Hydrodynamic jet incident on an uneven wall

Abstract

The axially symmetric free surface problem of an ideal incompressible jet issuing from a nozzle and impinging on an uneven wall is investigated in this paper. More precisely, we show that given a semi-infinitely long axially symmetric nozzle, a mass flux [Formula: see text] in the inlet and a constant atmospheric pressure, there exists a unique incompressible impinging jet whose free surface goes to infinity and is close to the impermeable wall at far field. Moreover, the free surface of the impinging jet initiates at the edge of the semi-infinitely long nozzle and the pressure remains the constant atmospheric pressure on the free surface. The main ingredient to show the existence and the uniqueness of the impinging jet is based on the variational method developed in a series of the celebrated works [Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981) 105–144; Variational Principles and Free-Boundary Problems, Pure and Applied Mathematics (John Wiley & Sons, 1982)] by Alt, Caffarelli and Friedman. Furthermore, some important properties of the axially symmetric impinging jet, such as positivity of the radial velocity, asymptotic behavior of the impinging jet, and the optimal decay rate of the free surface and the impinging jet, are obtained. Moreover, the problem of the axially symmetric jet impinging on a hemispherical cup is also considered. Finally, we establish the well-posedness theory on the incompressible impinging jet in two dimensions.