Hydrodynamic Stability of Plane Poiseuille Flow of Non-Newtonian Fluids in the Presence of a Transverse Magnetic Field

Abstract

The linear stability of a plane Poiseuille flow of an electrically conducting viscoelastic fluid in the presence of a transverse magnetic field is investigated numerically. The fourth-order modified Orr-Sommerfeld equation governing the stability analysis is solved by a spectral method with expansions in Lagrange polynomials, based on collocation points of Gauss-Lobatto. The combined effects of a magnetic field and fluid's elasticity on the stability picture of the plane Poiseuille flow are investigated in two regards. Firstly, the critical values of a Reynolds number and a wavenumber, indicating the onset of instabilities, are computed for several values of a magnetic Hartman number, M, and at different values of an elasticity number, K. Secondly, the structure of the eigenspectrum of the second-order and second-grade models in the Poiseuille flow is studied. In accordance to previous studies, the magnetic field is predicted to have a stabilizing effect on the Poiseuille flow of viscoelastic fluids. Hence, for second-order (SO) fluids for which the elasticity number K is negative, the critical Reynolds number Re c increases with increasing the Hartman number M, for various values of the elasticity number K. However, for second-grade (SG) fluids (K > 0), the critical Reynolds number Re c increases with increasing the Hartman number only for certain values of the elasticity number K, while decreases for the others.

Stability Analysis of Plane Couette-Poiseuille Flow With Porous Walls Under Transverse Magnetic Field

“…The spectrum is plotted in figure 1 (a). It is worth noting, from the results in this figure, that the curves converge in excellent agreement toward that obtained by Dongorra et al [23] when they used the Chebyshev tau-QZ algorithm method and Hifdi et al [24] and Rafiki et al [19] by the Chebyshev spectral collocation method. Moreover, the critical Reynolds and wave numbers are converging, respectively, to Rec = 5772.2210, αc = 1.0205 [22] for 60 collocation points [see figure 1 (b)].…”

“…They have also shown that the wall's velocity has a stabilizing effect on the flow that became unconditionally stable at (Re, α) = (6000, 1) for C = 1. In the other hand, numerous studies have been devoted to illustrate the effect of an external uniform transverse magnetic field on the channel flow's stability when the considered fluid is assumed to be electrically conducting [12][13][14][15][16][17][18][19][20]. For highly electrically conducting fluids, i.e., very small magnetic Reynolds number, Rm, compared to unity, the governing equations of the plan Poiseuille flow's stability were simplified by Lock [12].…”