HALL EFFECTS ON UNSTEADY MHD OSCILLATORY FLOW OF BURGER S FLUID THROUGH A PIPE

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1 Inter national Journal of Pure and Applied Mathematics Volume 113 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu HALL EFFECTS ON UNSTEADY MHD OSCILLATORY FLOW OF BURGER S FLUID THROUGH A PIPE M. VeeraKrishna 1, Kamboji Jyothi 2 1,2 Department of Mathematics, Rayalaseema University Kurnool, Andhra Pradesh , India 1 veerakrishna maths@yahoo.com Abstract In this paper, MHD oscillating flow of Burger s fluid through porous medium in a pipe with Hall current effects has been considered. The solution. The solution in closed form is obtained by Fourier transform technique and discussed computationally with reference to governing parameters. AMS Subject Classification: 76A05, 76E06, 76N20, 76S05, 76W05. Key Words and Phrases: Berger s fluid, Fourier transforms, Hall effects, non-newtonian fluids and porous medium. 1 Introduction A great deal of work has been done for the formulation of constitutive equations describing the mechanical behavior of various classes of visco-elastic fluid. Among the non- Newtonian fluids, the rate type fluids are those which take into account the elastic ijpam.eu

2 and memory effects. Hartmann flow is a classical problem that has many important applications in magnetohrdrodynamics (MHD) generators and pumps. Hartmann [1] first studied an incompressible viscous electrically conducting fluid flow between two infinite parallel non- conducting stationary disks under the action of a transverse magnetic field. Under different physical conditions it was considered by Hughes and Young [2], Cowling [3] and Pai [4]. The solutions for the velocity fields in closed form were studied [5], [6], [7] under different physical effects. Due to the growing use of non-newtonian fluids material in many manufacturing and processing industries, considerable efforts have been directed towards understanding their flows. Krishna and M.G.Reddy [8] discussed MHD free convective rotating flow of visco-elastic fluid past an infinite vertical oscillating plate. Krishna and G.S.Reddy [9] discussed unsteady MHD convective flow of second grade fluid through a porous medium in a Rotating parallel plate channel with temperature dependent source. Krishna et al. [10], [11], [12], [13] discussed MHD flow with heat and mass transfer. The considered fluid model is a visco-elastic model and has been used to characterize food products such as cheese [6], soil [7]. 2 Formulation and solution of the problem We consider MHD flow of an incompressible electrically conducting Burger s fluid through a porous medium in pipe under the influence of uniform transverse magnetic field taking Hall current into account. The physical configuration of the problem as shown in Figure 1. The governing equations are given in non-dimensional form as, ijpam.eu

3 Figure 1: Physical Configuration of the Problem ) 2 u (1 + λ + β t t 2 t = Q 2 0 (1 + λ + β t t ( ) [ 2 2 u λ r t r + 1 ] u 2 r r ( σm 2 1 im + 1 ) (1 + λ k The corresponding boundary conditions are ) e iω 0t t + β 2 t 2 ) u (1) u(±1, t) = 0 (2) In order to solve the governing problem we define the temporal Fourier transform pair as ψ(r, ω) = u(r, t)e iωt dt (3) u(r, t) = 1 ψ(r, ω)e iωt dω (4) 2π Taking Fourier transform to Eqs. (1) and (2) and then solving the resulting problem, we have the following general solution ψ(r, ω) = Q { 0(1 βω0 2 + iλω 0 1 J } 0(ξr) δ(ω ω ψ 2 0 ) (5) (1 + iλ r ω J 0 (ξ) ijpam.eu

4 Where J 0 (.) is the zeroth-order Bessel function, δ(.) is the dirac delta function and ( M ξ 2 2 = 1 im + 1 ) [ ] 1 βω 2 K + iω + iλω 1 + iλ r ω The Fourier inversion of Eq. (5) after using the property of delta function gives u(r, t) = Q { 0(1 βω0 2 + iλω 0 1 J } 0(ξ 0 r) e iω0t. (6) ψ 2 (1 + iλ r ω 0 J 0 (ξ 0 ) Where ξ 0 = ξ. ω=ω0 3 Results and Discussion This flow is governed by the non-dimensional parameters like, M Hartmann number, K permeability parameter, m Hall parameter, β the rheological parameter and ω the frequency of oscillation. For computational purpose we are fixing some parameters λ = 5, λ r = 1, Q 0 = 2, t = 1. Special attention has been given to examine the velocity profiles r 0for five different kinds of non-dimensional parameters, which are depict in the Figures (2-6). We noticed that from the Fig.2, in the presence of magnetic force, an increase of the magnitude in the magnetic parameter M reduces the velocity profiles monotonically due to the effect of the magnetic force against the flow direction. This is in accordance to the fact that the magnetic field is responsible to reduce the velocity. From the Figure (3), both the real and imaginary parts of the velocity enhance with increasing the permeability parameter K throughout the fluid region. We also observe that lower the permeability lesser the fluid speed in the entire fluid medium. Similar behaviour is observed with increasing Hall parameter m. Figure 4 shows that the increase of Hall parameter m for fixed magnetic parameter M increases the velocity profiles. Moreover, the velocity increases. Further, when the magnetic Reynolds number is very small, the flow pattern with Hall effect is remarkably analogous to that for the non-conducting flow. obviously, the supposition of very small magnetic Reynolds number will be ijpam.eu

