ON THE ENTROPY CHANGE FOR A FLUID CONTAINING SWIMMING MICRO-ORGANISMS

We consider the two-dimensional hydrodynamical theory of nearly circular biological objects (cilia) in the creeping approximation, focusing on the time rate of entropy production in the fluid as calculated by means of functional integration methods. It is pointed out that the formalism leads to an entropy change in the fluid which is always negative. This change has, for thermodynamic reasons, to be compensated for, by a change of entropy for the micro-organism itself which is positive, and which is large enough to outweigh the entropy change in the fluid. We consider also briefly the generalization to nonuniform closed objects.

The Taylor–Saffman problem for a non-Newtonian liquid

The Taylor–Saffman problem concerns the fingering instability which develops when one liquid displaces another, more viscous, liquid in a porous medium, or equivalently for Newtonian liquids, in a Hele-Shaw cell. Recent experiments with Hele-Shaw cells using non-Newtonian liquids have shown striking qualitative differences in the fingering pattern, which for these systems branches repeatedly in a manner resembling the growth of a fractal. This paper is an attempt to provide the beginnings of a hydrodynamical theory of this instability by repeating the analysis of Taylor & Saffman using a more general constitutive model. In fact two models are considered; the Oldroyd ‘Fluid B’ model which exhibits elasticity but not shear thinning, and the Ostwald–de Waele power-law model with the opposite combination. Of the two, only the Oldroyd model shows qualitatively new effects, in the form of a kind of resonance which can produce sharply increasing (in fact unbounded) growth rates as the relaxation time of the fluid increases. This may be a partial explanation of the observations on polymer solutions; the similar behaviour reported for clay pastes and slurries is not explained by shear-thinning and may involve a finite yield stress, which is not incorporated into either of the models considered here.