Influence of Inclined MHD on Unsteady Flow of Generalized Maxwell Fluid with Fractional Derivative between Two Inclined Coaxial Cylinders through a Porous Medium

Abstract "This paper presents a study of inclined magnetic field on the unsteady rotating flow of a generalized Maxwell fluid with fractional derivative between two inclined infinite circular cylinders through a porous medium. The analytic solutions for velocity field and shear stress are derived by using the Laplace transform and finite Hankel transform in terms of the generalized G functions. The effect of the physical parameters of the problem on the velocity field is discussed and illustrated graphically.


Introduction
The modeling of the equations governing the non-Newtonian fluids gives rise to a nonlinear differential equation .Such nonlinear fluids are considered to play a more important and appropriate role in technological applications in comparison with Newtonian fluids. Recently, the most important applications in differential equations, integral equations, physics, fluid mechanics, viscoelasticity, mathematical biology, and electrochemistry can be described by using the subject of fractional

ISSN: 0067-2904
Kadhim and Abdulhadi Iraqi Journal of Science, 2020, Vol. 61, No. 7, pp: 1705-1714 6071 calculus. There is no doubt that the fractional calculus is becoming an interesting spot and an exciting new way to solve the diverse problems in mathematics, science and engineering. In fluid mechanics, many models have been proposed to describe the response of the fluids, which are classified as fluids with a mathematical model of differential type with fractional derivatives. The applications of non-Newtonian fluids in engineering problems such as magnetohydrodynamic (MHD ) have become the focus of an extended number of studies; for example plastics, polymer fluids, exotic lubricant, food stuffs and polymers are handled extensively by chemical industries, whereas biological and rheological properties of many materials are described by their constitutive equation.
In recent years, the fractional derivatives are found to be quite flexible in describing the behaviors of the viscoelastic fluid and are studied by many mathematicians considering various motions of such fluids . In these studies, the constrictive equation for generalized non-Newtonian fluids are modified from the well-known fluid models by replacing the time derivative of an integer order by precisely non-integer order integrals or derivatives. The fractional derivative models of the viscoelastic fluids are obtained by researchers. Especially, the problems of the motion of a fluid in rotating or translating cylinder are of interest to both theoretical and practical domains . The first exact solution for non-Newtonian fluids which flows in cylindrical problems are those of Srivastava [9] for Maxwell fluids and Waters and King [8] for Oldroyd -B fluids in a straight circular tube . Fetecau [2] studied some helical flows of Maxwell and Oldroyd -B fluids within an infinite cylinder .The exact solutions of generalized Maxwell fluid flow due to oscillatory and constantly accelerating plate were obtained by Zheng [4]. Also, Hayat and Asgher [6] studied the exact solution for MHD flow of a generalized Oldroyd -B fluid with modified Darcyʼs law. Nazar and Corina Fetecau [7] considered a note on the unsteady flow of a generalized second-grade fluid through a circular cylinder subject to a timedependent shear stress. Liancun and Zhang [8] studied the unsteady rotating flows of a viscoelastic generalized Maxwell fluid with oscillating pressure gradient between coaxial cylinders. Sundos and Ahmed [10] studied the effects of MHD on the unsteady rotating flow of a generalized Maxwell fluid with oscillating gradient between coaxial cylinders.
In this paper, we studied the effects of inclined magnetic field on the unsteady rotating flow of a generalized Maxwell fluid with fractional derivative between two inclined infinite straight circular cylinders. The velocity field and the shear stress are obtained by means of discrete Laplace transform and finite Hankel transform. The exact solutions for the velocity field and the shear stress were obtained by integral and series form in terms of the generalized G functions. Graphs were plotted to show the effects of the fractional parameter on the fluid dynamic characteristics with MHD on the velocity field.

