The melting of a body in the neighbourhood of a critical point in a plane and axi-symmetric stream of gas

Abstract

If a body is moving at a supersonic velocity, or if it is in a hot stream of gas, some parts of its boundary can be heated to melting temperature T ∗ or higher, so that a melting film forms on the surface of the body and under the action of friction of the viscous gas begins to be drawn into motion. We can obtain the conditions under which the temperature of the surface will reach melting temperature taking into account the heat conduction of the body by solving simultaneously the boundary layer equations and the heat conduction equation in a solid body. In many cases which are important in practice these conditions are easily obtained. If, in a steady flow state, the body begins to melt, then it follows from numerous experiments (see [1], [2]) under certain conditions (such as if the body has high heat conduction coefficients or is sufficiently thick near a critical point) the melting state is also stationary and the front of the melting wave near the critical point moves with constant velocity. This steady state of melting is obtained in [3] for a given heat flow from the gas side with an arbitrary dependence between the thermal properties of the solid body and the temperature. In [4] an approximate solution of the problem of a steady melting state of an icy body near a critical point is obtained on the assumption that the thermal properties of ice are constant. The resulting system of ordinary differential equations is solved by Pohlhausen's method (given the velocity profile and the temperatures in the gas and the melting film) with additional simplifications connected with the specific properties of water. In addition, a certain average tangential friction and heat stream from the side of the gas for the whole range of temperature ratios T ∗ T ∞ is taken, where T ∞ is the temperature outside the boundary layer. The comparison of the results of [4] with those of the present article shows that the error in determining the velocity of melting of ice in a stream of air with T ∞ = 600 °C reaches 88 %, and in the determination of the thickness of the melting film it reaches 46 % (Section 5). We shall give a solution of the problem of the steady melting state of a solid body in a stream of gas near a critical point with an arbitrary dependence between the thermal properties of the body and the temperature. In practice the determination of the melting velocity and the thickness of the melting film for given values of the defining parameters in the problem reduces to the solution of two simultaneous transcendental equations. It is shown that when there is a definite connection between the defining parameters of the problem in the axi-symmetric case there exists a linear velocity profile in the melting film (exact solution) and the melting velocity and thickness of the melting film can in this case be found in finite form. Asymptotic formulae for the melting velocity are obtained for a region of values of the defining parameters which has practical interest (Section 4). Diagrams showing the melting velocity and thickness of the film are obtained for some values of the parameters in the problem. It is shown that for bodies with a small Prandtl number in the liquid phase σ 1 < 1 or for small values of the parameter A it is possible to express the melting velocity in terms of that of a model system. (Section 6).