Stability of plane Poiseuille flow of viscoelastic fluids in the presence of a transverse magnetic field

The linear stability of plan Poiseuille flow of an electrically conducting viscoelastic fluid in the presence of a transverse magnetic field is investigated numerically. The fourth-order Sommerfeld equation governing the stability analysis is solved by spectral method with expansions in lagrange’s polynomials, based on collocation points of Gauss-Lobatto. The critical values of Reynolds number, wave number and wave speed are computed. The results are shown through the neutral curve. The main purpose of this work is to check the combined effect of magnetic field and fluid’s elasticity on the stability of the plane Poiseuille flow. Based on the results obtained in this work, the magnetic field is predicted to have a stabilizing effect on the Poiseuille flow of viscoelastic fluids. Hence, it will be shown that for second-order fluids (K < 0), the critical Reynolds numbers Rec increase when the Hartman number M increases for different values of elasticity number K, which is a known result. The more important result we have found, concerning second-grade fluids (K > 0) is that the critical Reynolds numbers Rec increase when the Hartman number M increases for certain value of elasticity number K and decrease for others. The latter result is in contrast to previous studies.