The formation and growth of shock waves in the one-dimensional motion of a gas

Abstract

In many compressible fluid flow problems the classical solution breaks down completely owing to the formation of regions of infinite acceleration in the flow field. The actual behaviour of the fluid in such cases does not seem to have been investigated mathematically and this is largely due to the difficulties which enter with non-uniform shocks. It is with these difficulties that this paper is principally concerned.The way in which discontinuities may arise mathematically in a flow field is first discussed. The equations governing the one-dimensional motion of a gas due to an accelerating piston are then set up. It is shown that when allowance is made for varying entropy conditions due to the presence of non-uniform shocks these differential equations reduce (outside the shock wave) to three first order quasilinear ones.The initial solution breaks down when a point of infinite acceleration occurs in the flow field. From this point onwards a shock wave grows in the fluid and behind it three different sets of characteristics are required to describe the flow. By working in the plane of two quantities that are constant along two different sets of characteristics, we can use the shock-jump conditions to determine the equation of the shock-line in this plane and to reduce the equations of the characteristics to three differential equations, which would be linear if the relation between the entropy and other flow variables were known.In the case of a constantly accelerating piston a first approximation is found by neglecting reflexions and entropy variations behind the shock. Using this as a basis we then find a second approximation for the entropy in the neighbourhood of the initial portion of the shock-line and show that the problem reduces to the solution of a second order linear partial differential equation. The introduction of a Riemann function and the satisfaction of the boundary conditions at the shock lead to an integral equation whose solution enables us to determine the position of the shock as a function of the time. The solution is in the form of a power series and is valid provided the shock wave does not become too strong.Finally, it is shown that if the piston is given a constant terminal velocity a reflected wave from the shock is reflected again from the piston and eventually overtakes the shock and reduces its velocity to a final steady value which is in agreement with the value arising from an impulsive start.