Perturbation methods are one of the fundamental tools used by all applied mathematicians and theoretical physicists. In this book, the author has managed to present the theory and techniques underlying such methods in a manner which will give the text wide appeal to students from a broad range of disciplines. Asymptotic expansions, strained coordinates and multiple scales are illustrated by copious use of examples drawn from all areas of applied mathematics and theoretical physics. The philosophy adopted is that there is no single or best method for such problems, but that one may exploit the small parameter given some experience and understanding of similar perturbation problems. The author does not look to perturbation methods to give quantitative answers but rather to give a physical understanding of the subtle balances in a complex problem.

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The inertial migration of a small sphere in a Poiseuille flow is calculated for the case when the channel Reynolds number is of order unity. The equilibrium position is found to move towards the wall as the Reynolds number increases. The migration velocity is found to increase more slowly than quadratically. These results are compared with the experiments of Segré & Silberberg (1962 a, b).

The dynamic deformation of a solid elastic sphere which is immersed in a viscous fluid and in close motion toward another sphere or a plane solid surface is presented. The deformed shape of the solid surfaces and the pressure profile in the fluid layer separating these surfaces are determined simultaneously via asymptotic and numerical techniques. This research provides the first steps in establishing rational criteria for predicting whether a solid particle will stick or rebound subsequent to impact during filtration or coagulation when viscous forces are important.

The effect of rotary Brownian motion on the rheology of a dilute suspension of rigid spheroids in shear flow is considered for various limiting cases of the particle aspect ratio r and dimensionless shear rate y/D. As a preliminary the probability distribution function is calculated for the orientation of a single, axisymmetric particle in steady shear flow, assuming small particle Reynolds number. The result for the case of weak-shear flows, y / D < 1, has been known for many years.After briefly reviewing this limiting case, we present expressions for the case of strong shear where (r3 + r3) < y/D, and for an intermediate regime relevant for extreme aspect ratios where 1 < y / D < ( r 3 + r 3 ) . The bulk stress is then calculated for these cases, as well as the case of nearly spherical particles r N 1, which has not hitherto been discussed in detail. Various non-Newtonian features of the suspension rheology are discussed in terms of prior conljinuum mechanical and experimental results.

We study how an axisymmetric drop of inviscid fluid breaks under the action of surface tension. The evolution of various initial shapes is calculated numerically using a boundary-element method, and finite-time breakage is observed in detail. The pinchoff region is shown to have lengths scaling as t 2͞3 , where t is the time remaining until pinchoff, and is found to adopt a unique shape with two cones of angles 18.1 ± and 112.8 ± , independent of the initial conditions. The velocity potential in the intermediate region between the small pinchoff region and the large bulk of the drops is shown to take the form Ar 1͞2 P 1͞2 ͑cos u͒ 1 Bt͞r 1 . . . . [S0031-9007(97)05092-8]

A three-dimensional computer simulation of a concentrated emulsion in shear flow has been developed for low-Reynolds-number finite-capillary-number conditions. Numerical results have been obtained using an efficient boundary integral formulation with periodic boundary conditions and up to twelve drops in each periodically replicated unit cell. Calculations have been performed over a range of capillary numbers where drop deformation is significant up to the value where drop breakup is imminent. Results have been obtained for dispersed-phase volume fractions up to 30% and dispersed- to continuous-phase viscosity ratios in the range of 0 to 5. The results reveal a complex rheology with pronounced shear thinning and large normal stresses that is associated with an anisotropic microstructure that results from the alignment of deformed drops in the flow direction. The viscosity of an emulsion is only a moderately increasing function of the dispersed-phase volume fraction, in contrast to suspensions of rigid particles or undeformed drops. Unlike rigid particles, deformable drops do not form large clusters.

In the absence of Brownian motion, inertia and inter-particle forces, two smooth spheres collide in a simple shear flow in a reversible way returning to their initial streamlines. Because the minimum separation during the collision can be less than 10−4 of the radius, quite a small surface roughness can have a significant irreversible effect on the collision. We calculate the change between the initial and final streamlines caused by roughness. Repeated random collisions in a dilute suspension lead to a diffusion of the particles across the streamlines. We calculate the shear-induced diffusivity for both self-diffusion and down-gradient diffusion.