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In present paper, an investigation has been made on the fluctuating flow of a non-Newtonian second grade fluid through a porous medium over a semi-infinite porous plate in presence of a transverse magnetic field B0. The governing equations have been solved analytically and the expressions for the velocity and stress fields are obtained. The free stream velocity
*U*(
*t*) fluctuates in time about a non-zero constant mean. The effects of the permeability parameter K and magnetic field parameter
*M* on velocity field have been analyzed quantitatively with the help of figures. It is noticed that the velocity field asymptotically approaches free stream velocity as it goes far away from the plate.

The fluctuating flows of viscous incompressible fluid through porous medium over an infinite porous plate have been investigated rigorously by many researchers because of its wide application in different fields. The phenomenon of flows through porous medium has been a subject of interest of many researchers because of its wide range of application in different fields such as petroleum engineering, chemical engineering etc. In petroleum engineering, it is dealt with the movement of natural gas and oil through reservoirs. Further, the study on underground water resources, seepage of water in river bed is also related to the flow through porous medium.

Soundalgekar and Puri [

In the present study, we have investigated the fluctuating flow of non-Newtonian fluid in a semi-infinite region through a porous medium bounded by a porous plate in presence of transverse magnetic field. The non- Newtonian fluid considered here is of second grade type and fluctuating flow related to such type of fluid is different from all those works that have been done earlier. The analytical solutions for the velocity and stress fields have been obtained by Lighthill’s method. The effects of permeability parameter K, Hartmann number M, frequency of fluctuation

Let us consider the flow of a second grade fluid of density ρ and viscosity µ through a porous medium of permeability K occupying a semi infinite region of the space bounded by a porous plate in presence of a transverse magnetic field

to a pressure gradient

perpendicular to the porous plate. It is assumed that

The equation of motion for the incompressible viscous generalized second grade fluid can be written as

The equation of continuity

For free stream

Let the fluctuating stream and suction velocities of the form

where

The boundary conditions can be written as:

No slip boundary condition

And

Eliminating

Let us introduce the following non-dimensional quantities

where

The Equation (8) in terms of non-dimensional variables reduces to (Dropping

where

In non-dimensional variables Equation (5) and Equation (6) becomes

The boundary conditions (7) in non-dimensional variables can be written as

Let the velocity in the neighborhood of the plate be

Substituting for u and U from Equation (13) and Equation (11) respectively in Equation (10) and separating harmonic and non-harmonic terms and neglecting the squares of ε we obtain

subject to the boundary conditions

Using the boundary conditions the solution of Equation (14) and Equation (15) are given by respectively

where

Further

Therefore the velocity field u is given by

where

The non-dimensional form of the skin friction at the plate is given by

where

In the present study, the fluctuating flow of an unsteady incompressible fluid of second grade type through a porous medium occupying a semi-infinite region of the space bounded by a porous plate has been discussed. The analytical solution for the velocity field has been obtained by the method of Lighthill. The skin friction is also obtained at the plate

Graphical representations have been illustrated to see the effects of Hartmann number M, permeability parameter K and normal stress moduli α on the fluctuating parts

which the skin friction fluctuates at the plate is plotted against

DhimanBose,UmaBasu, (2015) MHD Fluctuating Flow of Non-Newtonian Fluid through a Porous Medium Bounded by an Infinite Porous Plate. Applied Mathematics,06,1988-1995. doi: 10.4236/am.2015.612176