Unsteady Flow of a Dusty Conducting Fluid through porous medium between Parallel Porous Plates with Temperature Dependent Viscosity and Heat Source

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1 Volume Issue3 3- June ISSN Unsteady Flow of a Dusty Conducting Fluid through orous medium between Parallel Porous Plates with Temerature Deendent Viscosity and Heat Source Shalini Prof. (Dr.) M. S. Saroa 3 Prof. (Dr.) Rajeev Jha Research Scholar in Maharishi Markandeshwar University Mullana (Ambala) shalini_bharti7@yahoo.co.in) Det. of Mathematics Maharishi Markandeshwar University Mullana (Ambala) 3 Deartment of Mathematics Alied College of Management and Engg. Palwal (Haryna) (dr.rajeevjha@yahoo.co.in jhadrrajeev@gmail.com) ABSTRACT The urose of resent roblem is to studies of effects of variable viscosity and heat source on unsteady laminar flow of dusty conducting fluid between arallel orous lates through orous medium with temerature deendent viscosity. The fluid is considered as unsteady laminar flow through orous medium and acted uon by a constant ressure gradient and an external uniform magnetic field is alied erendicular to the lates. It is assumed that the arallel lates are orous and subjected to a uniform suction from above and injection from below. It is also considered the viscosity is temerature deendent. The governing nonlinear artial differential equations are solved finite difference aroximation. The results for temerature field and velocity for fluids and dust articles have been obtain numerically and dislayed grahically. Key words: MHD orous medium heat source heat transfer transient state two-hase flow dust articles finite differences. I. INTRODUCTION The flow of a dusty and electrically conducting fluid through a channel in the resence of a transverse magnetic field through orous medium has imortant alication in such areas as magneto hydrodynamic generators ums accelerators cooling systems centrifugal searation of matter from fluid etroleum industry urification of crude oil electrostatic reciitation olymer technology and fluid drolets srays. On the other hand flow through orous medium have numerous engineering and geohysical alications for examles in chemical engineering for filtration and urification rocess; in agriculture engineering to study the underground water resources; in etroleum technology to study the movement of natural gas oil and water through the oil reservoirs in view of these alication the object of the resent aer is to study the effect of arallel lates through a orous medium with temerature deendent viscosity erformance and efficiency of these devices are influenced by the resent of susended solid articles in the form of ash or soot as a result of the corrosion and wear activities and/or the combustion rocesses in MHD generators and lasma MHD accelerators. The hydrodynamic flow of dusty fluids has been studied by Saffman in (96) on the stability of laminar flow of dusty gas. Soundalgekar (973) studied on free convection effects on the oscillatory flow ast on infinite vertical orous lates. Kim () discussed on unsteady MHD convective heat transfer ast a semi-infinite vertical orous late with variable suction. Attia () studied on influence of temerature deendent viscosity on MHD coquette flow of dusty fluid with heat transfer. Ramanathan and Suresh (4) discussed on effect of magnetic field deendent viscosity and anisotroy of orous medium free convection. Attia (5) have studied on unsteady flow of a dusty conducting fluid between arallel orous lates with temerature deendent viscosity. Seeton (6) studied on viscosity temerature correlation for liquid. EI-Amin et. al (7) have discussed on effect of thermal radiation on natural convection in a orous medium. Attia (8) studied on unsteady hydromagnetic Coquette flow of dusty fluid

