Ain Shams Engineering Journal (2014) 5, 187-192

Ain Shams University Ain Shams Engineering Journal

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MECHANICAL ENGINEERING

MHD squeezing flow between two infinite plates

Umar Khan a, Naveed Ahmed a, Z.A. Zaidi ab, Mir Asadullah b, Syed Tauseef Mohyud-Din a *

a Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan b COMSATS Institute of Information Technology, University Road, Abbottabad, Pakistan

Received 4 April 2013; revised 30 July 2013; accepted 7 September 2013 Available online 10 October 2013

KEYWORDS

Magneto hydrodynamic (MHD) flows; Squeezing flow; Variation of Parameters Method (VPM); Flow parameters

Abstract Magneto hydrodynamic (MHD) squeezing flow of a viscous fluid has been discussed. Conservation laws combined with similarity transformations have been used to formulate the flow mathematically that leads to a highly nonlinear ordinary differential equation. Analytical solution to the resulting differential equation is determined by employing Variation of Parameters Method (VPM). Runge-Kutta order-4 method is also used to solve the same problem for the sake of comparison. It is found that solution using VPM reduces the computational work yet maintains a very high level of accuracy. The influence of different parameters is also discussed and demonstrated graphically.

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1. Introduction

Because of its many industrial and practical applications, squeezing flow between parallel plates is an important area of study. These include polymer process industry, compression, injection modeling and the squeezed films in power transmission. The first ever work this regard was done back in 19th century by Stefen [1]. Later Reynolds in 1886 [2] and Archibald [3] in 1956 studied the same problem for elliptic plate and rect-

* Corresponding author. Tel.: +92 3235577701, 514908154; fax: +92 514908145.

E-mail addresses: umar_jadoon4@yahoo.com (U. Khan), nidojan@g-mail.com (N. Ahmed), zzaidi@ciit.net.pk (Z.A. Zaidi), assadullah@-ciit.net.pk (M. Asadullah), syedtauseefs@hotmail.com (S.T. Mohyud-Din).

Peer review under responsibility of Ain Shams University.

angular plates respectively. Since then many researchers have contributed their efforts concerning solution to this problem in different geometries [4-11].

In many physical situations the fluid under consideration is electrically conducting and even a slight presence of magnetic or electromagnetic field can change the behavior of flow. It was therefore essential to discuss the flow under the influence of magnetic field to see how it affects the flow behavior.

Effects of magnetic field on squeezing flow for different geometries have been considered in [12-15] by different researchers. Islam et al. [16] considered it and worked on squeezing flow problem by studying the effects of magnetic field. They used Optimal Homotpic Asymptotic Method (OHAM) to solve the governing equation. However in OHAM we face a lot of laborious work and determination of Cis is not an easy task. So an easier and improved solution is needed. For the fulfillment of this purpose we present this study and employ a technique known as variation of parameters (VPM).

Variation of Parameters Method (VPM) is an analytical technique used to solve different types of problems. It is free

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from round off errors, calculation of the so-called Adomian's polynomials, perturbation, linearization or discretization. It uses only the initial conditions which are easier to be implemented and reduces the computational work while still maintaining a higher level of accuracy. Ma et al. [17,18] employed Variation of Parameters Method (VPM) for solving involved non-homogeneous partial differential. Recently, Mohyud-Din et al. used VPM to solve different problems involving ordinary differential equations, partial differential equations and integral equations [19-26].

In this article, VPM is successfully applied to solve the squeezing flow problem. Comparison of analytical results obtained by this study to the numerical ones is also provided which shows effectiveness of this technique. There is a significant decrease in computational work using this method, yet the results are more accurate.

2. Mathematical formulation

The equations of motion for the flow are given by [12], V-V = 0, (5)

@7 + (V 'V)V

= V • T - fB

where, V is velocity vector, q constant density and T is the Cauchy Stress tensor given by,

T —-pI + Ai, Ai = (VV) + (VV)T.

Further fB is a source term arising due to applied magnetic field, i-e., the so called magnetic or Lorentz force. This force is known to be a function of the imposed magnetic field B, the induced electric field E and the fluid velocity vector V, that is

fB = r(E + V*V)*B.

Consider a steady axisymmetric flow where V is represented as V = [u(r, z, t), 0, w(r, z, t)] and the generalized pressure p and the vorticity X(r, z) are given by,

P** — P + p |V|2,

Introducing the stream functions by

. 1 d w x 1

u(r,z, t) —-—-, w(r,z, t) —--— .

r dz r or

Using Eqs. (10) and (5), continuity equation is identically satisfied. Also, by using Eqs. (7)-(10) in Eq. (6), we have,

dp* , p d2W _ E2W l dE2W , rB^ dW

dr ' r dtdz p dr r2 p dz ' r dz

dp* q 0 W @W E2W l 0E2W dr r dtdz q dz r2 r dz

Eliminating pressure from Eqs. (11) and (12), we get the compatibility equation,

i dE±_ d(w, EW)

r @ t @ ( r, z)

— l E4W - ^ dW,

r r dz

, , d2 1 d d2 A where,E2 — — --tt + tt^ — 0.

