poiseuille flow with couple stresses effect and no-slip boundary conditions

In this paper, the problem of poiseuille flow with couple stresses effect in a fluid layer using the linear instability and nonlinear stability theories is analyzed. also, the nonlinear stability eigenvalue problems for x,z and y,z disturbances are derived. the chebyshev collocation method is adopted to arrive at the eigenvalue equation, which is then solved numerically, where the equivalent of the orr-sommerfeld eigenvalue problem is solved using the chebyshev collocation method. the difficulties which arise in computing the spectrum of the orrsommerfeld equation are discussed. the critical reynolds number rc, the critical wave number ac, and the critical wave speed cc are computed for wide ranges of the couple stresses coefficient m. it is found that the couple stresses coefficient m has great stabilizing effects on the fluid flow where the fluid flow becomes more unstable as m increases.