Unsteady Flow of a Casson Fluid between Two Orthogonally Moving Porous Disks: A Numerical Investigation

The purpose of the present paper is to study the unsteady two dimensional laminar incompressible flow of non-Newtonian fluid between two orthogonally moving porous disks using the Casson fluid model that is used to characterize the non-Newtonian fluid behavior. Governing partial differential equations are reduced into nonlinear ordinary differential equations by using suitable similarity transformation and then solved numerically by Runge-Kutta-Fehlberg technique. The numerical results are found excellent agreement with the literature published before. Trusting to this validity, effects of different physical parameters are discussed. The study reveals that velocity of the fluid particles increases with an increase in the disk expanding ratio.


Introduction
Transportation of the fluid flow problems in a channel or pipe with expanding or contracting walls have significant applications in the field of engineering, science and medical field.Blood flows in arteries, vessels, blood flow in artificial kidneys, circulation in the respiratory system and regression of the burning plates in rocket motors are some most prominent applications of the flow in a porous channel with expanding or contracting walls [1].The first attempt of study on viscous flow inside a permeable pipe of contracting cross area was analyzed by Uchida and Aoki [2].Later, this problem was illuminated by Bujurke et al. [3] both numerically and analytically.Goto and Uchida [4] presented theoretical framework of unsteady incompressible laminar flows in a pipe.Suction or injection was taken place at the walls of pipe and the radius of the pipe is variant with respect to time.In addition, Majdalani et al. [5], Dauenhauer and Majdalani [6], Majdalani and Zhou [7] investigated the problem of laminar flow in a channel with porous expanding walls numerically as well as asymptotically.Later, analytical solutions were obtained by Rahimi et al. [8] for the case of expanding and contracting porous channel walls.Reddy et al. [9] were used perturbation technique to analysis the effects of heat and mass transfer on the asymmetric flow in a porous channel with http://www.ispacs.com/journals/cna/2017/cna-00291/International Scientific Publications and Consulting Services expanding or contracting walls.Series solution of uniformly expanding or contracting walls in a semi-infinite rectangular porous channel was investigated by Mohyud-din et al. [10].An analytical solution was investigated by Magalakwe and Khalique [11], for the flow and heat transfer between slowly expanding or contracting walls.Xinhui et al. [12] and [13] analyzed the flow of non-newtonian fluid in a porous channel with expanding or contracting walls.Recently, Raza et al. [14] investigated the multiple solutions of Copperwater nanofluids in a porous channel with expanding or contracting walls.Shooting method was employed in order to investigate the different branches of the solution.Numerical investigation was carried out in order to find the multiple solutions of micropolar fluid in a channel with expanding or contracting walls by Raza et al. [15].The study revealed that velocity profile increased near the center of the channel for expanding walls and decreased for contracting walls of the channel.Due to the various applications in the fields of food processing, metallurgy, drilling and biomathematics, Casson fluid gained much more attention of researchers [16]- [17].Moreover, the study of Casson fluid has the implications on the field of manufacturing of medical products, paints, coal in water and biological fluids [18].Due to the enormous applications in industry, engineering and medical sciences many researchers focused on Casson fluid in a channel [19], [20], [21], [22].However, none of these studies dealt with the case of expanding and contracting walls.Therefore, this study attempts to investigate the Casson fluid flow between slowly expanding and contracting walls which has not been studied before.Combined effects of hydrodynamic boundary layer and electromagnetic field lead us to describe the magnetohydynamics (MHD) phenomenon which has many applications in many engineering problems for intent in MHD generators, accelerators and blood flow measurements.Joneidi [23], Pourmahmoud [24] and Morely et al. [25] gerenalized the work of hydromagnetic flows in a channel under the various fluid flow conditions.All above mentioned implications of magnetohyrodynamics (MHD) phenomenon leads us to also consider this study.To our best knowledge no researcher has yet considered the proposed problem.Therefore, in this paper, we consider unsteady laminar incompressible non-Newtonian Casson fluid between two orthogonal moving porous disk under the influence of magnetic field.The mathematical formulation of law of conservation, momentum and heat equations are derived in section-II, numerical investigation of the governing equations are presented in the section-III, results and discussions are in section-IV and at the end some eminent conclusions are obtained and presented in section-V.

