An incompressible liquid approaches a porous flat surface with an accelerating external…

An incompressible liquid approaches a porous flat surface with an accelerating external velocity (U_e(x)=Kx^{1/3}), where (K=3.20\ \mathrm{m^{2/3}\,s^{-1}}) and (x) is measured from the leading edge. Uniform suction into the surface has magnitude (V_s=2.00\times10^{-3}\ \mathrm{m\,s^{-1}}), so the wall-normal velocity at the wall is (v_w=-V_s). The liquid has kinematic viscosity (8.50\times10^{-7}\ \mathrm{m^2\,s^{-1}}) and density (990\ \mathrm{kg\,m^{-3}}). Model the laminar boundary layer with (u/U_e=2\eta-2\eta^3+\eta^4), where η=y/δ for (0\le\eta\le1). Starting from the momentum-integral equation with wall mass transfer, derive the differential equation governing (\delta(x)) and integrate it numerically from (x_0=1.00\ \mathrm{mm}) to (x=0.500\ \mathrm{m}). Use the initial condition that at (x_0), (\delta) equals the no-suction similarity value obtained by retaining the given velocity profile and setting (V_s=0). Report (\delta), displacement thickness, momentum thickness, wall shear stress, and local skin-friction coefficient at (x=0.500\ \mathrm{m}). Also determine the percentage change in wall shear stress relative to the corresponding no-suction result at that location.

In the momentum-integral equation, suction enters through \(v_w/U_e=-V_s/U_e\).