Research Article Innovation: International Journal of Applied Research; ISSN: (Volume-2, Issue-2) ISSN: (Volume-1, Issue-1)

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1 Free Convective Dusty Visco-Elastic Fluid Flow Through a Porous Medium in Presence of Inclined Magnetic Field and Heat Source/ Sink 1 Debasish Dey, 2 Paban Dhar 1 Department of Mathematics, Dibrugarh University, Assam, India 2 Department of Mathematics, Gauhati University, Assam, India Abstract An unsteady free convective flow of dusty visco-elastic fluid through a porous medium in presence of inclined magnetic field, heat source and porosity has been investigated. The visco-elastic fluid flow is haracterized by Walter s liquid (for short relaxation memories). A magnetic field of strength B 0 is applied in a direction making an angle θ with the vertical. The porous medium is bounded by two non conducting parallel walls, where one wall of the channel is fixed and the other is oscillating with time about a constant non- zero mean. Initially, both the walls are kept at same temperature T s. The dust particles gain heat from fluid by conduction through their spherical surface. The equations of the governing fluid motion are solved analytically by using perturbation technique and the velocity profiles for fluid and dust particles are discussed graphically and shearing stress is analyzed numerically for various values of flow parameters involved in the solution. Keywords Free convection, Walters liquid, Saffman Model, porous medium, shearing stress, heat source, Grashoff number I. INTRODUCTION Nowadays researchers from various backgrounds like engineering, applied mathematics etc have shown their interest on dusty visco-elastic fluid flow because of its uses in various fields of engineering and environmental sciences. One of the classes of visco-elastic fluids used in Manufacture of spacecrafts, aeroplanes, tyres, belt conveyers, ropes, cushions, seats, foams, plastic engineering equipments, contact lens etc is Walters liquid (Model B ) and its constitutive equations have been discussed by Walters [1, 2]. The motion of dust particles in a laminar flow was first studied by Saffman [3]. Michael and Miller [4] have investigated the motion of dust particles in a plane parallel flow. Oscillating dusty flow through a rigid pipe has been discussed by Nayfeh [5]. Behaviour of unsteady dusty fluid flow in a channel under various physical situations have been analyzed by Gupta and Gupta [6], Singh [7], Singh and Ram [8], Prasad and Ramacharyulu [9], Gupta and Gupta[10], Ajadi [11]. Govindrajan[12] has discussed the problems of dusty viscous fluid flow using various physical properties. Free convection effects on the Stokes problem for an infinite vertical plate in a dusty fluid flow have been discussed by Ramamurthy [13]. Attia[14] have studied the problem of Unsteady MHD Couette flow and heat transfer of dusty fluid with variable physical properties. Alle et al. [15] have investigated the motion of dusty visco-elastic fluid in an inclines channel. Heat transfer cases in the motion of dusty viscous fluid with exponential decaying pressure gradient have been shown by Attia et al. [16]. Unsteady flow of a dusty viscous liquid between two parallel plates in presence of a transverse magnetic field has been studied by Kalita [17]. Sandeep and Sugunamma [18] have analysed free convective flow of dusty fluid through a porous medium in presence of inclined magnetic field and volume fraction. Two dimensional steady free convection flow of an electrically conducting viscous fluid through a porous medium bounded by two stationary infinite vertical porous plates in the presence of heat source and chemical reaction has been studied by Ahmed and Bhattacharyya [19]. Effects of chemical reaction on hydro-magnetic unsteady walter s memory flow with constant suction and heat sink have been investigated by Sekhar and Reddy [20]. In this paper we have analysed the problem of unsteady free convective flow of dusty visco-elastic fluid (Walters liquid, Model B ) through a porous medium in presence of applied inclined magnetic field, heat source and permeability of the medium II. MATHEMATICAL FORMULATION We consider an unsteady flow of two dimensional dusty visco-elastic fluid through a porous medium in presence of inclined magnetic field and external heat agent. A magnetic field of strength B 0 is applied in a direction making an angle θ with the vertical. The porous medium is bounded by two non conducting parallel walls, where one wall of the channel is fixed and the other one is oscillating with time about a constant non- zero mean. Initially, the system is at rest and both the walls are kept at same temperature T s. At time t > 0, the temperature of the upper plate is instantaneously raised to a temperature oscillating with time. To study the governing dusty fluid motion, we use the following assumptions: a. The dust particles are assumed to be electrically non-conducting and spherical in shape. b. The induced magnetic field is neglected by assuming very small values of magnetic Reynolds number. Debasish Dey et al. Page 28

