Spectral Relaxation Method for Powell-Eyring Fluid Flow Past a Radially Stretching Heated Disk Surface in a Porous Medium

Abstract: In the present analysis a spectral relaxation method is employed to analyze heat transferral in axisymmetric slip flow of a MHD Powell-Eyring fluid across a radially stretching surface implanted in porous medium with viscous dissipation. The transfigured ruling class of non-linear differential equations was resolved numerically by employing the spectral relaxation method which was projected to resolve the non-linear boundary layer equations. Outcomes had been acquired for the skin friction coefficient and for the local Nusselt number in addition to the velocity and temperature profiles for the same values of the governing physical and fluid parameters. On resemblance with restricted instances of earlier examines within the literature, outcomes were authenticated. It is established that the advocated method is an efficacious numerical algorithm with reliable convergence and it performs as a replacement to prominent numerical techniques to resolve non-linear boundary value problems. It is proved that the convergence rate of the spectral relaxation method is remarkably refined through the usage of the technique in together with the successive over – relaxation method.

Nomenclature

• Specific heat at constant pressure (J/kg K)
• Dimensionless stream function
• Velocity component in x-direction (m/s)
• Velocity component in z-direction (m/s)
• Temperature of the fluid (0C)
• Temperature at the stretching surface
• Ambient fluid temperature
• Stretching velocity along x – direction
• Permeability
• Time index during navigation
• Scale
• Time
• Number of grid points
• Prandtl number
• Velocity vector

Ec – Eckert number

• Local Reynolds number
• Skin friction coefficient
• Non-dimensional temperature
• Local Nusselt number
• Surface heat flux

Greek Symbols

• Thermal conductivity (W/m K)
• Thermal viscosity (N s/m)
• Fluid density (kg/m3)
• Wall shear stress
• Kinematic viscosity of the fluid
• Similarity variable
• Porous parameter
• Extra stress tensor
• Time dependent material constant

Subscript

• Condition at the surface
• Condition at infinity

Super script

Differentiation With Respect to

Over the decades, in the area of fluid mechanics, the attentiveness of many researchers is on the prominence of non-Newtonian fluids (i.e. the relation betwixt the stress-tensor and deformation-tensor is not linear). Micro polar fluid, Carreau fluid, Reiner-Philip off fluid, Casson fluid, Prandtl fluid, Prandtl-Eyring fluid, Power law fluid and Eyring-Powell fluid are instances for such fluids. On deeming the movement of greases, clay slips and paints, Richard Powell, Henry Eyring [1] discovered that the stream velocity relies exponentially on stress at moderate stresses; however, velocity reaches straightly with stress at a higher position of stresses in the flow. According to that, the stream which incorporates the breaking of a minimum of 2 kinds of bonds is summarized as category 1 which comprises robust bonds thus the fluid flows in accordance to non-Newtonian law at moderate stresses (the exponential law) and category 2 is frail bonds which obeys Newtonian law (such flow is proportional to stress-linear law). Manisha Patel, M.G. Timol [2] emphasized that Powell–Eyring model has sure benefits than Power-law model.

Afterwards Sirohiet al. [3] described the movement of Eyring–Powell fluid nearby an accelerated plate with the help of three distinct techniques of solution. After that in the same year, Yoon and Ghajar [4] liberated an extensive note on the Powell–Eyring model. In addition to that, by taking into account the Eyring-Powell fluid, Nadeem et al. [5] proffered their final observation on peristaltic flow . In that analysis, it was proved that the development in size of Eyring–Powell parameter enriches the peristaltic pumping area. Agbajeet al. [6] analysed the unsteady boundary-layer flow of an incompressible Powell-Eyring nano fluid over a shrinking surface. By employing a multi-domain bi-variate spectral- quasilinearization approach, the non-linear P.D.E’s which evokes the transport processes are acquired. In recent times, JayachandraBabuet al. [7] and Hayat et al. [8] emphasized double stratification effects in the flow of Eyring-Powell fluid. Rahimiet al. [9], employed The collocation technique and resolved a related problem of steady boundary layer flow of an Eyring-Powell non-Newtonian fluid across a linear stretching sheet. Later Ibrahim [10] numerically solved the problem which addressing the 3D rotating flow of Powell –Eyring nano fluid under the effect of non Fourier’s heat flux and non-Fick’s mass flux theory. Parandet al. [11] employed the quasi linearization-collocation method based on hermit functions to solve the boundary layer flow of Eyring-Powell fluid problem. Rauf et al. [12] analyzed on Cattaneo-Christov heat and mass flux theories on chemically reactive Powell- Eyring fluid flow. Farooq et al. [13] explored the MHD boundary layer flow of Eyring-Powell fluid in convectively curved configuration. Hassainet al. [14] developed the problem on MHD Prandtl- Eyringfluid flow across a stretching sheet and compared the result numerically through Keller-box technique. Makinde et al. [15] studied the consequences of convective heating on stagnation position of a reactive fluid across a stretching surface with slip.

