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Research Article
General Science
Mathematics

Unsteady MHD flow and heat transfer of third grade fluid past on oscillating inclined belt

Saleem Nasir1, Taza Gul1, Saeed Islam1, Rehan Ali Shah2

Abstract

In this article, we study the Magneto-hydrodynamic (MHD) unsteady thin film flow of a non-Newtonian fluid on inclined oscillating belt in presence of heat transfer. The analytical solutions of velocity and temperature profiles have been obtained by using optimal homotopy asymptotic method (OHAM) and homotopy perturbation method (HPM). The identical results obtained from both techniques are compared numerically and graphically. The effects of various physical parameters have been studied and presented graphically.

Keywordsunsteady thin film flow, MHD, third grade fluid, OHAM, HPM, heat and inclined belt

Author and Article Information

Affiliation

1 Department of mathematics, Abdul Wali Khan University Mardan, KPK Pakistan

2 Department of mathematics, UET Peshawar, Khyber Pakhtunkhwa, Pakistan

RecievedNov 26 2014 Accepted Jan 22 2015 PublishedFeb 27 2015

Citation Nasir S, Gul T, Islam S, Shah RA (2015) Unsteady MHD flow and heat transfer of third grade fluid past on oscillating inclined belt. Science Postprint 1(2): e00045. doi:10.14340/spp.2015.02A0003.

Copyright ©2014 The Authors. Science Postprint published by General Healthcare Inc. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs 2.1 Japan (CC BY-NC-ND 2.1 JP) License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

Funding Recently no one organization supporting us financially.

Competing interest We submit this manuscipt on behalf of ourselves and there is no funder which support our cause.

Corresponding authorTaza Gul

Address Abdul Wali Khan University Mardan Main Campus, 23200 Mardan, Pakistan.

E-mail tazagulsafi@yahoo.com

Peer reviewersMohammad Mehdi Rashidi1 and Revieweer B

1: Shanghai Automotive Wind Tunnel Center, Tongji University, Shanghai, China and ENN-Tongji Clean Energy Institute of advanced studies, Shanghai, China

Introduction

In recent year’s non-Newtonian fluid have become very much significant in a number of industrial manufacturing processes especially with the improvement of polymer industry, petroleum industry, food, micro fluid, fiber, and paper industries. Due to the complexity of non-Newtonian fluids it is extremely difficult to investigate a single model that exhibits all behavior of non-Newtonian fluids. The third grade fluid model contained all the properties of visco-elastic fluids. Siddiqui et al. 1, 2 discussed the exact solution of thin film flow of PTT fluid through inclined surface and vertically moving belt. Tshela 3 studied the fluid flow through inclined plane and effect of heat. Raunge Kutta integration method has been used to obtained solution. Sahoo and Poncent 4, 5 investigated the heat transfer analysis and steady flow of third order fluid. For heat analysis two processes are discussed which are prescribed surface temperature (PST) and prescribed heat flux (PHF). Aiyesimi et al. 6, 7 studied the steady MHD layer of non-Newtonian fluid and heat transfer over inclined plane. The problem is solved for three different flows by using perturbation technique. Similar and latest work can also be seen in 8-15. Shah 16 discussed the unsteady fluid flow between wire and die. Numerically the problem is solved by OHAM and the ideas of OHAM applied on linear, nonlinear differential and partial equations. Hayat and Sajid 17 discussed the series result of thin film flow of fourth grade fluid by HAM and HPM. Gbadeyan et al. 18 also discussed the film flow of third order fluid and heat transfer through inclined plane. The velocity and temperature distribution were solved by using regular perturbation method and OHAM. Numerically and graphically they compare the result for velocity and temperature. Makinde et al. 19 examine hydro-dynamically third order fluid film through isothermal inclined palate. Hermite Pade method used for approximate solution of velocity and temperature. Islam et al. 20 investigated the approximate solution of three different type non Newtonian fluid using OHAM. Gul et al. 21, 22 studied the MHD thin film flow of third order fluid through vertical belt. ADM and OHAM techniques were used to obtain the numerical as well as graphical solutions of problems. Yao and Liu 23 discussed the solution of three kinds of unsteady flows of second order fluid over flat plates by means of sine and cosine transformation methods. Erdongn and Imrak 24 examine the characteristics of unsteady flow of second grade fluid. Different examples have been given to explain the unsteady coutte, poiseuille and generalized coutte flows. Abdulhameed et al. 25 investigated the unsteady flow of of non-Newtonian fluid over oscillating plates. The exact solutions of problem are derived by using Laplace transformation, perturbation and variable separation methods. Ali et al. 26 calculated the solution of unsteady second grade fluid through vertical oscillating surface using Laplace transformation method. Zaman et al. 27 discussed the analytical solution of stocks first problem of unsteady MHD fourth grade fluid using HAM. The influence of various physical parameters on velocity field is discussed. Gamal 28 studied the effect of MHD on thin film flow on moving belt, inclined plane and vertical cylinder. The problems are solved by using shooting method. Mahmood and Khan 29 discussed the film flow of non-Newtonian fluid through porous inclined plane. HPM method is used to solve the problem.