Problem formulation
Consider the flow of non-Newtonian Casson fluid between two orthogonal porous disks.The flow is supposed to be unsteady, laminar and incompressible under the influence of magnetic field.Lower disk is located at  = −() and upper disk at  = (), so the distance between two disk is 2().The porous disks have same permeability and move down to up and up to the down uniformly at the time dependent rate ̇().Cylindrical coordinate system is used for the proposed problem and origin is located at the middle of the disks (see Fig. 1).The constitutive equation for the Casson fluid can be written as [26] where   is the plastic dynamic viscosity of the non-Newtonian fluid,   is the yield stress of the fluid,  is the product of the component of deformation rate with itself, and   is critical value ofbased on non-Newtonian model.Under these assumptions the governing equations for the proposed model are: http://www.ispacs.com/journals/cna/2017/cna-00291/ International Scientific Publications and Consulting Services ) ( ) It is important to mention that Eq. (2.2) is the continuity equation, Eqs.(2.3) and (2.4) correspond to the rand z-components of the momentum equation respectively.In the above equations u and w are, respectively, the velocity components in the r-and z-directions, p is the pressure, and B 0 is the strength of the magnetic field.Moreover  and  are the dynamic viscosity and density of the fluid respectively.
We take a as the reference length.The boundary conditions for the present problem are  = 0,  = 2  = ̇ for  = ()  = 0,  = 2  = −̇ for  = −(), whereas 2v w is taken to be the measure of the suction velocity at the disks, and  = 2  ⁄ is the measure of the wall permeability.The factor 2 is used in the boundary conditions for simplification in the later manipulation of the governing equations.Motivated by Uchida and Aoki [2], we use the following transformation: where  is the kinematic viscosity.After eliminating pressure terms from Eqs. (2.3) and (2.4), we use Eq.
For self-similar solution, we consider  =   by the transformation introduced by Uchida and Aoki [2], Dauenhauer and Majdalani [6].This can leads us to consider the case  is a constant and  = ().Therefore,   = 0.So Eq. (2.7) becomes: ) Subject to the boundary conditions )

Numerical solution with Maple 18
Runge-Kutta-Fehlberg method guarantees the accuracy in solution of the initial value problem   = (, ), (  ) =   using appropriate step size.For each step, two different approximations to the solution are computed and then checked by comparing calculated value with the given terminal point.The step size discretized into much smaller if the compared numerical value are not asymptotic to the required accuracy [29].In each step the following six steps are required to compute: Here,  1 ,  2 and  3 are unknown initial conditions.We have to shoot these initial conditions with some arbitrary slope such that solution of the system (3.13)satisfies the given conditions at the boundary.Hit and trail approach is acquire in order to find the unknown initial conditions.Once slope of  1 ,  2 and  3 assumed then numerical integration is made for the initial value problem and accuracy of missing initial conditions is then checked by comparing calculated value with the given terminal point.

Results & discussions
Present section is fermented for presenting our numerical solution of the proposed problem in the form of tables and graphs.The physical dimensionless parameters of our interest are wall expansion ratio , Casson parameter , Reynolds number , magnetic parameter  and Prandtl number .Numerical values of skin friction  ′′ (−1) and heat transfer rate  ′ (−1) at the lower disk is presented in Table 1 for the variations of above mentioned physical parameters.Increase values of suction ( > 0) increases the magnitude of the skin friction  ′′ (−1) and heat transfer rate  ′ (−1) for the fixed values of  = 0.5,  = 1,  = 0.5 and  = 0.7.Numerical values of skin friction increases monotonically and heat transfer rate decrease for the variation of Casson parameter  and magnetic parameter .To verify the accuracy of our numerical scheme, a comparison of the computed values of skin friction  ′′ (−1) at the lower disk for wall expansion ratio  = 1 and  = −1 is made to that of Ghaffar et al. [30] for  = 0 and  → ∞ in Table 2   elucidates the effect of disk expansion ratio  > 0 on velocity profile by setting  = 5,  =  = 0.5 and  = 0.7.Velocity profile  ′ () increases near the center of the region  ≈ 0 and decreases near the disks by increasing the values in disk expansion ratio  > 0. This is because fluid moves freely near the center of the region due to the space generated by the disk expansion, so therefore fluid velocity  ′ () increases gradually near the center of the region  ≈ 0. Effect of disk contraction  < 0 on velocity profile  ′ () for the fixed values of Reynolds number  = 5, magnetic parameter  = 0.5, Casson parameter  = 0.5 and Prandtl number  = 0.7 depicts in the Fig. 7. From this profile it is noticed that decreasing values of disk contraction ratio −5 ≤  ≤ 0, the velocity profile  ′ () decreases gradually near the center of the region and totally opposite behavior can be seen near the region adjacent to the disks.Physically we can say that disk contraction  < 0 provide less space for the fluid to flow so therefore fluid velocity near the center of the channel decreases, flow towards the region  ≈ 0 becomes more noticeable.The present study is motivated to investigate numerical solution of unsteady, laminar and incompressible flow of Casson fluid between two orthogonal moving porous coaxial disks with suction.The following conclusions can be drawn.

