On plane, steady, creeping flow, generated around an arbitrary cylinder by a translating flat plate (Q1885385)

The authors consider plane steady creeping flow around an arbitrary cylinder placed in a vicinity of a flat plate, when the flow is generated by translation of this plate along itself with constant speed perpendicular to generatrices. The streamfunction of the flow satisfies the biharmonic equation, so that all properties of the flow -- such as velocity and pressure fields as well as fields of stress tensor components -- are expressed in terms of Goursat functions. Hence, the problem of determination of the flow reduces to the determination of these functions. The two-step approach is applied to the solution of this problem. The first step consists in conformal mapping of the original domain of solution onto an annulus -- by means of a suitable set of mapping functions. The second step consists in expansion of two Goursat functions in Laurent series extended by two logarithmic terms. The most important part of the results is a formula which represents the complex conjugate velocity of fluid as continuous and differentiable function of two independent variables introduced in the annular domain. The formula contains numerical coefficients representing the solution of a system of linear algebraic equations stemming from boundary conditions. This system is arrived at by means of the pseudo-spectral method. The so obtained velocity field is applied to generation of regular and singular streamline patterns. An ordinary differential equation for generation of such streamlines is formulated. Accuracy of the solution is checked by evaluation of errors in boundary conditions on both boundary circles, at selected points not coinciding with collocation points.