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The unsteady incompressible viscous flow of a Generalised Maxwell fluid between two coaxial rotating infinite parallel circular disks is studied by using the method of integral transforms. The motion of the fluid is created by the rotation of the upper and lower circular disks with different angular velocities. A fractional calculus approach is utilized to determine the velocity profile in series form in terms of Mittag-Leffler function. The influence of the fractional as well as the material parameters on the velocity field is illustrated graphically.

The study of fluid flow between two parallel disks is of practical importance in many fields such as machine storage devices, computer devices, crystal growth processes, turbine engines, radial diffusers, lubrication, viscometry etc. The rotating disc problem was ﬁrst formulated by von Kármán [

In the present paper we have considered unsteady incompressible visco-elastic flow of a generalized Maxwell fluid between two rotating infinite coaxial circular disks. In the aforesaid problems, time derivative of integer order has been considered in the Navier-Stokes equation but in the present problem we have considered the constitutive equation for Maxwell fluid with fractional order time derivative instead of integer order time derivative. In the constitutive equation the time derivative of integer order is replaced by the Caputo fractional calculus operator. We have obtained the analytical solution to the velocity field in series involving Mittag-Leffler function and illustrated graphically the dependence of the velocity field on the fractional and material parameters.

The constitutive equation of a Generalised incompressible Maxwell fluid can be written as,

where, is the shear stress, is a relaxation parameter, G is the shear modulus, α and β are fractional parameters such that and is the shear strain. and are Caputo operators given by

For the Equation (1) gives Ordinary Maxwell fluid model and for, , a Classical Newtonian fluid model is recovered.

The Equation (1) can be rewritten as

where is the shear rate.

The equation of motion in the absence of the body force can be written as

where is the density of the fluid, is the fluid velocity, is the material derivative, is the stress tensor.

The equation of continuity is given by

Let an incompressible viscous Generalised Maxwell fluid be bounded by two coaxial infinite parallel circular disks at a distance “d” apart and the fluid as well as the disks are initially at rest as shown in

The momentum equation is

Eliminating between the Equations (6) and (7) we get the basic equation as

The Equation (8) is the governing equation of the flow of a Generalised Maxwell fluid between two rotating infinite parallel circular disks considered in the present problem.

The boundary conditions are given by

at

at

“s” is some constant.

The initial condition is given by

Now let us introduce the dimensionless variables

Then the governing Equation (8) in non-dimensional variables is given by (for simplicity the dimensionless mark “'” will be neglected hereinafter).

where,

The boundary conditions in non-dimensional variable becomes

and

Let us consider the transformation given by

Then in terms of new variable the governing equation becomes

Subject to the boundary conditions

and

and initial condition

Taking Laplace transformation and using initial condition we get from Equation (11)

where, is the Laplace transformation of defined by

where “p” is Laplace transform parameter.

Taking finite Fourier sine transformation we get from the Equation (12)

where, is the finite Fourier sine transformation of defined by

where

Taking Laplace transformation of the boundary conditions we get,

and

Using the above conditions we get from Equation (13)

The Equation (14) can be written as

In order to avoid the lengthy procedure of residues and contour integrals, we rewrite the Equation (15) into series form given by

(16)

Now we have an important Laplace transformation of the nth order derivative of Mittag-Leffler function given by

Taking inverse Laplace transformation we get from Equation (16)

Taking inverse finite Fourier sine transformation we get from Equation (19)

Changing the variable to by the transformation we get the expression for the velocity field as follows,

Case-I If then the equation of motion is

Subject to the boundary conditions

and

and initial condition

The Equation (21) is the damped wave equation and it represents the governing equation of an Ordinary Maxwell fluid.

Then we get the velocity profile from the Equation (20) as

(22)

Case-II If, the equation of motion is given by

subject to the boundary condition

and

and initial condition

The Equation (23) is diffusion equation and it represents the governing equation of a Classical Newtonian Fluid.

Then we get the velocity profile from the Equation (20) as

In the present paper we have found out the analytical solution to the velocity field by integral transform in series form in terms of Mittage-Leffler function for the unsteady incompressible flow of a Generalised Maxwell fluid between two rotating infinite parallel coaxial circular disks. We have got the solutions to the velocity fields for ordinary Maxwell fluid and Classical Newtonian fluid as the limiting cases of the solution of Generalised Maxwell fluid. In the constitutive equation for the Maxwell fluid the time derivative of integer order is replaced by Riemann-Liouville operator. The dependence of the velocity field on the fractional as well as material parameters has been illustrated graphically.

In

In

ferent values of material parameter “η” in

the lower disk) at which the velocity gradient is zero. The velocity curves for Classical Newtonian and Generalised Maxwell Fluids are almost parallel to the horizontal axis compared to the velocity curve for the Ordinary Maxwell Fluid. It can be seen that the velocity curve for the Generalised Maxwell Fluid has a point of local maximum.