Figs. 2
Figs. 2 and 3 depict the behavior of Casson parameter  on velocity profile  ′ () for expanding  > 0 and contracting  < 0 disk respectively for the fixed values of  = 5,  = 0.5 and  = 0.7.Enhancement in the numerical values of Casson parameter  gives rice to the dynamic viscosity.Due to the rice of the viscosity, resistance in the flow field increases and velocity of the fluid decreases.This can be seen at the center of the two disks because flow near the upper and lower disk is under the influence of suction and injection respectively.The effects of permeability Reynolds number  on velocity profile  ′ () for expanding and contracting disks are presented in Figs.4 and 5respectively.It is notice that the velocity profile increases near the disks while forming the parabolic nature in the middle of the domain.Fig.6elucidates the effect of disk expansion ratio  > 0 on velocity profile by setting  = 5,  =  = 0.5 and  = 0.7.Velocity profile  ′ () increases near the center of the region  ≈ 0 and decreases near the disks by increasing the values in disk expansion ratio  > 0. This is because fluid moves freely near the center of the region due to the space generated by the disk expansion, so therefore fluid velocity  ′ () increases gradually near the center of the region  ≈ 0. Effect of disk contraction  < 0 on velocity profile  ′ () for the fixed values of Reynolds number  = 5, magnetic parameter  = 0.5, Casson parameter  = 0.5 and Prandtl number  = 0.7 depicts in the Fig.7.From this profile it is noticed that decreasing values of disk contraction ratio −5 ≤  ≤ 0, the velocity profile  ′ () decreases gradually near the center of the region and totally opposite behavior can be seen near the region adjacent to the disks.Physically we can say that disk contraction  < 0 provide less space for the fluid to flow so therefore fluid velocity near the center of the channel decreases, flow towards the region  ≈ 0 becomes more noticeable.

Figure 3 :
Figure 3: Effect of Casson parameter  on velocity profile  ′ () for contracting disk

Figure 4 :
Figure 4: Effect of Reynolds number  on velocity profile  ′ () for expanding disk

Figure 5 :
Figure 5: Effect of Reynolds number  on velocity profile  ′ () for contracting disk

Figure 8 :
Figure 8: Effect of magnetic parameter  on velocity profile  ′ () for expanding disk

Figure 9 :
Figure 9: Effect of magnetic parameter  on velocity profile  ′ () for contracting disk

Figure 10 :
Figure 10: Effect of Reynolds number  on temperature profile () for expanding disk

Figure 11 :
Figure 11: Effect of Reynolds number  on temperature profile () for contracting disk

Figure 12 :
Figure 12: Effect of Prandtl number  on temperature profile () for expanding disk

Table 1 :
and close agreement is found.Comparison show an excellent agreement for each value of Reynolds number .Therefore, we are confident that the present results are very accurate.http://www.ispacs.com/journals/cna/2017/cna-00291/International Scientific Publications and Consulting Services Effect of dimensionless Reynolds number, magnetic parameter, Casson parameter, wall expansion ratio and Prandlt number on skin friction and heat transfer rate http://www.ispacs.com/journals/cna/2017/cna-00291/ International Scientific Publications and Consulting Services

Table 2 :
Comparison of dimensionless skin friction  ′′ (−1) for the various values of Reynolds number for  = 0 Effect of Casson parameter  on velocity profile  ′ () for expanding disk http://www.ispacs.com/journals/cna/2017/cna-00291/ International Scientific Publications and Consulting Services