@ r2 r @ r @ z2

Consider the axisymmetric squeezing flow of an incompressible viscous fluid between two infinite parallel plates. For small values of velocity of the plates, gap 2H changes slowly with the time and flow can be assumed to quasi-steady [27]. Thus, from Eq. (13) we get

d W, EW

— l E4W - dW,

r r dz

with associated auxiliary conditions z — H, then u — 0, w — — V,

z — 0, then w — 0, du — 0.

We can now define stream function as

W(r, z)—r2f(z). (16)

Consuming Eq. (16), the compatibility Eq. (14) reduces to

(15a) (15b)

f(z)+21f(z)^"(z)-r^f(z) —0, l ir

Subject to the boundary conditions

f(0) — 0, f '(0) —0,

f(H) — VV, f (H) — 0.

Eqs. (17) and (18) can be made dimensionless by introducing following the non-dimensional parameters

z* = —, Re =

V/2 ' H i/V' y i

After omitting the '*' for the sake of simplicity, we may obtain

F(z) + ReF(z)F''(z)- M2F(z) — 0, (20)

with boundary conditions,

F(0) — 0, F(0)—0, F(1) — 1, F (1) —0.

3. Solution procedure for VPM

Following the procedure proposed for variation of parameters method [19-26] to solve Eq. (9) with the associated boundary conditions (10), we have

F„+1 (z) — Â1 + A2z + A3 — + A4 — 26

3! - 2T + ÎT - 3. (ReFn(S)Fn(s)

- M2Fn(s)) ds, with n = 0, 1, 2, ...

Using a part of boundary conditions given in Eq. (10), we have Ai =0, A3 = 0. After setting A2 = A and A4 = B Eq.

(22) can be written as

Fn+1 (z) = Az + B-6

- Lz(f! - + zs2 - 3!) (ReFn(s)Fn'(s)

M2Fn(s)) ds, n

A, B can be calculated by using the conditions F(1) = 1 and F(1) = 0, respectively.

The first two iterations of the solution extracted by Eq. (23) are given by

F1 (z) = Az + B— - f—ReAB - M2B )z5 6 \120

--1—ReB2z7,

F2(z) = Az + B- +—ReABz5 6 120

+ (— Re2A2B--1—ReB2--—.

^V1680 5040 1260

ReABM2 z7

22680 1

Re2AB2 Re2B3

30240 1

1108800 1

38438400 1

3962649600*

Re3 AB3

Re3 B4z1'

1900800 1

ReB2M2J z9 Re3A2B2 -

38438400

1900800

Re2B3M2 z13

ReB2M4 z

Similarly, other approximations for F(z) can also be calculated.

4. Results and discussions

Figure 3 Effects of magnetic number on the velocity profile for Re = 1.

Determination of basic flow parameter F(z) makes it easier to examine the flow behavior in terms of it. Axial velocity is directly characterized by F(z) while F (z) symbolizes the radial velocity. Influences of Reynolds number Re and Magnetic number M on

the velocity profile are demonstrated in Figs. 1-4. In Fig. 1, effects of increasing Re on axial velocity are shown for fixed M = 0.2. Figure clearly displays the axial velocity to be an function of Re. It is worth mentioning that increase in Re is credited

= 0,1,2,...

Table 1 Convergence of VPM solution for different values of Reynold's number.

Order of approximation A = f (0) R =1, M =1 B = /"(0) A = f (0) R = 2, M = 2 B = /"■■(0) A = /"(0) R = 4, M = 4 B = f'(0)

1 1.509388 -3.124286 1.470834 -2.666944 1.369780 -1.447342

2 1.507976 -3.110485 1.475536 -2.709380 1.366139 -1.421508

3 1.508142 -3.111912 1.474693 -2.702554 1.365997 -1.427149

4 1.508125 -3.111779 1.474831 -2.703601 1.365919 -1.425997

5 1.508127 -3.111790 1.474810 -2.703451 1.365963 -1.426220

6 1.508127 -3.111789 1.474813 -2.703469 1.365964 -1.426187

8 1.508127 -3.111789 1.474813 -2.703469 1.365964 -1.426187

10 1.508127 -3.111789 1.474813 -2.703469 1.365964 -1.426187

Numerical (RK-4) 1.508127 -3.111789 1.474813 -2.703469 1.365964 -1.426187

to either increase in distance between the plates and speed by which they move or it is associated to the decrease in kinematic viscosity. Fig. 2 shows the influence of Re on radial velocity for fixed magnetic number. It is evident that radial flow near the walls is delayed while in center it is accelerated.

Figs. 3 and 4 reveal the upshots of magnetic number M on axial and radial velocities respectively. Fig. 3 depicts that for increasing M axial velocity is a decreasing function of M. Noti-cable reduction in axial velocity can be observed near the center while near the walls this deceleration is slight. According to Fig. 4 accelerated radial flow is observed near the walls for increasing M while this pehonomena is reversed in vicinity of center of the channel and decrease in axial velocity is reported.