The central strategy of the recent effort is to study the approximate analytical solution of unsteady MHD thin film of non-Newtonian fluid with heat transfer on oscillating inclined belt using optimal homotopy asymptotic method (OHAM) and homotopy perturbation method (HPM). Mabood et al. 30 studied the solution of Raccati differential equation by using OHAM. Kashkari 31 discussed the solution of Kawahara equation by using OHAM, HPM, VHPM and VIM methods. He 32, 34 discussed the fundamental introduction of OHAM method. Anakira 35 studied the OHAM solution of delay differential wquation. Marinca et al. 36, 37 studied the solution of fourth grade fluid using OHAM. Iqbal et al. 38 solved the linear and nonlinear Klein Gordon equation using OHAM.

Basic equation

The constitutive equations for third grade fluid model is

where T is the Cauchy stress tensor I is identity tensor, p is fluid pressure, μ is coefficient of viscosity, α1, α2 are material constants, A1, A2 and A3 are Rivlin-Ericksen tensor and S is extra stress tensor. The extra stress tensor and Rivlin-Ericksen tensor for third grade fluid is define as

where β1, β2 and β3 are material coefficients for third grade fluid.

The continuity, motion and energy equations governing the flow of MHD incompressible third grade fluid are

where u = (u, v, w) is the velocity vector , ρ is the fluid density, J is the current density, B = (B0, 0, 0) is the uniform magnatic filed, σ is the electrical conductivity, μ0 is the magnetic permeability, E is electric field, D/Dt is the material time derivative, g is the external body force, Θ is temprature, k is the thermal conductivity and cp is specific heat. For small Reynolds number the magnetohydrodynamic force invvoved in equation (6) take the form

Problem formulation

Consider a thin film of third grade fluid flowing down on an oscillating inclined belt. The belt is stationary and only oscillating. The thin film is of uniform thickness δ. A uniform magnetic field B is applied transversely to the belt. In problem x-axis is taken parallel to the belt surface and y-axis is perpendicular to the belt. Assuming that the flow is unsteady, laminar and the pressure is atmospheric.

The corresponding velocity field is

subject to boundary conditions

here ω is used as frequency of the oscillating belt.

The equation of motion and energy equation reduce to the form

the component Txy of third grade fluid is

Inserting equation (15) in equations (13, 14)

introducing non-dimensional variables as

where

M = δ2σB0/μU is the magnetic parameter, m = ρδ2 gsinθ/μU is gravitational parameter,

β = β3U2/μδ2 is the non-Newtonian parameter, Br = V2μ/(k(Θ1 –Θ0)) is Brinkman number,

Pr = cpμ/k is Prandtl number and α = α1/ρδ2 is non-Newtonian effect.

Using the above dimensionless variables in equation (16, 17) and dropping bars we obtain

and the boundary conditions are

Solution techniques

Basic theory of OHAM

To discussed the fundamental theory of OHAM, we consider the general form of nonlinear partial differential problem

where L is the linear operator in differential equation, N is non linear term, h is source term, x is independent variable and B is boundary operator. According to OHAM

where p is embedding parameter and P ∈ [0, 1] , the auxiliary function H(P) is define as

c1, c2, c3,..., is called auxiliary constants.

Here ψ(y, t, P) is unknown function. Clearly when p = 0 and p = 1 it holds that

inserting equation (27) into equation (24) collecting the same power of P and equating each coefficient of P to zero. We obtained solution of the zero, first and second component problems. The general solution of equation (27) is of the form

substituting equation (28) in equation (23) we obtained the residual

There are different method like Galerkin’s method, Ritz method, least squares method and collocation method to find the optimal value of auxiliary constants cj, j = 1, 2, 3,…. Here we use the method of least squares in the present problem

where a, b are two values depending on the problem.

The auxiliary constants cj, j = 1, 2, 3,… . n can be identified from the given conditions

The auxiliary constants cj can be finding from the other methods. At the last the approximate solution and the auxiliary constants are well determined.

Basic idea of HPM

To illustrate the basic concept of HPM for solving the nonlinear partial differential equation we consider the following general equation

where u(y, t) is the unknown function, M(y, t) is the known analytic function, Q is the boundary operator and T is the general differential operator which is express in linear part L(u(y, t)) and nonlinear part N(u(y, t)) as

Therefore equation (32) can be written as

Now according to the homotopic method define as

or we can write equation (35) as

here P∈[0, 1] is the embedding parameter and u0(y, t) is the initial approximation of equation (32) satisfying the boundary condition. Now from equation (35) and (36) we have

By the variation of P from 0 to 1, u(y, t, P) change from u0(y, t) to u(y, t) which is called deformation. L(u(y, t)) - L(u0(y, t)) and T(u(y, t)) - M(y, t)) are called homotropic.

