Glenn Fulford

SWIMMING OF SPERMATOZOA IN A LINEAR VISCOELASTIC FLUID

A modified resistive force theory is developed for a spermatozoon swimming in a general linear viscoelastic fluid. The theory is based on a Fourier decomposition of the flagellar velocity, which leads to solving the Stokes flow equations with a complex viscosity. We use a model spermatozoon with a spherical head which propagates small amplitude sinusoidal waves along its flagellum. Results are obtained for the velocity of propulsion and the rate of working for a free swimming spermatozoon and the thrust on a fixed spermatozoon. There is no change in propulsive velocity for a viscoelastic fluid compared to a Newtonian fluid. The rate of working does change however, decreasing with increasing elasticity of the fluid, for a Maxwell fluid. Thus the theory predicts that a spermatozoon can swim faster in a Maxwell fluid with the same expenditure of energy for a Newtonian fluid.

Continuous growth and decay models

In this chapter some problems of growth and decay will be studied for which differential equations, rather than difference equations, are the appropriate mathematical models. Such problems include: the growth of large populations in which breeding is not restricted to specific seasons, the absorption of drugs into the body tissues, the decay of radioactive substances. The differential equations which arise from the above problems are all of the first order. The two methods of solution which we explain are sufficient to solve all the differential equations which arise in the next three chapters. The theoretical background for these two methods is contained in Chapter 5. The first of the two methods, which applies only to linear differential equations, is very similar to the method already given in Section 8.1 for solving linear difference equations. The continuous models used in this chapter are similar to the discrete models discussed in Chapter 9. First-order differential equations The two types of differential equations which you need to be able to solve in this chapter are called linear with constant coefficients and variables separable differential equations. The former arise from problems of unrestricted growth, while the latter appear when the growth is restricted. How to recognize and solve the two types of differential equations will now be explained.