It is important to ensure that the series solution in Eq. (25) is convergent. In Table 1, numerical values of the constants A and B are calculated to check the convergence of VPM solution. Only sixth order approximation is enough for a convergent solution. It is worth noticing that error between successive approximation becomes minimal with increasing order of approximations and after sixth approximation onwards it becomes zero. It can also be seen from the same table that with each increasing iteration solution become nearer to the numerical solution obtained by employing Runge-Kutta (RK-4) method, both solution match excellently after six approximations.

A graphical comparison between both the numerical and analytical solution is presented in Fig. 5. It shows, VPM and numerical results agree exceptionally well. Graph displays the curves associated to F(z) under different values of parameters Re and M. In each case results are compatible.

Numerical values of skin friction coefficient F (1)are displayed in Table 3. Absolute of Skin fircition coefficient decreases

Table 2 Comparison of VPM solution and numerical solution for different values of Reynold's number.

z Re = 1 VPM M = 1 Numerical Re = 2 VPM M = 2 Numerical Re = 4 VPM M = 4 Numerical

0 0 0 0 0 0 0

0.1 0.150294 0.150294 0.147030 0.147030 0.136356 0.136356

0.2 0.297480 0.297480 0.291350 0.291350 0.271249 0.271249

0.3 0.438466 0.438466 0.430220 0.430220 0.403064 0.403064

0.4 0.570188 0.570188 0.560845 0.560845 0.529885 0.529885

0.5 0.689623 0.689623 0.680342 0.680342 0.649343 0.649343

0.6 0.793795 0.793795 0.785714 0.785714 0.758455 0.758455

0.7 0.879779 0.879779 0.873813 0.873813 0.853453 0.853453

0.8 0.944695 0.944695 0.941309 0.941309 0.929595 0.929595

0.9 0.985706 0.985706 0.984648 0.984648 0.980929 0.980929

1 1 1 1 1 1 1

with increasing Re. On the other hand absolute coefficient of skin friction is an increasing function of magnetic number M.

5. Conclusions

Variation of Parameters Method (VPM) for MHD squeezing flow is discussed here. Effects of involved parameters on velocity are demonstrated graphically and a comprehensive discussion is provided. Convergence of the series solution is assured for different values of parameters in Table 1. A numerical solution is also obtained for the same problem by Runge-Kutta or-der-4 method. Comparison between the results is provided in Table 2 and also demonstrated graphically in Fig. 5. It is clear that our results agree exceptionally well with the numerical results. From Figs. 1-5, we may conclude that:

(I) Increase in Reynolds number increase the velocity of fluid.

(II) Effect of magnetic number is opposite to that of Reynolds number.

(III) Only sixth order approximation is enough to obtain a convergent solution.

(IV) Solutions obtained by VPM are in complete agreement with the numerical solution of the problem.

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Umar Khan completed his master degree from COMSATS Institute of Information Technology, Abbottabad, Pakistan. He is pursuing Ph.D. in the field of Applied Mathematics at Hitec University Taxila Pakistan. He is currently working as Visiting Faculty member at COMSATS Institute of Information Technology, Abbottabad. He has a teaching experience of 1 year and a research experience of 3 years.

Naveed Ahmed completed his master degree from COMSATS Institute of Information Technology, Abbottabad, Pakistan. He is pursuing Ph.D. in the field of Applied Mathematics at Hitec University Taxila Pakistan. He is currently working as Dean of Mathematics at Iqra Academy Abbottabad. He has a teaching experience of 9 years and a research experience of 5 years.

Zulfiqar Ali Zaidi completed his master degree from COMSATS Institute of Information Technology, Abbottabad, Pakistan. He is pursuing Ph.D. in the field of Applied Mathematics at Hitec University Taxila Pakistan. He is currently working as Assistant Professor COMSATS Institute of Information Technology, Abbottabad. He has a teaching experience of 18 years and a research experience of 5 years.

Mir Assadullah completed his Ph.D degree from University of Essex in 1984in the field of Fluid Mechanics. He is currently working as an Advisor at COMSATS Institute of Information Technology, Abbottabad. His field of interest is Newtonian and non-Newtonian Fluid Mechanics He has a teaching experience of over 40 years and a research experience of 30 years.

Syed Tauseef Mohyud-Din completed his Ph.D. degree from COMSATS Institute of Information Technology, Islamabad, Pakistan. He is currently working as Dean faculty of Sciences & Chairperson at Titec University Taxila Cantt, Pakistan. He has won Presidents Pride of Performance in 2011. Also won best research paper award from Higher Education Commission Pakistan in 2010. His field of interest includes Analytical and numerical solutions of initial and boundary value problems related to the applied/engineering sources using Variational iteration, Homotopy Perturbation, Variation of Parameters, Modified Variation of Parameters, Variational Iteration using He's polynomials, Variational Iteration using Adomain's polynomials, exp function, iterative, decomposition and finite difference schemes. He has a teaching experience of 18 years and a research experience of 5 years.