2 Volume Issue3 3- June ISSN with temerature deendent viscosity and thermal conductivity. Ezzat et. al () discussed on Sace aroach to the hydro-magnetic flow of dusty fluid through a orous medium. Chauhan and Agarwal () studied on MHD through a orous medium adjacent to a stretching sheet numerical and an aroximation solution. Recently Attia(5) have studied on Unsteady flow of a dusty conducting fluid between arallel orous lates with temerature deendent viscosity. In the resent work the effects of variable viscosity and heat source on the unsteady laminar flow of an electrically conducting viscous incomressible dusty fluid variable viscosity and heat transfer between arallel non-conducting orous lates is studied. The fluid is flowing through orous medium between two electrically insulating infinite lates maintained at two constant but different temeratures. An external uniform magnetic field is alied erendicular to the lates. The magnetic Reynolds number is assumed small so that the induced magnetic field is neglected. The fluid is acted uon by a constant ressure gradient and its viscosity is assumed to vary exonentially with temerature. The flow and temerature distributions of both the fluid and dust articles are governed by the couled set of the momentum and energy equations. The joule and viscous dissiation terms in the energy equation are taken into consideration. The governing couled nonlinear artial differential equations are solved numerically using the finite difference aroximations. The effects of heat source and the temerature deendent viscosity on the time develoment of both the velocity and temerature distributions are discussed. II. DESCRIPTION OF THE PROBLEM The dusty fluid is assumed to be flowing through orous medium between two infinite horizontal lates located at the y = ±h lanes as show in Figure. The dusty articles are assumed to be uniformly distributed throughout the orous fluid. The two lates are assumed to be electrically non-conducting and ket at two constant temeratures: T for the lower late and T for the uer late with T > T. A constant ressure gradient is alied in the x- direction and the arallel lates are assumed to be orous and subjected to a uniform suction from above and injection from below. Thus they comonent of the velocity is constant and denoted by υ. A uniform magnetic field B is alied in the ositive y- direction. By assuming a very small magnetic Reynolds number the induced magnetic field is neglected [5]. The fluid motion start from rest at t = and the no-sli condition at the lates imlies that the fluid and dust articles velocities have neither a z-axis nor an x- comonent at y = ±h. The initial temeratures of the fluid and dust articles are assumed to be equal to T and the fluid viscosity is assumed to vary exonentially with temerature. Since the lates are infinite in the x- and z-direction the hysical variables are invariant in these directions. The flow of the fluid through orous medium is governed by the Navier- Stokes equations [5]. u u dp u t y dx y y K B u u KN u u () Where ρ is the density of clean fluid µ is the viscosity of clean fluid u is the velocity of fluid u is the velocity of dust articles σ is the electric conductivity is the ressure acting on the fluid N is the number of dust ' K is orosity arameter and articles er unit volume K is a constant. The first five terms in the right hand side are resectively the ressure gradient viscous force Lorentz force terms and Porous medium. The last term reresents the force term due to the relative motion between fluid and dust articles. It is assumed that the Reynolds the force term due to the relative velocity is small. In such a case the force between dust and fluid is roortional to the relative velocity []. The motion of the dust articles is governed by Newton s second law [] via

3 Volume Issue3 3- June ISSN u m KN u u t () Where m is the average mass of dust articles. The initial and boundary conditions on the velocity fields are resectively given by t ; u u...(3) For t > the no-sli condition at the lates imlies that y h: u u y h: u u...(4)...(5) Heat transfer takes lace from the uer hot late towards the lower cold late by conduction through orous medium the fluid. Also there is a heat generation due to both the joule and viscous dissiations. The dust articles gain heat energy from the fluid by conduction through their sherical surface. Two energy equations are required which describe the temerature distribution for both the fluid and dust articles and are resectively given by [6] T T T u c c k t y y y T t T Cs Bu Q T T T T...(6) T T T (7) Where T is the temerature if the fluid T is the temerature of the articles c is the secific heat caacity of the of fluid at constant ressure C s is the secific heat caacity of the articles k is the thermal conductivity of the fluid The term Q ( T T ) is assumed to be amount of heat generated or absorbed er unit volume Q is a constant T is the temerature 3Pr / Cs c relaxation time is the velocity s / 3 relaxation time D s is the material 3 / D KN density of the dust articles righthand side of Eq.(6) reresent the viscous dissiation the joule dissiation heat generation/absortion and the heat conduction between the fluid and dust articles resectively. The initial and boundary conditions on the temerature fields are given as t : T T (8) t y h: T T T t y h: T T T () The viscosity of the fluid is assumed to deend on f T temerature and is defined as. By assuming the viscosity to vary exonentially with temerature the function f (T) Takes the form [3 4] bt T f T c where the arameter b has the dimension of T and such that at T = T f(t ) = and then. This means that is the velocity coefficient at T = T. The arameter b may take ositive values for liquids such as water benzene or crude oil. In some gases like air helium or methane a may be negative i.e. the coefficient viscosity increases with temerature [3 4]. The roblem is simlified by writing the equations in dimensionless form. The characteristic length is taken to be h and the characteristic time is /h (9) h / while the characteristic velocity is. Thus we define the following non-dimensional quantities: t t h h Ph d dx x y z x y z h ' K K

