Linearity, consequences creeping flow

First, and most important, nonlinear dynamics provides an intellectual framework to pursue the consequences of nonlinear behavior of transport systems, which is simply not possible in an intellectual environment that is based upon a linear mentality, characterized by well-behaved, regular solutions of idealized problems. One example that illustrates the point is the phenomenon of hydrodynamic dispersion in creeping flows of nondilute suspensions. It is well known that Stokes flows are exactly reversible in the sense that the particle trajectories are precisely retraced when the direction of the mean flow is reversed. Nevertheless, the lack of reversibility that characterizes hydrodynamic dispersion in such suspensions has been recently measured experimentally [17] and simulated numerically [18], Although this was initially attributed to the influence of nonhydrodynamic interactions among the particles [17], the numerical simulation [18] specifically excludes such effects. A more general view is that the dispersion observed is a consequence of (1) deterministic chaos that causes infinitesimal uncertainties in particle position (due to arbitrarily weak disturbances of any kind—... 

B. SOME GENERAL CONSEQUENCES OF LINEARITY AND THE CREEPING-FLOW EQUATIONS... 

A very important consequence of approximating the Navier-Stokes equations by the creeping-flow equations is that the classical methods of linear analysis can be used to obtain exact solutions. Equally important, but less well known, is the fact that many important qualitative conclusions can be reached on the basis of linearity alone, without the necessity of obtaining detailed solutions. This, in fact, will be true of any physical problem that can be represented, or at least approximated, by a system of linear equations. In this section we illustrate some qualitative conclusions that are possible for creeping flows. 

B. Some General Consequences of Linearity and the Creeping-flow Equations... 

Problem 7-1. General Consequences of the Linearity and Quasi-Steady Nature of Creeping Flow. 

In the diagram illustrating creep under constant load recovery curves are also displayed. We will presently show that the recovery behaviour is basically similar to the creep behaviour if we neglect the quantity C3, the Newtonian flow. This is a direct consequence of linear viscoelastic behaviour. 

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