Two dimensional sink flow of a viscous, heat-conducting, compressible fluid; cylindrical shock waves
The steady two-dimensional sink-type flow of a viscous, heat-conducting perfect gas is investigated. An approximate solution of this problem is given for the case of large Reynolds number Re (cf. the definition given in the text). In obtaining the present solution the values of Prandtl number and the ratio of the first and second viscosity coefficient may be arbitrary. The result shows that the solution has two branches, both of physical significance. On the subsonic branch of the solution the flow speed starts from the stagnation point at infinity, increases monotonically for decreasing radial distance and eventually terminates with maximum speed at a certain point inside the inviscid sonic circle. The solutions of the supersonic branch, which start with the maximum speed at infinity, all contain cylindrical shocks. Within the shock the flow speed assumes a minimum value and after the shock all solutions tend asymptotically to the subsonic branch. In contrast to the plane shock, the cylindrical shock strength is limited to the order O(Re^-1/3), and the shock-thickness, of O(Re^-2/3) The latter quantity implies that the thickness of the region in which the viscous effects are important is thinner, in order of magnitude, than that of ordinary boundary layer (of O(Re^-1/2)) but is thicker than that of plane normal shock (of O(fRe^-1)). It is found that the entropy of the super sonic branch rises to a maximum within the shock while for the subsonic branch, the entropy increases monotonically with the radial distance. The total variation of the entropy across the shock is found to be of O(Re^-2/3) which is in general greater than that across a plane normal shock (~O {shock strength}^3). The effect on the flow quantities due to the variation in viscosity coafficients, assumed to depend on the local temperature, is found to be at most of O(Re^-2/3).