Determination of the form of a thin axisymmetric cavity on the basis of an integrodifferential equation

Abstract

With the principle of “the independence of expansion” proposed in [1] as a basis, simple differential equations are obtained which describe the form both of a steady and of an unsteady cavity [2–4], and solutions are obtained for some problems [1, 5] by the use of empirical constants and dependences. The mathematically closed formulation of the problem was proposed in [6–8]. At the present time, different variants of the equations for the form of a steady thin axisymmetric cavity are known together with their solutions (for example, [8–11]), and so are the numerical solutions of the steady problem in the exact formulation (without any assumption about the thinness of the cavity [12–14]). In the unsteady case all that is known is integrodifferential equations for the form of the cavity [3, 15] and the solutions to the equations of the first approximation [15]. These equations provide a good qualitative description of the form of steady and unsteady cavities [11, 15]. In order to achieve quantitative agreement, it is essential to solve the integrodifferential equations. In the present study, the solution of this type of equation is reduced to the integration of a chain of linear first-order differential equations. If one restricts oneself to the first two in this chain, the second approximation equations so obtained provide a high degree of accuracy and are fairly simple. In particular, analytical equations can be obtained in the steady case for the radius of the cavity during flow of a weightless, and of a heavy, fluid round an object. In the case of unsteady cavities, an analytical expression is obtained for the second derivative of the radius.