Transient Newtonian Poiseuille flow in a square channel with dynamic wall slip

We analytically solve the transient Newtonian Poiseuille flow in a square channel in the presence of wall slip, which obeys a dynamic equation that extends the static Navier slip law by means of a slip relaxation parameter. The Fourier method is employed considering the appropriate Sturm–Liouville problem in order to account for the presence of time derivative (and thus of the eigenvalues of the problem) in the boundary conditions deriving the proper orthogonality condition for this two-dimensional flow. All desired eigenvalues are determined numerically. The analytical solution yields available analytical solutions for certain special cases and is validated fully explicit finite difference solutions. Comparisons are also made with available results in the literature for one-dimensional Newtonian flows with dynamic wall slip. The effects of the slip relaxation parameter on the evolution of the velocity field and the volumetric flow rate are discussed, and the deceleration of the flow cessation with the slip relaxation parameter is analyzed. It is demonstrated that both slip and the slip relaxation parameter delay flow cessation and that the effect of the latter is more pronounced when wall slip is moderate.