MHD Effect on Viscoelastic Fluid through a Long Vertical Tube

Soffman (1962) has proposed simple analytical model for the motion of a dusty fluid in term of a large density number of very small particles (uniform in size and shape) distributed in a fluid assuming that the bulk concentration and also the Sedimentation are negligible. Later a large number of dusty flow problems have investigated in the literature and are well documented in a review by Marble (1963), Michael and Miller (1966), Michael and Norey (1968) have considered the unsteady flow problem of a dusty gas in different channels. Dutta (1985) discussed the Temperature field in the flow over stretching surface with uniform heat flux. Khani et al. (2009) gave the Analytic solution for heat transfer of a third grade viscoelastic fluid in non-Darcy porous media with thermo physical effects. In the present paper we consider the unsteady MHD flow of dusty elastic-viscous liquid through a long uniform tube whose cross section is curvilinear quadrilateral bounded by the arcs and radii of two concentric circles r=l, r=b and θ=0, θ=α under the influence of time varying pressure gradient. Initially the liquid particles are at rest. Some particular cases for different pressure gradient have also been discussed in detail.

Pik*- stress tensor u - Velocity of liquid, λ0 - Elastic coefficient, μ - Viscosity of liquid, P - Pressure k - The stokes resistance coefficient, N0 – number of density of the particles, v - Kinematic coefficient of viscosity, ρ - The density of the fluid, B0 – Magnetic induction, r, θ, z – cylindrical coordinates z-axis,

According to Kuvshiniski (1951), rheological equations satisfied by viscoelastic liquid are We consider cylindrical polar coordinates (r, θ, z) with the z-axis along the axis of tube. Let and of the components of liquid velocity and The equations (1) to (4) combined with boundary condition we get the following equation of motion of dusty viscoelastic liquid Introducing the following non dimensional quantities are Using the boundary the equation (5) and (6) becomes The initial boundary conditions are

Putting in equation (7) and (8) we get The boundary conditions are After taking finite Fourier’s the transformation equations (9) and (10) reduce in the form Where Taking finite Hankel transformation of (11) and (12) and applying boundary conditions We get Where

is the ratios of the equation Taking Laplace transform of the equations (13) and (14) we get And Where and are the Laplace transform of and respectively and Now we obtain u and v from the equation (16) and (17), first invert the Laplace transform by inversion theorem, then applying version formulation for Hankel and sine transformation we get Where the root of the cubic education

Case I: flow under constant pressure gradient, let us consider f (t) =c Where c is the positive constant now substituting f (t) =c in equation (18) and (19) get we

Case II: flow under exponentially pressure gradient, let consider Substitute in (18) and (19), we get

If the mass of the dust particles are small then there influence and the fluid flow is reduced as m → 0 then fluid becomes ordinary viscous. If we put α=0, B0=0 then all the results are in agreement with these of Gupta (1979).

Department of Mathematics, Government P.G. College, Lalitpur (UP)