Mathematical Formulation
The constitutive equations of an unsteady inclined magnetic hydrodynamic incompressible flow of generalized Maxwell fluid with fractional derivative between two inclined coaxial cylinders through a porous medium are given by [2,5] where T is the Cauchy stress tensor, -pI is the indeterminate spherical stress, S is the extra-stress tensor , is the fluid density , is the external body force, A = L + L T is the first Rivlin-Ericksen tensor with L=grad V, B is the magnetic field, and is the current density (or conduction current). is the dynamic viscosity of the fluid, is the material constant, and is defined by where  is gradient operator, α is the fractional calculus parameter such that and is the fractional differential operator based on Riemann-Liouville's, defined as ∫ where is the Gamma function and We assume that the velocity and shear stress in cylindrical coordinates are given by where is the unit vector in the  -axis and is the velocity. Since is dependent on and t, we also assume that S depends only on and  . If the fluid is being at rest at t=0, then According to our problem, is the shear stress .
We consider a generalized Maxwell fluid between two inclined infinite circular cylinders through a porous medium .The pressure gradient in the θ axial direction and the balance of the linear momentum leads to the relevant and meaningful equation where is the constant density of the fluid , is the permeability and is the porosity of the porous medium . Now, by eliminating ) , ( t r  between Eqs, (7) and (8) [ ] where f is constant.

3.Calculation of the velocity field
In this section ,the velocity field will be calculated for different cases of pressure gradient.

Case one
To find the solution for Eq(10) subject to boundary condition and , we frist take the Laplace transformation to Eqs. (10)-(12) and, using the Laplace transform of the sequential fractional derivatives [6], we find that  (14) and (15) as well as the identity By taking inverse Hankel transformation of Eq.(19), we get  [3] .
Finally, we get the expressions for the velocity field as follows:

Case two
By a similar method and using , we can obtain the solution in the following form

Calculation of the shear stress
In this section, the shear stress is calculated for different types of pressure gradient.

Case one
The shear stress can be calculated from Eq (7) and , by taking Laplace transform of Eq (7), we get By applying discrete inverse Laplace transform for Eq(28), we obtain the shear stress in the following form

Case two
By proceeding in the same approach as before and using , we can find the solution in the following form

Numerical Results and Conclusions
In this paper, we have established the influence of inclined magnetic field on the unsteady rotating flow of a generalized Maxwell through two inclined infinite straight circular cylinders. The analytical solutions for velocity field and shear stress are obtained using Hankel and Laplace transforms of sequential fractional. The solution was determinedand written under integral and series from in terms of generalized G-function. Diagrams are plotted to show the behavior of different parameters involved in the expressions of the velocity field .
The velocity and shear stresses are plotted about the case using the Mathematica package. Figure-1 is sketched to show the velocity field of Maxwell model with fractional derivative at different values of time. From this figure, it is obvious that velocity is decreasing as the time increases. Figure-2 displays the impact of the parameter on the velocity field. It can be seen that the velocity becomes similar with increasing the value of . Fig (3) is depicted to show the changes in the velocity with the parameter and magnetic field. The velocity decreases with increasing the parameter . Figure-4 displays the behavior of the parameter with the magnetic field. It is observed that the velocity is increasing in the interval but it maintains the same value when It can be also seen that the velocity is decreasing in the interval . Figure-5 provides the graphical illustrations for the effects of the non-integer fractional parameter  on the velocity fields. The velocity is increasing with the increase of the parameter . Figure-6 displays the behavior of parameter with the magnetic field. It is observed that the velocity keeps similar values with the increase of the parameter . Figure-7 is prepared to show the effect of the kinematic viscosity on the velocity field with the magnetic field. The velocity decreases with the increase of . Figure-8) is prepared to show the effect of the parameter  on the velocity field with the magnetic field. The velocity decreases with the increase of the parameter . Figure-9 is established to show the behavior of different values of time with =8, 1 and magnetic field. The velocity field decreases with the increase of time. Figure-10 provides the graphical illustration for the effects of different values of r with the magnetic field. The velocity field is increasing with the increase of r. Figures-(11, 12) provide the graphical illustration for the effects of different values of and respectively with the magnetic field. The velocity field decreases with the increase in the values of and respectively. Figure-13) provides the graphical illustration for the relationship of velocity field with parameter . The velocity field is increasing with the increase of . Figure-14 is established to show the effects of different value of g with the magnetic field on the velocity field. The velocity field is decreasing with the increase of g.