Abstract
Unidirectional unsteady flows of the incompressible Burgers’ fluids between two infinite horizontal parallel plates are analytically studied when the magnetic and porous effects are taken into consideration. The fluid motion is induced by the two plates, which move in their planes with time-dependent velocities. Exact general expressions are established both for the dimensionless velocity and shear stress fields as well as the corresponding Darcy’s resistance in the channel using the Laplace transform. If both plates move with equal velocities in the same direction, the fluid motion becomes symmetric with respect to the mid-plane between them. Otherwise, its motion is non-symmetric. To bring to light the behavior of the fluid, the dimensionless velocity profiles versus the spatial variable as well as its time evolution are presented both for the symmetric and asymmetric case. Finally, for comparison, similar graphical representations are presented together for the velocities of the incompressible Oldroyd-B and Burgers’ fluids. For large values of the time t, as expected, the behavior of the two different fluids is almost identical. The Darcy’s resistance against y is also graphically represented for the symmetric flow at different values of the time t. The influence of the magnetic field on the fluid motion is graphically revealed and discussed.
Highlights
General expressions have been previously determined for the dimensionless velocity, shear stress and Darcy’s resistance, corresponding to the unsteady MHD flow of the incompressible Burgers’ fluids between two infinite horizontal parallel plates embedded in a porous medium
Exact general solutions were established both for the dimensionless velocity and the shear stress fields as well as the corresponding Darcy’s resistance when the magnetic and porous effects were taken into consideration
In order to bring to light some physical insight of the results that were obtained here, a few graphical representations were provided for the symmetric and asymmetric flows with respect to the median plane of the channel
Summary
Burgers [1] developed a one-dimensional linear model whose constitutive equation is given by the relation. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. T = − pI + S, conditions of the Creative Commons. Where σ is the stress, ε is the one-dimensional strain and λ1 , λ2 , η1 and η2 are the material constants. This model was used to characterize the behavior of different viscoelastic materials such as soil, asphalt and food products such as cheese [2,3]. Lee and Markwick [4]
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