Time-Dependent MHD Flow of Non-Newtonian Generalized Burgers' Fluid (GBF) Over a Suddenly Moved Plate With Generalized Darcy's Law

Abstract

Time-dependent magnetohydrodynamic (MHD) motion of a generalized Burgers' fluid (GBF) is investigated in this article. GBF is a highly complicated non-Newtonian fluid and is of highest degree in the class of rate type fluids. GBF is taken electrically conducting by using the restriction of small magnetic Reynolds number. Darcy's law has been used here in its generalized form using the GBF constitutive relation; hence, the medium is made porous. The impulsive motion in the fluid is induced due to sudden jerk of the plate. Exact expressions for velocity as well as for shear stress fields are obtained using the Laplace transform method. The solutions for hydrodynamic fluid (absence of MHD) in a non-porous medium as well as those for a Newtonian fluid (NF) executing a similar motion are also recovered. Results are sketched in terms of several plots and discussed for embedded parameters. It is found that the Hartmann number and porosity of the medium have strong influence on the velocity and shear stress fields.

A Strange Result Regarding Some MHD Motions of Generalized Burgers’ Fluids with a Differential Expression of Shear Stress on the Boundary

“…Velocity vector for isothermal motions of the incompressible Newtonian or non-Newtonian fluids between two infinite horizontal parallel plates can be provided by the relation [13][14][15][16][17][18][19][20][21]:…”

“…The form of boundary conditions (19) and (20) and the fact that the fluid has been at rest up to the initial moment t = 0, tells us that the respective motions become steady or permanent in time. The fluid behavior in such motions can be characterized by the start-up (starting) solutions w c (z, t), η c (z, t), R c (z, t) or w s (z, t), η s (z, t), R s (z, t) sometime after its initiation.…”

“…As previously mentioned, the steady state solutions w cp (z, t), η cp (z, t), R cp (z, t) and w sp (z, t), η sp (z, t), R sp (z, t) have to satisfy the governing Equations ( 14)-( 16) and the boundary conditions (19) and (20), respectively. In addition, they correspond to the steady or permanent state and are valid for any value of the time t ∈ ℜ.…”

“…It is worth to point out the fact that the expressions that have been obtained for w cp (z, t), η cp (z, t), R cp (z, t) and w sp (z, t), η sp (z, t), R sp (z, t) satisfy governing Equations ( 14)-( 16) and boundary conditions (19) and (20), respectively. In addition, if b = c = d = 0, they take simpler forms that have been recently obtained by Vieru and Fetecau [25] for incompressible Maxwell.…”

“…where T is the stress tensor, T e is the extra-stress tensor, D is the rate of deformation tensor, In the existing literature there are many studies regarding motions of IGBFs in which, velocity, non-trivial shear stress or a differential expression of this shear stress is prescribed on the boundary. Among them we mention those of Zheng et al [12], Jamil [13], Sultan et al [14], Khan et al [15], Khan et al [16], Sultan and Nazar [17], Abro et al [18], Alqahtani and Khan [19], Hussain et al [20] and Fetecau et al [21] which study motions over an infinite flat plate or between two infinite horizontal parallel plates. Early enough, Renardy [22,23] remarked that differential expressions of shear stresses have to be prescribed on the boundary in order to formulate well-posed boundary value problems for motions of rate type fluids like Maxwell and Jeffrey fluids.…”