2 c. Volume fraction of dust fraction is neglected here. d. The energy dissipation and Ohmic heating is neglected in the energy equation. g The governing equations of fluid motion are: Figure 1: Physical Configuration of the Problem Where is the temperature of the system in static case. The boundary conditions are where & are velocities of fluid particles and dust particles respectively, the displacement variable, the time, g be the acceleration due to gravity, β be the co-efficient of volume expansion, ρ & the densities of fluid particles and dust particles respectively, be the limiting viscosity at small shear rate, K T the thermal conductivity of the fluid, K 1 be the permeability of the medium, B 0 be the strength of the applied magnetic field, m be the mass of the dust particles, Q 0 be the external heat source, & specific heats of fluid and dust particles, be the temperature relaxation time, the viscosity, N 0 the number of dust particle per unit volume, K= 6πµa (a= radius of dust particle) be the Stokes constant, k 0 be the visco-elastic parameter. III. METHOD OF SOLUTION Let us introduce the following non-dimensional parameters in the governing equations of motion (2.2) - (2.5), Then the dimensionless equations of the governing motion are Debasish Dey et al. Page 29

3 The boundary conditions in the dimensionless form are, To solve the equations (3.2) (3.5), we assume that the velocity and temperature to be of the form as follows The zero-th order and first order equations of motion for fluid particles and dust particles are given by The relevant boundary conditions are,, (3.16) IV. RESULTS AND DISCUSSIONS Solving equations (3.8) to (3.15) subject to the boundary conditions (3.16), the velocity profiles of fluid particles and dust particles, temperature fields of fluid and dust particles are obtained as The shearing stresses formed at the plates are given by The problem of dusty visco-elastic fluid characterized by Walter s liquid (Model B ) through a porous medium in presence of inclined magnetic field and heat source has been discussed analytically. The non-zero values of k characterize the visco-elastic fluid and its zero value represents the phenomenon of Newtonian fluid motion. Figure 2: Velocity profile u against y for Pr=3, Gr= 5, k = 0.2, L=1, ωt=π/2, S=1, θ=π/6 Debasish Dey et al. Page 30

4 Figures 2 to 7 characterize the nature of velocity profiles against the displacement variable y over entire height of the channel and it is seen that fluid accelerates with the increasing values of y. Effect of applied magnetic field is exhibited through the magnetic parameter M and figure 2 and 3 states that increasing values of M retards the fluid motion and as well as the motion of dust particles. Grashoff number represents the effect of free convection on the governing fluid motion and its higher values reduce viscosity and as a result the fluid motion is accelerated and so as the motion of the dust particles imposed in the mixture (figure 4 and 5). Figure 3: Velocity profile V against y for Pr=3, Gr= 5, k = 0.2, L=1 ωt=π/2, S=1, θ=π/6. Figure 4: Velocity profile u against y for Pr=3, M= 1, k = 0.2, L=1 ωt=π/2, S=1, θ=π/6 In simultaneous momentum and heat diffusion, the importance of Prandtl number cannot be neglected and its higher value increases the momentum diffusion and as a consequence the speed of the fluid motion is decelerated (figure 6). Figure (7) depicts the velocity profiles for different values of strength of heat source (s) and it can be concluded that during the enhancement of heat source, the fluid will accelerate. Figure 5: Velocity profile V against y for Pr=3, M= 1, k = 0.2, L=1 ωt=π/2, S=1, θ=π/6 Figure 6: Velocity profile u against y for Gr=5, M= 1, k = 0.2, L=1 ωt=π/2, S=1, θ=π/6 Debasish Dey et al. Page 31

5 Tp T Innovation: International Journal of Applied Research; Figure 7: Velocity profile u against y for Pr=3, Gr=5, M= 1, k = 0.2, L=1, ωt=π/2, θ=π/6 cases M Pr Gr Shearing stress For Lower Plate(at y=0) Shearing stress For Upper Plate(at y=1) k=0 k=0.1 k=0.2 k=0 k=0.1 k=0.2 I II III IV Table 1: Shearing stresses at the two plates for θ = 0.6, L=1, ωt=π/2, S=1 Viscous drags or shearing stresses at the two plates are calculated for various cases given in table 1. The table enables the fact that during the growth of visco-elasticity of the governing fluid motion, the shearing stresses at the two plates experience an increasing trend. The growths in Hartmann number (Cases I and II), Prandtl number (I and III) have a retarding effect on shearing stresses at the lower plate but a reverse phenomenon is seen in case of shearing stress formed at the upper plate. Enhancement of Grashoff raises the magnitude of shearing stresses at both the plates (Cases I and IV) Pr=3 Pr=5 Pr =7 Pr = Figure 8: Temperature of fluid particlest against y for, Gr=5, M= 1, k = 0.2, L=1, ωt=π/2, θ=π/6, L o = y 0.6 Pr = 3 Pr = 5 Pr = 7 Pr = Figure 9: Temperature of dust particles Tp against y for, Gr=5, M= 1, k = 0.2, L=1, ωt=π/2, θ=π/6, L o = 0.5. Debasish Dey et al. Page 32 y

