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Advances in Pure Mathematics, 2011, 1, 187-192 doi:10.4236/apm.2011.14033 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Unsteady Flow of a Dusty Visco-Elastic Fluid through a Incliend Channel Geetanjali Alle1, Aashis S. Roy2, Sangshetty Kalyane1, Ravi M. Sonth3 1Department of Physics, Singhania Un i ve rs i ty, Rajasthan, India 2Department of Materials Science, Gulbarga University, Gulbarga, India 3Department of Mat hem at i cs, K.C.T. Engineering College, Gulbarga, India E-mail: principalkct@rediffmail.com Received March 1, 2011; revised April 26, 2011; accepted May 10, 2011 Abstract The present discussion deals with the study of an unsteady flow of a dusty fluid through an inclined channel under the influence of pulsatile pressure gradient along with the effect of a uniform magnetic field. The ana- lytical solutions of the problem are obtained using variable separable and Fourier transform techniques. The graphs drawn for the velocity fields of both fluid and dust phase under the effect of Reynolds number. The velocity profiles for the liquid and the dust particles decreases at different values of time t increases. As the visco-elastic parameter increases the velocity of the liquid and the dust particles deceases. When relaxa- tion time parameter increases, the velocity of the liquid and dust particles decreases. Keywords: Dusty Fluid, Pulsatile Pressure Gradient, Velocities of Dust and Fluid Phase, Inclined Channel, Reynolds Number 1. Introduction In recent years many authors have studied the flow of immiscible viscous electrically conducting fluids and their different transport phenomena. These fluid also known as non-Newtonian fluids are molten plastics. Plups, emulsion etc., and large variety of industrial product having visco-elastic behavior in their motion. Such fluids are often embedded with spherical non- conducting dust particles in the form of impurities. This fluid also called dusty Rivlin-Ericksen second order fluid. The influence of dust particles on visco-elastic fluid flow has its importance in many applications such as extrusion of plastic in the manufacture of rayon and Nylon, purifi- cation of crude oil, pulp oil, pulp, paper industry, textile industry and in different geophysical cases etc. In these cases stratification effect is often observed which are under the action of geomagnetic field. Saffman et al., (1962) studied the stability of a laminar flow of dusty gas with uniform distribution of dust parti- cles. Michel (1965) considered the Kelvin-Helmholtz instability of the dusty gas. Michael and Miller (1965) discussed the motion of the dusty gas enclosed in the same infinite space above a rigid plane boundary. We have studied the unsteady dusty visco-elastic liquid in a channel bounded by two parallel plates. The change in velocity profiles for dust and liquid particles has been depicted graphically. 2. Theory Formulation and Solution of the Problem The X-axis is taken along the plate and the Y-axis nor- mal to it. The basic equations of hydro magnetic flow are 2 111 1 021 1' ' uuu Pu t kN uu (1) 222 12 ' ' uk uu uu tm 1 div 0u (2) (3) 2 div 0u (4) where the 1 , 2 u u denotes the velocity vector of fluid and dust particles respectively: the pressure: p the density of the fluid: t kinematic coefficient of viscosity: the time: m, the mass o f the dust particles: , the number density of dust particles: K, the stokes 0 N 188 G. ALLE ET AL. resistance coefficient which for spherical particles of radius a is a: , the coefficient of viscosity of fluid particles. In the present analysis, the following important as- sumptions are made: 1) The dust particles are spherical in shape are uni- formly distributed. 2) Chemical reaction, mass transfer and radiation be- tween the particles and fluid are not considered. 3) The temperature is uniform with in a particle. 4) Interaction between particles themselves is not con- sidered. 5) The flow is fully developed. 6) The buoyancy force is neglected. 7) The number density of the dust particles is constant throughout the motion. 8) The displacement current is zero, since the flow velocity is small relative to the speed of light. 9) The Hall effects are negligible. 10) The fluid is electrically neutral, i.e., no surplus electrical charge distribution is present in the fluid. 11) Only the electromagnetic body forces are present. 12) Fluid properties are invariable. 