The approximate solution of equation (32) can be expressed as a series of the power of p as

setting P = 1, then the approximate solution of equation (39) becomes

Solution of optimal homotopy asymptotic method (OHAM)

Write equation (19, 20) in standard form of OHAM then we obtained zero, first and second component problems.

Zero component problem:

Solution of zero, first component problem of velocity and temperature distribution using boundary condition in equation (21, 22) is

The solution of second component of velocity distribution is too large. So derivation are given up to first order while, graphical solutions are given up to s{econd order.

The value of cj for the velocity components u0(y, t), u1(y, t, c1), and u2(y, t, c2) are

c1 = -0.20743738104347828, c2 = -0.14134043204452848.

The values of cj for the temperature distribution are

c1 = 0.9452610, c2 = 0.79699391, c3 = 0.0009547247523, c4 = - 0.005014082.

Solution of homotopy perturbation method (HPM)

Applying HPM on equations (19, 20) we obtained zero, first and second component problems of velocity and temperature distribution.

Zero component problem:

Solution of zero, first component problem of velocity and temperature distribution using boundary condition in equation (21, 22) is

The second component solution of velocity and temperature distribution is too massive. So second component solutions are not given and graphical solutions are given up to second order.

Discussion and Conclusions

The approximate solutions of velocity and temperature distribution of MHD unsteady thin film flow of third grade fluid through an oscillating inclined belt have been obtained. The leading non-linear partial differential equations are solved by using OHAM and HPM methods and compared the results for velocity and temperature distribution. The numerical comparisons and absolute error are shown in Table 1 and 2. In Figure 1 we deliberated the geometry of the problem. Figure 2 indicates the comparison of both methods is in excellent agreement. These figures show that OHAM and HPM have approximately similar solution. The influence of different time level on velocity and temperature distribution is shown in Figure 3 and 4 illustrates the graphical representation of velocity and temperature distribution for different value of model parameters. It is clear from these figures that the flow of fluid is due to the oscillation as of the plane, so the velocity and temperature distribution of the fluid will be high at the surface of the plane comparatively to the remaining domain and will decrease gradually for the fluid away from the surface of the plane. Further we discussed the effect of various physical parameters on velocity and temperature distribution. Figure 5 shows the effect of magnetic parameter M on velocity and temperature. Increase in transverse magnetic field reduces fluid motion during oscillation. It is clear that the boundary layer thickness is reciprocal to the transverse magnetic field and velocity decreases as flow progresses towards the surface of the fluid. Figure 6 represent the effect of the gravitational parameter m. Increasing the value of m resulted to increase the velocity. Figure 7 and 8 show the effect of non-Newtonian parameter α, β. The fluid motion increases with increasing non-Newtonian parameters. Figure 9(a) shows the effect of Br. The fluid motion increases as the Br increases and becomes more trampled for higher values of Br. The effect of similarly Figure 9(b) show the effect of Prandtl number pr. Prandtl number pr is reciprocal to other physical parameters. So increase in Prandtl number at larger level decreases the temperature distribution.

Table 1: Comparison of OHAM and HPM for the lift velocity profile, by taking, ω = 0.2, α = 0.02, St = 0.5, M = 0.2, β = 0.5, t = 20

Table 2: Comparison of OHAM and HPM for temperature distribution by taking, ω = 0.2, α = 0.02, St = 0.5, M = 0.2, β = 0.5, t = 20

Figure 1 Geometry of problem

Figure 2 Comparison of OHAM and HPM methods for (a) velocity profile by taking ω = 0.2, α = 0.02, St = 0.5, M = 0.2, β = 0.5, t = 20; (b) temperature distribution by taking ω = 0.2, α = 0.02, St = 0.3, M = 0.2, β = 0.5, t = 10

Figure 3 Influence of different time level on (a) velocity profile and (b) temperature distribution

Figure 4 Velocity distribution of fluid (a) and temperature distribution of fluid (b) by taking ω = 0.2, α = 0.02, St = 0.5, M = 0.9, β = 0.5

Figure 5 Effect of the magnetic parameter M on (a) the velocity profile and (b) temperature distribution by taking St = 0.5, α = 0.2, β = 0.5, ω = 0.02, t = 10

Figure 6 Effect of the gravitational parameter m on (a) the velocity profile and (b) temperature distribution by taking M = 0.4, α = 0.02, β = 0.5, ω = 0.02, t = 10

Figure 7 Effect of the non-Newtonian parameter β on (a) velocity profile and (b) temperature distribution by taking M = 0.9, α = 0.2, St = 0.5, ω = 0.02, t = 10

Figure 8 Effect of the non-Newtonian parameter α on (a) velocity profile and (b) temperature distribution by taking M = 0.9, α = 0.2, St = 0.5, ω = 0.02, t = 10

Figure 9 Effect of (a) the Brinkman number Br and (b) Prandtl number Pr on temperature distribution by taking M = 0.9, α = 0.2, St = 0.5, ω = 0.02, t = 10

Author Contributions

All authors have same contribution in the present work.

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