Statistical Mechanics, Diffusion, and Self-Assembly

Equations for random wandering of particles are derived, and the phenomenon of entropic ordering is explained. Boltzmann’s particle-based approach to diffusion is compared to Maxwell’s continuum hypothesis, and van ’t Hoff’s formula for osmosis is obtained. Other topics include diffusivity, the distribution of energy, and the applications of Maxwell–Boltzmann statistics.

Pulsatile Flows

Flow solutions in the presence of pulsation (e.g. from the heart) are developed. Bessel functions are introduced as an aside. The concepts of shocks and solitary waves (solitons) are then discussed as examples of nonlinear effects. The strategy for dealing with intrinsic nonlinearity is described in terms of mode coupling and the Korteweg–de Vries (KdV) equation.

Flow through Elastic Tubes

Fluids equations from Chapter 1 and elasticity equations from Chapter 2 are combined to establish how flow through elastic tubes (as in the vasculature) must occur. Unstable versus stable flows are considered, and concepts from complex analysis are introduced. The main topic is flow in elastic-walled tubes, whereas complex analysis is dealt with as an aside.

Rheology in Complex Fluids 2

Blood flow is described, including changes in viscosity in narrow tubes and effects of shear-induced migration. Segregation of particles based on size is considered, and ordering of red blood cells is used as a model for spontaneous ordering in other systems. Unexpected consequences of diffusion are introduced.

Entrances, Branches, and Bends

Effects of boundaries are described, including those associated with entrances, branches, and bends in tubes. Couette flow between rotating cylinders, modelling of volume-preserving flows, and chaotic flows are described. Model building is considered, and the effects of out of plane bending are discussed in the context of static mixers.

Self-Assembly and Beyond

Effects of combining reaction with diffusion are examined, and the resulting self-assembly of ordered patterns is overviewed. Turing patterns and limit cycle oscillations are shown to result from these considerations, and future avenues for research into these topics are briefly discussed. Additional topics include reaction-diffusion equations, and limit cycles wave solution, and the limit cycle.

Diffusion

The diffusion equation is derived and solved for simple geometries. Fourier series are described, and superposition is used to combine simple solutions into more complicated ones. Advection is combined with diffusion, and compartment models defining diffusion between contacting systems (e.g. a pill, the gut, the bloodstream and tissues) are described.

Rheology in Complex Fluids 1

Complex flows are described, including shear thinning, shear thickening, and yield-stress. Mechanisms of changing viscosity in dense suspensions are explored, including the relevance of the lubrication approximation, dilatency, and the spaghetti model of polymers. Liquid crystal alignment is discussed, and model equations are introduced for flows in packed beds. The viscosity of synovial fluid is described, and equations to combine viscous and elastic behaviors are obtained.

Shearing Flows around Cylinders and Spheres

Novel flows between rotating cylinders, of materials settling in tubes, and in roller bottles are described. Low speed flow around a sphere is derived, and paradoxical settling behaviors are mentioned, including the effects of red blood cells. Stokes drift and Magnus force on falling bodies near boundaries. A first example from scientific ethics is raised. The streamfunction and biharmonic equation are derived and applied to flow past a sphere.

Elastic Surfaces

In this chapter, the effects of elasticity are overviewed, the Young–Laplace equation is introduced, and the importance of nondimensionalization is emphasized. Topics covered include the Plateau–Rayleigh instability and dimensional analysis.