Navier-Stokes flow down an inclined plane: Downward periodic motion
Let us consider the motion of a viscous fluid stratum flowing on an inclined plane. Its bottom is at smooth contact with the plane, and its top is free. Let the fluid be subjected to the gravity force, so that the motion is due to the component of this force parallel to the inclined plane. A steady motion, defined by a constant velocity \(u_ 0\) parallel to the slope of the bottom, is a solution of the equations of the flow. Let us now suppose that another motion, periodic with period \(2\pi\) in the direction of the flow, may be superimposed on the fundamental steady motion. We aks whether such a motion is possible. The problem can be mathematically treated in the following way. Through a diffeomorphism we may reduce the domain to a stratum contained between two parallel planes. Let us also assume that the solution is smooth enough, although this restriction can be removed by using the method of mollification. Then we can derive a priori estimates, necessary for treating the Navier-Stokes equations. Under this respect, the paper is an application of a previous result by \textit{T. Nishida} [Commun. Pure Appl. Math. 39, Suppl., S221--S238 (1986; Zbl 0615.76078)].
scientific article; zbMATH DE number 567078