Linear stability and energy stability of plane Poiseuille flow with isotropic and anisotropic slip boundary conditions

The linear stability and energy stability of the plane Poiseuille flow with the isotropic and anisotropic slip boundary conditions are theoretically analyzed and numerically calculated in this paper. The asymptotic expansions of the critical parameters for the linear stability and energy stability are derived from the eigenvalue equations characterizing the least stable modes. The critical Reynolds number for the linear stability based on 1.5 times of the bulk mean streamwise velocity is found to be Rl2Dapproximate to (1+2741l2)R02D when the non-dimensional isotropic slip length l << 1, where R02Dapproximate to 5772 is the critical Reynolds number under the no-slip boundary condition. The critical Reynolds numbers for the linear stability are calculated for a wide range of anisotropic slip lengths and are found to be no larger than their counterparts in the isotropic slip cases with the same streamwise slip lengths. In the energy stability analyses of the two-dimensional and three-dimensional plane Poiseuille flows with the isotropic slip boundary condition, the critical Reynolds numbers are found to be RlE2Dapproximate to (1+14.95l2)R0E2D and RlE3D approximate to (1+8.37l2)R0E3D, where R0E2D approximate to 87.6 and R0E3D approximate to 49.6 are their counterparts under the no-slip boundary condition. In the three-dimensional plane Poiseuille flow with the anisotropic slip boundary condition, the critical Reynolds number for the energy stability increases with the increase in streamwise slip length l(x) and with the decrease in spanwise slip length l(z), and its first-order approximation is Rlx,lzE3D approximate to [1+2.41(lx-lz)]R0E3D.