid through a vertical porous channel. Hameed and Nadeem [4]investigated the focal point of study is the problems useful in experimental determination of viscoelastic fluid and presented results for the difference between hydrodynamic non-Newtonian second order flow and classical viscous fluid in unsteady magneto hydrodynamics non-Newtonian flows. The low Reynolds and transverse magnetic field on unsteady flow over porous parallel plates is examined by Ganesh and Krishnambal [5]. Stamenkovic et al. [6] investigated the effect of heat transfer of two immiscible fluids between to moving plates in magneto hydrodynamics case. Verma and Mathur [7] studied the effect of Hartmann number in hydromagnetic Couette type flow and they noted that when the Hartmann number increases then coefficient of Skin friction and pressure decreases. Electric conductivity and variable viscosity on unsteady magnetohydrodynamic dusty fluid over two parallel plates is studied by Attia[8]. Faisal and Alam[9]investigated unsteady magnetic hydro dynamics flow of viscous incompressible Caisson fluid bounded by non-conducting porous plates with Joule heating and Hall current. Kiemaet al.[10] studied the steady magneto hydrodynamic scouette flow between infinite plates with transverse magnetic field. Soundalgekar and uplekar[11] proposed to study the effect of heat-transfer of Couette flow over two plates were assumed to be linear variation of temperatures. Moniem and Hassanin[12] illustrated the effects of oscillatory suction velocity and permeability variation in mass transfer and free convective flow of a viscous fluid through porous infinite plates. Dhiman Bose et al.[13]depicted the effects of magnetic field and the permeability on velocity field analyzed in a non-Newtonian fluid flow through semi-infinite porous plates.Thamizhsudar and Pandurangan[14] studied the hall current and radiation on Magnetic hydro dynamics flow past an exponentially accelerated plate in presence of rotation. Most of the research works analyzed the Hartmann flow under different physical effects[15-20].

By keeping into view above literature in this article, the unsteady hydro dynamic flow through the parallel platesunder the influence of magnetic field and assess the physical quantities of longitudinal, transverse velocity and pressure on the solution of Hartmann number. The governing partial equations are solved by analytically and the problems are discussed with the help of graph.

MATHEMATICAL ANALYSIS WORK:

An incompressible viscous fluid flow between two parallel plates y=-h and y=h bounded by parallel plates with uniform suction. Let X, Y-plane lie along the plates.Let u and v be the velocity components in the x and y directions respectively. In the analysis,it is assumed that the unsteady hydro dynamic flow between non-conducting two parallel plates in that one plate is in uniform motion and the stationary plate at rest with uniform suction. Viscosity of the fluid is assumed as constant and assumed u and v are velocity components be taken in x and y directions respectively.

From the previous section discussion,let us choose

(4)

are the solutions of governing equations (1)-(3) respectively, with the boundary conditions

(5)

Let be kinematic viscosity and be the dimensionless distance and the equations (1)-(3) become

(6)

(7)

(8)

The corresponding boundary conditions are changed into

(9)

Let be the stream function and it is defined as

(10)

and, (11)

Solving equations (2),(3),(9) and (10) we get

(12)

(13)

Differentiating the equations (12) and (13) with respect to x andrespectively

(14)

(15)

Solving equations (14) and (15) we get

(16)

Integrating above equation we get

(17)

Where and

Equation (17) reduces to ordinary differential form

(18)

The corresponding boundary conditions are, , , (19)

Solving equation (18) with boundary conditions (19) we get the solution is

(20)

Where and

Value is substituting in the stream function

Hence the longitudinal velocity becomes

(20)

The transverse velocity becomes

(21)

From equations (7),(8) and (20), the pressure drop is

NUMERICAL DISCUSSIONS AND RESULTS:

Table 1:Comparison of the present results with the results of A.M. Ismail et al [20] for B₀=0.5,t=0.2,u₀=0.5 and v₀=0.5 by taking (1/k)=0

Values of u

Values of v

Present values

A.M. Ismail et al [20]

Present values

A.M. Ismail et al [20]

3.2961

1.3503

0.4649

0.0621

-0.1212

3.3113

1.3552

0.4663

0.0624

-0.1211

2.8171

0.9262

0.0957

-0.2523

-0.3808

2.8324

0.9312

0.0971

-0.2520

-0.3807

Table 2: Comparison of the present results with the known results of A.M. Ismail et al [20] for B₀=0.5,t=0.2,u₀=0.5 and v₀=0.5,M=1 by taking (1/k)=0

Values of P

Present values

A.M. Ismail et al [20]

1.5

1.215

0.375

0.135

0.015

1.5

1.215

0.96

0.735

0.54

Table 1 shows the comparison of numerical values of longitudinal velocity(u) and transverse velocity(v) with magnetic field in the present analysis with the corresponding numerical values of A.M. Ismail et al [20] in the absence of porous medium ((1/k)=0) and the results are seen to be in good agreement. From Table 1, it is observed that for transverse velocity and longitudinal velocity decreases when magnetic field increases. Table 2 shows a comparison of numerical values of pressure and density in the present analysis with the corresponding numerical values of A.M. Ismail et al [20] in the absence of porous medium ((1/k)=0) and it is observed that pressure decreases when density increases,the magnitude of the lower and upper plates are same .

In this analytical study, analytical solutions are obtained and the outcomes are discussed graphically. From figures 1-6 shows the characters of significant physical parameters on the average entrance density, velocity, Hartmann number,time and pressure distributions. In this problem the following computations are considered as input values for required graphs.x=1,y=-1 to 1,h=2,u₀=0.5,v₀=0.5,n=0.5,M=1,B₀=0.5,t=0.2,=0.5, =1, =0.5. Form figures 1-6 show the effect of transverse velocity of fluid, longitudinal velocity of fluid and pressure distribution respectively.

CONCLUSION:

In this article, Results are evaluated by using similarity transformation method and analytically. The analytical and graphical solutions are obtained for the unsteady Magnetic hydro dynamic Couette flow between two parallel plates with uniform suction.The following conclusions are investigated from this problem.

Ø Transverse velocity of the fluid decreases as increase the density and magnitude (Hartmann number).

Ø The longitudinal velocity of the fluid decreases as increase the density, magnitude and time.

Ø The pressure distribution of the fluid decreases as density increases.

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