Couette flows over a rough boundary and drag reduction

We consider the Couette flow between two plates. The lower plate is fixed and has periodically placed riblets of the characteristic size E on it. In the limit epsilon --> 0 we find the effective Couette-Navier flow as an O (epsilon(2)) approximation for the effective mass flow and an O (epsilon(2)) L-1-approximation for the velocity. In the effective solution the effect of roughness enters through the Navier slip condition with the matrix coefficient in front of the effective shear stress, calculated using a boundary layer problem. Furthermore, an O(epsilon(2)) approximation for the tangential drag force is found. In all estimates explicit dependence on the kinematic viscosity nu, the velocity (U) over right arrow of the upper plate and the distance between the plates L-3 is kept. Also the uniqueness of the solution is expressed through a non-linear algebraic condition linking epsilon, nu, \(U) over right arrow \ and L-3. Then the result is applied to the viscous sub-layers around immersed bodies, strictly containing the surface riblets. It is found that for the riblets of the characteristic size epsilon, being of the order smaller or equal to nu(9/14), the approximation obtained for the tangential drag could be applied. We compare epsilon and nu(9/14) for realistic data and our results lead to the conclusion that the riblets reduce significantly tangential drag, which may explain their presence on the skin of Nektons.