Several engineering problems are related with the fusion of heat transferral in fluid saturated porous media. Extensive utilizations of civil, mechanical and chemical engineering are related to thermally driven flows in porous media. This authentic truth provokes the curiosity in this subject. Among the few foundational issues regarding heat transferral in porous media, the free convection boundary layer flow from a stretching/shrinking sheet in a porous medium filled with a Powell – Eyring fluid is also one. An extensive utilization of porous media in several pragmatic utilizations could be detected in Nield and Bejan [16], Sharma et al. [17], Prakash et al. [18], Khamiset al. [19], Chinyoka and Makinde [20]. A short time ago, Makinde and Eegunjobi [21] numerically explored the alloyed consequences of thermal radiation, magnetic field and velocity slip on Casson fluid flow through a porous medium. Rundora and Makinde [22] discussed the consequences of variable viscosity on unsteady reactive fluid flow with slip in a porous medium. Makinde and Moitsheki [23] employed the non-perturbative tehnique that relies on semi-numerical technique to analyze the effect of buoyancy force on fluid flow across a flat heated surface implanted in a porous medium.

In general, the effects of viscous dissipation are omitted in macro scale systems, especially in laminar flow, other than for extremely viscous fluids at relatively high velocities. On the other hand, general fluids at laminar Reynolds numbers, frictional effects in micro scale systems are able to alter the energy equation [24]. Koo and Kleinstreuer [25] are explored the impact of viscous dissipation on temperature field by means of dimensional analysis and pragmatically endorsed the computer simulations. In this study, 3 familiar usable fluids- methanol, isoropanol and water in dissimilar channel geometries are deemed. The authors accomplished that the dimension of the channel is a vital feature which ascertain the influence of viscous dissipation. Moreover, the consequences of viscous dissipation could be awfully essential for fluids with excessive viscosities and little certain heat potentials, even also for comparatively low Reynolds numbers. In view of that, the term viscous dissipation must be appraised in the micro scale systems. Shah et al. [26] numerically explored the MHD flow of Carreau fluid past a porous stretching sheet with viscous dissipation and established that rising values of Eckert number encourage the temperature profiles. Khader and Mziou [27] revealed the Chebyshev spectral method which is used to study the viscoelastic slip flow due to a permeable stretching surface implanted in a porous medium with viscous dissipation and non-uniform heat generation.

In view of the previous information, the authors can say that, there is no existing document regarding the axisymmetric slip flow of Powell-Eyring fluid through a radially stretching surface implanted in porous medium by taking into consideration the effect of viscous dissipation using the spectral relaxation method. Hence, the intend of the existing exploration is to examine the heat transferral features in a Powell-Eyring fluid driven by a radially impermeable stretching surface implanted in porous medium within the existence of viscous dissipation via spectral relaxation method.

Numerical Procedure

The present part of this article provides a short narration on the Spectral Relaxation method which is utilized to resolve the equations (9)- (12); observe as well Motsa and Makukula [28]. The Spectral Relaxation method is suggested for solving similarity boundary layer problems with exponentially deteriorating profiles. The algorithm of Spectral Relaxation method for self-similar boundary layer problems is outlined as:

Introduce the transformation in order to diminish the order of the momentum equation for. Now specify the primary equation in terms of.

Presume that is obtained from an earlier iteration (indicate with ). Now formulate an iteration program for by presupposing that simply linear terms in are to be appraised in the present iteration (indicate with ) and presume that the remaining terms (linear and non-linear) are obtained from the earlier iteration. Also the non-linear terms containing are to be appraised in the earlier iteration.

Formulate the new iteration program for the remaining ruling dependent variables in a like fashion with the upgraded solutions of the variables resolved in the earlier iteration.

The above mentioned program and the Gauss–Seidel thought of decoupling linear system of algebraic equations are equivalent. USAge of the present algorithm gives rise to a series of linear differential equations with variable coefficients. One can solve these equations effortlessly by employing the typical numerical techniques for resolving linear differential equations. In the present examination, via Chebyshev spectral collocation methods we discretized the differential equations (for example, see [29–30]). We ascertained that with the grid points via numerical experimentation, to provide sufficient precision for the spectral relaxation method.

Spectral methods are most popular due to their exceptional lofty precision and easy execution in discretizing and the succeeding solution of linear differential equations with variable coefficients with unwrinkled solutions over simple domains so that Spectral methods are preferred here.