6 T Innovation: International Journal of Applied Research; Figures 8-10, represent the nature of the temperature field of fluid particles and dust particles against the width of the channel in combination with other flow parameters involved in the solution. The figures states that during the increasing values of displacement variable, the temperature of the fluid particles and dust particles enhance and also it is noticed that the maximum discrepancy is seen in the neighbourhood of the upper plate. Increasing values of Prandtl number decreases the temperature of both fluid and dust particles of the governing dusty visco-elastic fluid motion (figure 8 and 9). Effects of Heat source/ sink on the temperature are seen on figure 10. As the strength of the external heat agent increases, a declined trend in the temperature of the dusty visco-elastic fluid is experienced along with the increasing height of the channel S = - 3 S = -1 S = 1 S = Figure 10: Temperature T against y for, Pr=3, Gr=5, M= 1, k = 0.2, L=1, ωt=π/2, θ=π/6, L o = 0.5. y V. CONCLUSIONS The behaviour of an unsteady dusty visco-elastic fluid governed by Walters liquid (Model B ) has been investigated under the influence of incline magnetic field and heat source/ sink. Some of the important conclusions are stated as below: Acceleration in fluid motion is dominant in the neighbourhood of the upper plate. Retarded motions of fluid and dust particles are seen during the growth of Hartmann number, Prandtl number but an opposite phenomenon is noticed during the enhancement of Grashoff number. The growth of visco-elasticity strengthens the shearing stresses at the two plates. Maximum variation in the temperature is seen in the neighbourhood of the oscillating plate. ACKNOWLEDGMENT The authors acknowledge Professor Rita Choudhury, Department of Mathematics, Gauhati University, for her encouragement throughout this work. REFERENCES [1] Walters K. (1960), The motion of an elastic-viscous liquids contained between co-axial cylinders, Quart. J. Mech. Appl. Math., 13(4), [2] Walters K. (1962), Non-Newtonian effects in some elastic-viscous liquids whose behaviour at small rates of shear is characterized by general linear equations of state, Quart. J. Mech. Appl. Math., 15(1), [3] Saffman P. G. (1962), On the stability of laminar flow of a dusty gas, Fluid Mech., 13, [4] Michael, D. H. and Miller, D. A. (1966), Plane parallel flow of a dusty gas, Mathematika, 13, [5] Nayfeh A. H., (1966), Oscillating two-phase flow through a rigid pipe, AIAAJ, 4(10), [6] Gupta, P. K. And Gupta, S. C. (1976), Flow of a dusty gas through a channel with arbitrary time varying pressure gradient, Journal of Appl. Math. and Phys., 27, 119. [7] Singh K. K. (1976), Unsteady flow of a conducting dusty fluid through a rectangular channel with time dependent pressure gradient, Indian J. Pure and Appl. Math., 8, [8] Singh, C. B. and Ram, P. C. (1977), Unsteady flow of an electrically conducting dusty viscous liquid through a channel, Indian J. Pure and Appl. Math., 8 (9), [9] Prasad, V. R. and Ramacharyulu, N. C. P. (1979), Unsteady flow of a dusty incompressible fluid between two parallel plates under an impulsive pressure gradient, Def. Sci, Journal, 38, 125. [10] Gupta, R. K. And Gupta K., (1990), Unsteady flow of a dusty visco-elastic fluid through channel with volume fraction, Indian J. Pure and Appl. Math. 21(7), [11] Ajadi, S. O., (2005), A note on the unsteady flow of dusty viscous fluid between two parallel plates, J. Appl. Math. and Computing, 18(1-2), [12] A.GOVINDARAJAN, (2008), Study of physical properties of two phase dusty fluid flow through porous medium, Thesis, July Debasish Dey et al. Page 33

7 [13] Ramamurthy, V., (1990), Free convection effects on the Stokes problem for an infinite vertical plate in a dusty fluid, J. Math. Phys., 24, 297. [14] Attia, H. A., (2006), Unsteady MHD Couette flow and heat transfer of dusty fluid with variable physical properties, Applied Mathematics and Computation, 177, [15] Alle, G., Roy, A. S., Kalyane, S. and Sonth, R. M., (2011), Unsteady flow of a dusty visco-elastic fluid through an inclined channel, Advances in Pure Math., 1, [16] Attia, H. A., Al-Kaisy, A. M. A. and Ewis, K. M., (2011), MHD Couette flow and heat transfer of a dusty fluid with exponential decaying pressure gradient, Tamkang J. Sci. and Engg., 14(2), [17] Kalita, B. (2012), Unsteady flow of a dusty conducting viscous liquid between two parallel plates in presence of a transverse magnetic field, Appl. Math. Sciences, 6(76), [18] Sandeep, N. & Sugunamma, V., (2013), Effect of inclined magnetic field on unsteady free convection flow of a dusty viscous fluid between two infinite flat plates filled by a porous medium, International Journal of Applied Mathematics and Modelling, 1, 1, [19] Ahmed, N. & Bhattacharyya, D. J., (), Free Convection in MHD Couette Flow with Heat Source and Chemical Reaction, Applied Mathematical Sciences, 8(50), [20] Sekhar D. V., & Reddy, G. V., (2012), Chemical reaction effects on mhd unsteady free convective walter s memory flow with constant suction and heat sink, Advances in Applied Science Research, 2012, 3 (4): Debasish Dey et al. Page 34