4 Volume Issue3 3- June ISSN h u w u w h u w u w T T T T T T T T T T b T T T at f T e e Where a is the viscosity arameter H Bh a Where H a is the Hartmann number and R KNh is the article concentration arameter G m hk is the article mass arameter h Qh S c Pr is the suction arameter is the heat source arameter c k is the Prandtl number Ec hc T T L is the Eckert number h T is the temerature relaxation time arameters. In terms of the above dimensionless variables and arameters equation equations ()-(5) (6)-(7) with boundary condition (8)-() take the following form (where we have droed the hats for convenience): u u u f T u a f T H a u Ru u t y y y y K () u G u u t t ; u u t y u u t y u u T T T u Ecf T t y Pr y y T R EcH au ST T T 3Pr t L T T () (3) (4) (5) (7)...(6) t ; T T (8) t y ; T T (9) t y ; T T. () Equations () () (6) and (7) reresent a system of couled and nonlinear artial differential equations which must be solved numerically under the initial and boundary conditions (3)-(5) and (8)-() using finite difference aroximations [7]. The nonlinear terms are first linearized and then an iterative scheme is used at every time-ste to solve the linearized system of difference equations. III. RESULTS AND DISCUSSIONS The exonential deendence of viscosity on temerature results in decomosing the viscous force term

5 Volume Issue3 3- June ISSN u y y in Eq. () into two terms. the variations of those resulting terms with the viscosity arameter a heat generation/absortion arameter (S) Magnetic field (H a ) ermeability of orous medium (K ) Concentration arameter (R) Suction arameter and Prandtl number (Pr) have an imortant effect on the flow and temerature fields. In the following discussion selected arameters are given the following fixed values G =.8 α = 5 Ec =.3 and L =.7. Numerical calculations have been carried out for dimensionless velocity of dusty fluid (u) velocity of dust article (u ) temerature rofiles T and temerature of dust article T for different values of arameters which are dislayed in Figures-() to (). Figures () and () deict that with the increase in Hartmann (H a ) the velocity of fluid and dust articles decreases. This agrees with the natural henomena because in the resence of magnetic field Lorentz force sets in which imedes the velocity of fluid and dust articles. Figures (3) and (4) deict that with increase in orosity arameter (K ) then increasing the value of the velocity of fluid and dust articles. Figures (5) and (6) indicate the variations of the velocity of fluid (u) and velocity of dust articles (u ) at the centre of the channel y = with time for different values of the viscosity arameter a. The figures show that increasing viscosity arameter a increases the velocities and the time required to aroach the steady state. This imlies that higher velocities are obtained at lower viscosities. The effect of the viscosity arameter a on the steady state time is more ronounced for ositive values of a than for negative values. Notice that the velocity of fluid (u) reaches the steady state more quickly than velocity of dust articles (u ). This is because the fluid velocity is the source for the dust articles velocity. Figures (7) and (8) deict that with increase in suction arameter of article increasing the value of velocity of fluid u and dust articles (u ) because; increase in suction arameter of article mass forces. reduces Figure (9) () () and () show the variations of the temeratures rofile of fluid (T) and fluid articles T at the centre of the channel y = with time for different values of Suction arameter (ξ) and Prandtl number (Pr). The figures show that increasing Suction arameter (ξ) and Prandtl number (Pr) decreases the temeratures and the steady state times. Figures (3) (4) (5) and (6) indicate the variations of the both velocities u and u at the centre of the channel y = with time and both temeratures T and T for different values of the concentration arameter (R). It is clear that the suction velocity decreases both u and u and both temeratures T and T are decreases with increasing the value of concentration arameter (R). Form figures (7) and (8) that the velocity of fluid (u) and the temerature rofile of fluid (T) is increasing with increasing the value of heat source (S). The boundary layer and thermal boundary layer thicknesses increase with increase the heat source (S). IV. CONCLUSIONS In this aer the effect of a temerature deendent viscosity thermal radiation heat generation/absortion arameter the article concentration arameter Suction arameter suction and injection velocity and an external uniform magnetic field on the unsteady laminar flow and temerature distributions of an electrically conducting viscous incomressible dusty fluid between two arallel orous late through orous medium has been studied. The viscosity was assumed to vary exonentially with temerature and the joule and viscous dissiations were taken into consideration. Also changing the viscosity arameter a leads to asymmetric velocity rofiles about the central lane of the channel (y = ) which is similar to the effect of variable ercolation erendicular to the lates. The effect of the suction velocity on both the velocity and temerature of the fluid and articles is more ronounced for higher values of the arameter a K and orosity arameter. REFERENCES. Saffman P. G.: The stability of laminar flow of dusty gas Journal of Fluid Mechanics 3 (96).-9.. Soundalgekar V. M.: Free convection effects on the oscillatory flow ast on infinite vertical

