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The effect of a uniform cross flow (injection/ suction) on the transient energy growth of a plane Poiseuille flow is investigated. Non-modal linear stability analysis is carried out to determine the two-dimensional optimal perturbations for maximum growth. The linearized Navier-Stockes equations are reduced to a modified Orr Sommerfeld equation that is solved numerically using a Chebychev collocation spectral method. Our study is focused on the response to external excitations and initial conditions by examining the energy growth function G(t) and the pseudo-spectrum. Results show that, the transient energy of the optimal perturbation grows rapidly at short times and decline slowly at long times when the crossflow rate is low or strong. In addition, the maximum energy growth is very pronounced in low injection rate than that of the strong one. For the intermediate crossflow rate, the transient energy growth of the perturbation, is only possible at the long times with a very high-energy gain. Analysis of the pseudo-spectrum show that the nonnormal character of the modified Orr-Sommerfeld operator tends to a high sensitivity of pseudo-spectra structures.

A linear stability analysis of a plane Couette-Poiseuille flow of an electrically conducting fluid with uniform cross-flow is investigated in the presence of a transverse magnetic field. The Chebyshev spectral collocation method is utilized to obtain eigenvalues of the modified classical Orr-Sommerfeld equation. The effect of cross-flow with its sense and transverse magnetic field on the stability are examined. The results show that crossflow acts to stabilize or destabilize the flow. The cross-flow's sense produces a significant influence on the stability. This effect becomes more important in the presence of a magnetic field. The dependence of the magnetic field's effect on the cross-flow's sense is also presented.

A linear stability analysis of a plane channel flow with porous walls under a uniform cross-flow and an external transverse magnetic field is explored. The physical problem is governed by a system of combined equations of the hydrodynamic and those of Maxwell. The perturbed problem of base state leads to a modified classical Orr-Sommerfeld equation which is solved numerically using the Chebyshev spectral collocation method. The combined effects of the cross-flow Reynolds number and the Hartmann number on the dangerous mode of hydromagnetic stability are investigated.The study shows that, the magnetic field tends to suppress the instability occurred by cross-flow. This stabilizing effect becomes perceptible when the magnetic field produces a mode transition from walls mode to that of the center.