A mathematical model has been devised for predicting the oxygen concentration gradient across the epithelium of ovarian follicles at pre-antral stages. Most dissolved oxygen entering the follicle by diffusion is evidently consumed in the outer layer of cells; little reaches the oocyte. Even when the rate of consumption or the diffusion coefficient of oxygen was adjusted by an order of magnitude to favour oxygen penetration, the concentration gradient into the follicle remained steep. On the basis of measurements of ovine granulosa cell respiration in vitro, the model predicts that a large pre-antral follicle with a radius of 0.15 mm consumes oxygen at the rate of 0.22 nmol min-1.

An integral equation for unsteady inviscid surface waves has been obtained. Existing known approximations are all derived from the one equation. The Boussinesq equation is obtained and criticized.

In this paper we consider the head-on collision of two equal solitary waves this being equivalent, in the absence of viscosity to the reflection of one solitary wave by a vertical wall. The perturbation expansion of the Euler equations, which lead to the Boussinesq equation at lowest order, is recast to obtain two weakly coupled KdV equations. We show analytically that the amplitude of the solitary wave after reflection is reduced. This change in amplitude is shown to be fifth order in ε, the amplitude of the wave. It is also shown that the experimentally observed transient loss of amplitude can be explained by the presence of the third-order dispersive tail.

Mammalian preimplantation embryos are conventionally grown in small, static droplets of medium. By contrast, embryos within the oviduct are subject to mixing forces arising from the action of cilia and the contractions of the myosalpinx. Such forces will minimize the buildup of unstirred layers around the embryos and facilitate the exchange of gases and metabolites. We have devised a mathematical model to investigate whether preimplantation mouse and human embryos grown in static culture can satisfy their requirement for oxygen solely by diffusion i.e. in the absence of stirring. The model incorporates the diffusion coefficients for oxygen in the medium outside and within the embryo, the oxygen tension of the culture medium, the size of the embryo and its oxygen uptake. Solutions of the model are provided for mouse oocytes and 2-cell embryos, mouse blastocysts and human morulae. In each case, the model is solved for two different values of the diffusion coefficient of oxygen within the cell and of the oxygen content of the medium (5 and 20%). The conclusion is that mouse embryos in static culture are likely to be able to satisfy their demands for oxygen by diffusion alone, but that human embryos may become marginally hypoxic, especially at lower oxygen levels.