6 Volume Issue3 3- June ISSN orous lates with constant suction. Proc. Royal Soc. Londan A Guta R. K. and Guta S. C.: Journal of Alied Mathematics and Physics 7 (976) Prasad V. R. and Ramacharyulu N. C. P.: Def. Sci. Journal 3 (979) Dixit L.A.: Indian Journal of Theoretical Physics 8 () (98) Klem K. Herwig H. and Sclamann M.: Proceedings of the Third International congress of Fluid Mechanics Cairo Egyt 3 (99) Attia H. A. and Kotb N. A.: Acta Mechanica 7 (996) Attia H. A.: Mechanica Research Communications 6() (999) Kim Y. J.: Unsteady MHD convective heat transfer ast a semi-infinite vertical orous late with variable suction International journal of engineering science.-38 () Attia H.A.: Influence of temerature deendent viscosity on MHD coquette flow of dusty fluid with heat transfer All Qasseem university buraidah 8999 Kingdom of Saudi Arabia ().. Ramanathan A. and Suresh G.: Effect of magnetic field deendent viscosity and anisotroy of orous medium free convection IJES vol. 4 (4) Seeton J. C.: Viscosity temerature correlation for liquid. Tribology letters vol.- Nov. (6). 3. EI-Amin M. F. Abbas I. and Gorla R. S. R.: Effect of thermal radiation on natural convection in a orous medium. Int. journal of fluid mechanics research 34 (7) Attia H. A.: Unsteady hydromagnetic coquette flow of dusty fluid with temerature deendent viscosity and thermal conductivity. I. J. of nonlinear mechanics vol.8 (8) Ezzat M.A. and EI-Bary A.A.: Sace aroach to the hydro-magnetic flow of dusty fluid through a orous medium. Journal of comuter and mathematics with alication. Vol.-59 Arail-(). P Chauhan D.S. and Agarwal R.: MHD through a orous medium adjacent to a stretching sheet numerical and an aroximation solution. The Euroean hysical journal. Vol. 6 (). 7. Attia H.A.: Unsteady flow of a dusty conducting fluid between arallel orous lates with temerature deendent viscosity. Turk. Journal of Phys. Vol. 9(5) u yu Ha= Fig. - : Velocity rofile of fluid for different values of H yu Ha = Fig. - : Velocity rofile of dust article for different values of H K = y 5 Fig. - 3: Velocity rofile of fluid for different values of K.

7 Volume Issue3 3- June ISSN u u K = 5 5 ξ = 4 yu Fig. - 4: Velocity rofile of dust article for different values of K yu a = Fig. - 5: Velocity rofile of fluid for different values of a yu a = Fig. - 6: Velocity rofile of dust article for different values of a y 5 Fig. - 7: Velocity rofile of fluid for different values of ξ y 5 Fig. - 8: Velocity rofile of dust article for different values of ξ yt ξ = ξ = Fig. - 9: Temerature rofile of fluid for different values of ξ.

8 T u T International Journal of Scientific Research Engineering & Technology (IJSRET) Volume Issue3 3- June ISSN ξ = - -5 y 5 Fig. - : Temerature rofile of dust article for different values of ξ yt Pr = Fig. -: Temerature rofile of fluid for different values of Pr Pr = y 5 Fig. - : Temerature rofile of dust article for different values of Pr yu R = Fig. - 3: Velocity rofile of fluid for different values of R y 5 Fig. - 4: Velocity rofile of dust article for different values of R R = 4 6 R = Fig. -5: Temerature rofile of fluid for different values of R.

9 T u T International Journal of Scientific Research Engineering & Technology (IJSRET) Volume Issue3 3- June ISSN R = y 5 Fig. - 6: Temerature rofile of dust article for different values of R S = y Fig. - 7: Velocity rofile of fluid for different values of S S = y 5 Fig. - 8: Temerature rofile of fluid for different values of S.