Model of a strong discontinuity for the equations of spatial long waves propagating in a free-boundary shear flow

Abstract

The long-wave equations describing three-dimensional shear wave motion of a free-surface ideal fluid are rearranged to a special form and used to describe discontinuous solutions. Relations at the discontinuity front are derived, and stability conditions for the discontinuity are formulated. The problem of determining the flow parameters behind the discontinuity front from known parameters before the front and specified velocity of motion of the front are investigated. Introduction.We consider a mathematical model which describes three-dimensional shear flows of a heavy incompressible ideal fluid with a free surface above an even bottom in the long-wave approximation. This model, which extends the classical shallow water model, reduces to a system of integrodifferential equations. Unlike for the classical model, the propagation of nonlinear wave perturbations for the integrodifferential model has been studied less extensively. New approaches to the solution of these questions were proposed in [1-4] using a new mathematical apparatus. In [1], generalized characteristics were found and hyperbolicity conditions were formulated for the system of integrodifferential shallow-water equations describing three-dimensional stationary shear flows of a free-boundary ideal fluid. The spatial simple waves described by the indicated system of equations were studied in [2]. A definition of discontinuous solutions for a mathematical model of shear plane-parallel incompressible flows was proposed in [3]. In the same paper, the properties of strong-discontinuity relations were analyzed. A similar analysis for a model of plane-parallel flows of barotropic fluids was performed in [4]. New approaches to the description of the interaction of shear flows of an incompressible ideal fluid were used in [5]. Problems of conjugation of various filtration and channel flows of a viscous incompressible fluid and various mathematical models of two-phase fluids were studied in [6,7].In the present paper, we consider the spatial problem of conjugation of non-one-dimensional flows of an ideal incompressible fluid with a free boundary. Relations at the discontinuity front and stability conditions for discontinuous flow are formulated.1. Formulation of the Problem. We consider the system of equations

Strong discontinuities in spatial stationary long-wave flows of an ideal incompressible fluid

“…In the present section, use is made of the discontinuity relations obtained in [5] for the equations of stationary flows (1.2). It is shown that the set of states behind the jump is determined by some curve similar to the (θ, p)-polar curve in gas dynamics.…”

“…System (1.2) cannot be reduced to divergent form. In [5], it is proposed to use the following form of system (1.2), which is equivalent to the initial one for smooth solutions: (2.1)…”

“…It has been shown [5] that in this class of functions, the notion of a solution with a strong discontinuity is defined correctly [i.e., Eqs. (2.1) are meaningful in the class of generalized functions] and system (2.1) defines discontinuity relations.…”

“…Square brackets denote the jump of the function across the discontinuity: [ϕ] = ϕ 2 − ϕ 1 (subscript 1 corresponds to the state ahead of the discontinuity front; subscript 2 to the state behind the front). Strong-discontinuity relations for the model of spatial shear flows were derived in [5]. In the case of stationary flows, these relations become…”

“…It has also been proved [5] that the flow parameters behind the front of the jump are determined uniquely, given the flow parameters ahead of the jump, the velocity of the front, and the normal vector to the jump front.…”