5 Figure 2: The velocity Profiles against M with K = 1, m = 1, β = 1, ω = 1.2 Figure 3: The velocity Profiles against K with M = 1.5, m = 1, β = 1, ω = 1.2 Figure 4: The velocity Profiles against m with M = 1.5, K = 1, β = 1, ω = 1.2 legitimate for flow of liquid metals or slightly ionized gas. Also from the Figure (5) appears that the velocity is an increasing function of the rheological parameter β of the Burgers fluid. But this result cannot be generalized for other chosen values of the rheological parameter β since the behaviour of β is non-monotonous. Moreover, the magnitude of the velocity also increases with frequency of ijpam.eu

6 Figure 5: The velocity Profiles against β with M = 1.5, K = 1, m = 1, ω = 1.2 oscillation ω (Figure 6). Figure 6: The velocity Profiles against ω with M = 1.5, K = 1, m = 1, β = 1 4 Conclusions 1. An increase of the magnitude in the magnetic parameter M reduces the velocity profiles monotonically due to the effect of the magnetic force against the flow direction. 2. Both the real and imaginary parts of the velocity enhance with increasing the permeability parameter K or Hall parameter m throughout the fluid region. 3. Lower the permeability lesser the fluid speed in the entire fluid medium. 4. The solution of Burgers fluid only contributes if there is a pressure gradient of the oscillatory nature. ijpam.eu

7 5. For limiting cases, the magnitudes of the real and imaginary velocities enhance with increasing Hall parameter m. References [1] J. Hartmann, Hydrodynamics, theory of the laminar flow of an electrically conductive liquid in a homogeneous field, Kgi Danske Videuskab, Sleskuh, Mat. Fys. Medd., 15 (1937). [2] W.F. Hughes, Y.J. Young, The Electromagnetodynamics of Fluids, John Wiley, New York (1966). [3] T.G. Cowling, Magnetohydrodynamics, Interscience Publishing, New York (1957). [4] S.I. Pai, Magneto-gas Dynamics and Plasma Dynamics, Springer-Verlag (1962). [5] G.W. Sutton, A. Sherman, Engineering Magnetohydrodynamics, McGraw-Hill (1965). [6] R.A. Alpher, Heat transfer in magnetohydrodynamic flow between parallel plates, Int. J. Heat Mass Transfer, 3 (1961), 108. [7] K.R. Cramer, S.I. Pai, Magnetofluid Dynamics for Engineers and Applied Physicist, McGraw-Hill, New York (1973). [8] M. VeeraKrishna, M. Gangadhar Reddy, MHD free convective rotating flow of Visco-elastic fluid past an infinite vertical oscillating porous plate with chemical reaction, IOP Conf. Series: Materials Science and Engineering, 149 (2016) doi: [9] M. VeeraKrishna, G. Subba Reddy, Unsteady MHD convective flow of Second grade fluid through a porous medium in a Rotating parallel plate channel with temperature dependent source, IOP Conf. Series: Materials Science and Engineering, 149 (2016), doi: ijpam.eu

8 [10] M. VeeraKrishna, B.V. Swarnalathamma, Convective Heat and Mass Transfer on MHD Peristaltic Flow of Williamson Fluid with the Effect of Inclined Magnetic Field, AIP Conference Proceedings, 1728 (2016), doi: [11] B.V. Swarnalathamma, M. Veera Krishna, Peristaltic hemodynamic flow of couple stress fluid through a porous medium under the influence of magnetic field with slip effect, AIP Conference Proceedings, 1728 (2016), doi: [12] D.V. Krishna, R. Siva Prasad, M. Veera Krishna, Unsteady Hydro Magnetic Flow of an Incompressible Viscous Fluid in a Rotating Parallel Plate Channel with Porous Lining, Journal of Pure and Applied Physics, 21 (2009), [13] M. Veera Krishna, S.V. Suneetha, R. Siva Prasad, Hall Effects on Unsteady Hydro Magnetic Flow of an Incompressible Viscous Fluid in a Rotating Parallel Plate Channel with Porous Lining, Journal of Ultra Scientist of Physical Sciences, 22 (2010), ijpam.eu

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