An Accurate MHD Flux Solutions of a Viscose Fluid and Generalized Burgers' Model fluxwithin an Annular Pipe Under Sinusoidal Pressure

The aim of this work presents the analytical studies of both the magnetohydrodynamic (MHD) flux and flow of the non-magnetohydro dynamic (MHD) for a fluid of generalized Burgers’ (GB) withinan annular pipe submitted under Sinusoidal Pressure (SP)gradient. Closed beginning velocity's' solutions are taken by performing the finite Hankel transform (FHT) and Laplace transform (LT) of the successivefraction derivatives. Lastly, the figures were planned to exhibition the transformations effects of different fractional parameters (DFP) on the profile of velocity of both flows.


Prevalent Equations
The constituent equations for an incompressible fractional(GB)fluid are agreed through in whose When (.)  is the Gamma function.The type diminished to the model of (GO-B) when 0 2   and if, addendum for that α = β = 1 the normal Oldroyd-B typeshall be earned.So, we suppose that shear stress and the field of velocity of the format When z e meant vector unit along the z-direction .Equation (5) substituted into (1) and takeover an account for the first condition S (r,0) = 0 (6) Obtain Thereafter the being gradient of pressure at z-direction, the motion equation provided next scalar equation: When  showed constant fluid density.The judge rz S amidst Eqs.(7), and (8), we earn the next fractional differential equation

First Problem of the Non-magnetohydrodynamic Flow
Regard that the fluxaffair of an incompressible (GB) fluid is firstly at rest in between two infinitely long coaxial cylinders of the radius 0 R and 1 fluid is generated because of(SP) gradient which acts on liquid in z-direction.Pointing to Eq. ( 9), the coinciding (DFP) equation that define such flux has the way The condition of initial and boundary relations are described as form For producing the accurate analytical solution of the previous problems ( 10)-( 12), First, we perform(LT) rule Garg et al. [15] through respect to t, we got and s denoted the parameter transform.
We imply the (FHT) Garg et al. [15], described as form While its inverse as Where i k are the positive roots of equation are the functions of Bessel of the first and second types of zero order.
Here using (FHT) to Eqs. ( 13)-( 14) through respect to r, we take Currently, lettering Eq. (17) In the form of a chain as . Whileits discrete inverse (LT) Garg et al. (2007) will yield the form showed generalized Mittag-Leffler function Garg et al. [15] andto earn Eq. ( 19), the following property of inverse (LT) is used (20) eventually, the inverse (FHT) obtains the analytic solution of velocity classification

The Special Cases
Working the limits of Eq.( 21) where 0   , 0 2   (b=0) , we obtain the distribution of velocity for a (GO-B) fluid.So the field of velocity decreases to .

Second Problem of the Magnetohydrodynamic (MHD) Flow
Moreover, it believes that showing fluid is prevailed by imposing magnetic field 0 [0, H , 0]  H which work in positive z-direction.In the calculation of the low-magnetic Reynolds number, H w  , when  indicated electrical accessibility of fluid.Now, by adding magnetic field to Eq.( 8) we get an Eq. ( 23): the judge rz S among Eqs.( 7) and ( 23), we make the next fractional differential equation In the same way as calculating the flux of the first problem we find Eq.( 24), the according fractional partial differential equation that term such fluxhas the shape Where To earn the accurate analytical solution of the previous problems ( 25)-( 12), First, we perform (LT) rule Garg et al. [15] through respect to t, we obtain Now using (FHT) to Eqs. ( 26) -( 14) through respect to r, we obtain Now, inscription Eq. ( 27) in sequence form as While its discrete inverse (LT) Garg et al [15] will yield the form the following property of inverse (LT) is used (20).finally, the inverse (FHT) gets the analytic solution of velocity classification

The Special cases
Working the limitsfor Eq.( 30

Results Discussion
In the present study, we have been discussed MHD flux of (GB) fluid that passed an annular pipe.The accurate solution for the field of velocity u is gotten by performing the (LT) and (FHT).Furthermore, figures were plotted to show the behavior of diverse parameters included the velocity expressions u .
A comparison between the effect of magnetic parameter (M≠0) (Panel (a)) and the effect of non-magnetic parameter (M=0) (Panel (b)) were also done graphically in figures (1)(2)(3)(4)(5)(6).figures ( 1) and ( 2) the velocity is increased with the increasing of the  with both cases (M=0 & M≠0), while it increased with  (M≠0) more than with  (M=0).figures (3), ( 4) and ( 5) showed the relaxation parameter effect 1  on the fields of velocity.Velocity is decreased for the incensement of 1  for (M≠0), and it did not affected with the increase of 1  for (M=0).Velocity is increased with the incensement of 2  when (M=0), and it oscillated with the increase of 2  for (M≠0).The velocity is decreased with the incensement of 3  when (M=0), and decreased more with the incensement of 3  for (M≠0).figure (6) has shown the effect of the magnetic parameter M inshort as well as in long time.It is detected that the velocity profile is increased with the increase of t = 0.5 -1.2 for (M≠0) more than for (M=0).Comparison displays that velocity sketch with the effect of magnetic field is greater when compared with velocity sketch without the effect of magnetic field.The result is demonstrated in long time.

,
 showed the efficient viscosity of fluid, 1  and 3  (< 1  ) are the relaxation, and the obstruction times, respectively, 2  is the modern item parameter of (GB)fluid, α and β the (DFP) calculus like that

D
are the (DFP)of order  and  be contingent on the definition of Riemann- , we obtain the distribution of velocity for (GO-B) fluid.So the field of velocity decreases to

Figure ( 1 )
Figure (1): Shows velocity for various values of  while remaining another parameters constant (a) M 3  , and (b) M 0