Shahid Chamran University of AhvazJournal of Applied and Computational Mechanics2383-45366Special Issue20201201Poiseuille Flow with Couple Stresses Effect and No-slip Boundary Conditions106910831523810.22055/jacm.2019.31964.1946ENAkil J.HarfashDepartment of Mathematics, College of Sciences, University of Basrah, Basrah, Iraq0000-0002-3738-4242Ghazi A.MeftenDepartment of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah, Iraq0000-0003-0900-9303Journal Article20191216In 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 <em>x,z</em> and <em>y,z</em> 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 Orr-Sommerfeld equation are discussed. The critical Reynolds number <em>R<sub>c</sub></em>, the critical wave number <em>a<sub>c</sub></em>, and the critical wave speed <em>c<sub>c</sub></em> are computed for wide ranges of the couple stresses coefficient <em>M</em>. It is found that the couple stresses coefficient <em>M</em> has great stabilizing effects on the fluid flow where the fluid flow becomes more unstable as <em>M</em> increases.https://jacm.scu.ac.ir/article_15238_158388bcde9e6ce061f1004095e742ae.pdf