Solutions of the continuity equation

Abstract

Solutions are obtained to the continuity equation for the transport of a quantity f ( r, t ) by a velocity v or the change of a distribution n ( r, t ) by a growth rate ṙ . When v is a separable function of r and t , explicit solutions are obtained from the formal solutions known in hydrodynamics theory which are extended to include contributions from source terms. When v is also a function of f , a general method is given for solving the equation, together with an illustrative example referring to porosity transport in nuclear fuel elements. In the case when ṙ depends on moments of the dis­tribution, a method of solution is suggested which involves an expansion in terms of the moments and which can be applied to void swelling in steels. Perturbation theory and the application of the results to more general problems are discussed.