In 2022the study about the unsteady magnetohydrodynamic (MHD) convective flow of the Rivlin–Ericksen fluid past an infinite vertical plate with Hall effects.The plate travels with the constant velocity in the direction of fluid flow, whereas the free stream velocity follows the exponentially enhancing small perturbed flow[10]. In2013 the unsteady laminar flow of dusty visco-elastic liquid through an open rectangular channel under the influence of time varying pressure gradient is investigated [11]. The expression of velocity has been obtained by using the Laplace Transform Techniques by taking the fluid and dust particles to be initially at rest, velocity distribution and flux of flow had been obtained. In 2013 the study of unsteady, MHD, heat absorbing and chemically reacting dusty viscoelastic (Walter’s liquid model-B) fluid Couette flow between vertical long wavy wall and a parallel flat wall saturated with porous medium subject to convective cooling and varying mass diffusion [12]. The perturbation method is employed to analyze the coupled equations involving nonlinear problem and solution for the velocity, temperature and concentration distributions are obtained analytically. In 2016 the found that, if the lines of motion are parallel to a fixed plane and if the velocity at corresponding points of all planes has the same magnitude and direction the motion is two dimensional and w=0 [13]. The present problem is concerned with heat and mass transfer in free convective flow of a visco elastic dusty gas through a porous medium induced by the motion of a semi-infinite flat plate moving with velocity decreasing exponentially with time heat source parameter [15]. It is clear that the existence of a stream function is a consequence of stream lines and equation of continuity for incompressible fluid. The expression for velocity distribution of dusty fluid, dust particle, temperature and concentration distribution are obtained. Source is a point at which liquid is continuously created and sink is a point at which liquid is continuously annihilated.

We consider the motion of a dusty visco-elastic fluid present in the space y>0. The flow is due to the motion of semi-infinite flat plate moving with velocity Ve-αt in x’-axis is taken along the plate, y’-axis perpendicular to it. Since the plate is infinite, all the physical quantities will be functions of y’ and t’ only. Then the flow of dusty visco-elastic fluid in presence of magnetic field can be shown to be governed by the following set of equations:

Where u^’(y^’, t^’), u^’_s(y^’, t^’) are respectively the velocities of fluid and dust particles, v being the kinematic coefficient, N₀ being the number density of the dust particle, ρ the density of the fluid, m^’ the mass of the dust particle, λ₁ stress relaxation time, λ₂ strain retardation time, σ₆ the electrical conductivity,  magnetic permeability, H_x is the x-component of magnetic field vector and H₀ applied magnetic field.

The boundary conditions are:

To make equations (1), (2) and (3) dimensionless, we use the following non-dimensional variables:

Where R is the Reynold’s number, M the Hartmann number, C the mass concentration of dust particles, and R_m the magnetic Reynold’s number λ₁, δ dimensionless stress relaxation time and dimensionless strain retardation time.

Thus equations (1), (2) and (3) will be:

The boundary conditions (4) are reduced to:

In order to solve the equations (5), (6) and (7), we assume

Using (9), the boundary conditions (8) are transformed to:

With the help of equation (9), the equations (5), (6) and (7) are transformed respectively to:

Now, eliminating G and h from (11), (12) and (13), we get

The exact solution of the hydromagnetic equations subject to boundary conditions (10) are as follows:

Real part of (17), (18) and (19) represent the expressions for the velocity of the fluid, dust particle and magnetic field of the dusty fluid, respectively. From these the velocities of the fluid and dust particles have been calculated for different values of time t, elastic parameter λ, δ, magnetic parameter M and mass concentration C.

The values of u and v coincide with the result of Mitra[7] in absence of magnetic field by taking, R=1, y=1 and λ=0, δ=0.

For clean fluid, mass concentration of the dust particles, C=0. The velocity of the clean fluid u_c is obtained from equation (17) by substituting C= f_y=0.

In order to discuss results, numerical computations have been made by taking typical values of various parameters. The result are shown in figs. 1 to 3, using R_m = 0.5, R = 1, y= 0.8, β = 0.5.

Fig, 1. Shows the velocity profiles of fluid and dust particle for several values of t. it is noticed that as t increases, the velocities of fluid and dust particle decreases and also the difference between them decreases.

Fig. 2. Indicates the velocity profiles of the fluid for different values of λ, δ, M and C for constant time t. It shows that the velocity decreases by increasing values of δ and velocity increases with the increases of M.

Fig. 3. Shows that for constant values of t, λ  and C, magnetic field decreases with the decreases of δ and M.