Unsteady dispersion of material from a continuous source which is non-uniformly distributed over the cross-section at the inlet of a tube in which fluid is in time-dependent laminar flow, is analysed. A superposition integral is developed for use in obtaining solutions for the unsteady concentration distribution created by time variable continuous sources from the solution for instantaneous sources when the coefficients in the convective diffusion equation are functions of time. This reduces to the conventional superposition integral when the coefficients are time-independent. Numerical results are obtained for the special case of a steady continuous source in steady laminar flow. A very good approximation to the mean concentration which considerably reduces the computational labour is proposed. For the steady-state problem of dispersion of material due to a non-uniform step change at the inlet of the tube, a boundary-layer analysis is presented for the inlet region and an orthogonal function expansion is developed for the downstream region. The steady-state orthogonal function expansion solution compares well with the present method. The boundary-layer solution is useful very close to the tube entrance where it is difficult to obtain numerical results from the other methods.
On the use of the profiled singular-function expansion in gravity gradiometry