13) Viscous dissipation is neglected. Maxwell’s equations, tog ether with Ohm’s law and th e law of electromagnetic conservation, are written in the case of zero-displacement and hall currents as: B J (5) t B E 1E (6) J VB 0 (7) B (8) 0 E (9) The usual Prandtl boundary layer assumptions along with assumptions (5)-(9) leads to the following reduction of the previous equations: 021 KN uu Puu tx t y 2 11 2 1 (10) 12 2 u K uu 12 0uu t (11) which are to be solved subject to the boundary conditions 0,t 1 0, P t x (c on s ta nt)C 12 ,0,0u u (12) yh Changing it into non dimension a l form by putting 2 12 22 ,, ,,, '' uh u yx tph yxtu vp hhh We have 2 2 1 up ul vu tx t y (13) vuv t (14) where 2 m K h Relaxation time parameter, 0 mN l Mass Concentration 2 h 0: =0,0tuv Visco-elastic parameter The boundary conditions are 0:=0, at1tu y =0, at1uy (15) take pC x 0 (constant) for t. Then the equation (2.2.13) becomes 2 2 1 uul Cvu tt y (16) Appling the Laplace Transform, we have from (14) and (16) 2 2 1 Cul SuSv u Sy (17) Svuv (18) where 00 ed,ed st st uutvvt The boundary conditions (15) are transformed to 0, 01uvaty (19) Solving equations (17) and (18) subject to the bound- ary conditions (19) we have 22 2 d 1 d uC uSS y (20) where 21 11 SSls SS (21) Finally 2 cosh 1cosh 1 Cy uSS (22) Copyright © 2011 SciRes. APM G. ALLE ET AL. Copyright © 2011 SciRes. APM 189 211 vSS S cosh 1cosh Cy (23) Applying Laplace Inversion formula 1ed 2 iSt i uut i (24) Here is greatest then the real part of all the Singu- larities of u 2 1 2c 1 i i C u iSS cosh 1ed osh St y t (25) Taking Inversion Laplace Transform and with the help of calculus of residues the above equations (22) and (23) yields. 1 2 11 22 1 e 1 e St St S S 2 2 133 2 133 22 cosh 1cosh 21 1cos 1 82 21 21 1cos 1 82 21 r r r r uQy CQ Q ryS rA ryS rA (26) and 1 2 11 22 1 e 1 e St St S S 2 133 133 22 cosh 1cosh 21 1cos 1 82 21 21 1cos 1 82 21 r r r r vQy CQ Q ryS rA ryS rA (2.2.25) where 22 2 111 2 111 S Sl S 23312 01 lS QatS ASS 22 2 222 2 222 S Sl S 33 12 1 lS ASS 11 22 21 22 XX rr 12 22 , 21 21 21 XX SS where 22 22 121 21 122 rr Xl 22 22 2 121 21 41 22 rr XX (27) 3. Results and Discussion The unsteady flow of a dusty visco-elastic fluid through a channel is studied. From Figures 1 and 2 it shows that the velocity profiles for the liquid and the dust particles decreases at different values of time t increases. As the visco-elastic parameter increases the velocity of the liquid and the dust particles deceases as shown in Figure 3 and 4. When relaxation time parameter increases the velocity of the liquid and dust particles decreases as shown in Figure 5 and 6. F rom Figure 7 and 8 it can be observed that as mass concentration increases the veloc- ity of the liquid and the dust particles deceases. In case when gravity or inclination angle 0 0 and visco- elastic parameter and adding the magnetic field term then the present model becomes that of Singh and Ram. Figure 1. Show the variation of velocity pr ofile of liquid for different value of time at fixed (σ = 0.8, λ = 0.5, I = 0.5). G. ALLE ET AL. Copyright © 2011 SciRes. APM 190 Figure 2. Show the variation of velocity profile of dust for different values of time at fixed (σ = 0.8, λ = 0.5, I = 0.5). Figure 3. Show the variation of velocity pr ofile of liquid for different values of time at fixed (σ = 0.8, λ = 0.5, I = 0.5). Figure 4. Show the variation of velocity profile of dust for different values of time at fixed (σ = 0.8, λ = 0.5, I = 0.5). Figure 5. Show the variation of velocity profile of liquid or different values of relaxation and at fixed (λ = 0.8, t = 0.5, I = 0.5). G. ALLE ET AL. Copyright © 2011 SciRes. APM 191 Figure 6. Show the variation of velocity profile of dust or different values of relaxation and at fixed (λ = 0.8, t = 0.5, I = 0.5). Figure 7. Show the variation of velocity pr ofile of liquid for different values of I mass concentration and at fixed (σ = 0.8, t = 0.5, λ = 0.5). Figure 8. Show the variation of velocity profile of dust for different values of I mass concentration and at fixed (σ = 0.8, t = 0.5, λ = 0.5). 4. References [1] D. H. Michael, “Kelvin-Helmholtz Instability of A Dusty Gas,” Proceeding Cambridge Philosophical Sciety, Vol. 61, No. 2, 1965, pp. 569-571. doi:10.1017/S030500410000414X [2] D. H. Michael and D. A. Millar, “Plane Parallel Flow of a Dusty Gas,” Mathematika, Vol. 13, No. 1, 1966, pp. 97- 109. doi:10.1112/S0025579300004289 [3] P. G. Saffman, “On stability of a Laminar Flow of a Dusty Gas,” Journal of Fluid Mechanics, Vol. 13, No. 1, 1962, pp. 120-128. doi:10.1017/S0022112062000555 [4] C. B. Singh and P. C Ram, “Unsteady Flow of an Elec- trically Conducting Dusty Viscous Liquid Through a Channel,” Indian Journal of Pure and Applied Mathe- matics, Vol. 8, No. 9, 1977, pp. 1022-1028. [5] G. C. Sharma, “Unsteady Flow of an Electrically Con- ducting Dusty Viscous Liquid between Two Parallel Plates,” International Journal of Pure & Applied Mathe- matical, Vol. 18, No. 12, 1987, p. 1131. [6] K. K. Singh, “Unsteady Flow of Conducting Dusty Vis- cous Liquid in an Annulas,” Acta Ciecia Indica, Vol. 3, No. 3, 1977, p. 264. [7] O. P. Varshney, “Flow of a Dusty Rivlin-Ericksen Fluid through a Channel. Ph.D Thesis,” Agra University, Agra, 1983. G. ALLE ET AL. Copyright © 2011 SciRes. APM 192 [8] M. L. Sharma, “MHD Flow of a Conducting Dusty Vis- cous Liquid through a Long Elliptic Duct with Pressure Gradient as Function of a Time,” Ph.D Thesis Agra Uni- versity, Agra, 1980. |