Non-inertial approximation

Except in Chap. 7, the non-inertial approximation is adopted throughout this work. Even for the very simple inertial problems considered in Chap. 7, it is apparent that there are significant difficulties associated with the retention of inertial terms. 

However, it is clear that the dynamical equation (1.8.11) cannot be re-expressed in terms of [Pg.42]

This approach is therefore most useful in the non-inertial approximation. The discussion will be confined to this case for the remainder of this section, and in fact on through later sections, with minor exceptions. 

For isotropic materials, in the non-inertial approximation, we can write down the solution... 

For the moment, let us confine ourselves to the non-inertial approximation. The dynamical equations (1.1.3) read ... 

The conclusion is also valid for viscoelastic bodies - if the non-inertial approximation applies. This follows immediately by invoking the Classical Correspondence Principle. Our object in this section is to generalize the result to the case of two viscoelastic bodies in contact. 

What has been shown here is that energy considerations for a viscoelastic medium in the non-inertial approximation give no more than a Griffith instability criterion similar to that for an elastic medium. One cannot therefore hope to obtain a condition determining crack velocity from a non-inertial energy equation. The status of conditions which emerge by using approximate solutions [Christensen (1979), Christensen and Wu (1981)] has been discussed by Christensen and McCartney (1983). 

All this work is based on the non-inertial approximation. However the practical utility of this approximation for the corresponding elastic problem has been demonstrated by Lord Rayleigh [Strutt (1906)], Hunter (1957) and Tsai (1968, 1971). If anything, the theory should be more realistic in the viscoelastic case. On the other hand. Aboudi (1979) who has developed a completely numerical... 

We now write down the equations governing the impact of an axisymmetric rigid indentor on a viscoelastic half-space, in the non-inertial approximation, using the results of the last section, specifically (5.2.22, 29). These hold for t>t, the time when the contact area is maximum. However, if we extend the definition of 6 (t) so that 01 (0 = if it can be shown, by reference to (5.2.5, 7), that they also hold for t[Pg.184]

Using the characteristic parameters shown in the figure, critical transition diameters were calculated. The values obtained were 570 microns for transition from non-inertial to inertial and 1140 microns from inertial to coating, and are seen to be within a factor of 1.5-2 of the experimental data which, in view of the approximate nature of these calculations, is quite remarkable. The constant rate of growth in the non-inertial regime also implies that only growth by nucleation occurred and that coalescence (see Fig. 12) was not prevalent. 

Let us restrict ourselves to the case of fast coagulation, assuming that each collision of bubbles results in their coalescence. The study of mutual approach of bubbles in a laminar flow is based on the analysis of trajectories of their relative motion. The equations of non-inertial motion of a bubble of radius a relative to a bubble of radius h in the quasi-stationary approximation are ... 

The major practical use of Equation 11.1 lies in testing whether a film of uniform initial thickness ho remains stable or eventually become unstable with time. The solution of linearized equations of motion incorporating the effect of intermolecular forces, can be simplified by the lubrication approximation (non-inertial laminar flow in thin films) and admit space periodic solutions for the film thickness h(x, t) = ho + E sin(kx) exp(cof) with ... 

We consider plane contact and crack problems in this chapter, without neglecting inertial effects. Such problems are typically far more difficult than the non-inertial problems discussed in Chaps. 3 and 4, and require different techniques for their solution. This is an area still in the development stage so that it will not be possible to achieve the kind of synthesis or unification which is desirable. We confine our attention to steady-state motion at uniform velocity V in the negative x direction. We begin by deriving boundary relationships between displacement and stress. These are applied to moving contact problems in the small viscoelasticity approximation, and to Mode III crack problems without any approximation. 

During the seventies, the work on non-inertial problems was consolidated. The main purpose of the present volume is to present a coherent, unified development of this topic, in particular of those problem classes which are not covered by the Classical Correspondence Principle. There has also been some progress on inertial problems. Typically however, to make progress on such problems it is necessary either to confine one s attention to the most idealized configurations or to introduce some approximation. Also, the mathematical techniques used have been generally rather sophisticated. We briefly discuss this work in the last chapter, and derive certain results by comparatively elementary methods. 

In Markovian approximation (zj =0) this quantity approaches the famous Debye plateau shown in Fig. 2.3 whereas non-Markovian absorption coefficient (2.56) tends to 0 when ft) — 0 as it is in reality. This is an advantage of the Rocard formula that eliminates the discrepancy between theory and experiment by taking into account inertial effects. As is seen from Eq. (2.56) and the Hubbard relation (2.28)... 

Direct comparisons can be made with the Schur transformation in the linear case. Other techniques based on the use of algebraic sets for the approximation of non-linear inertial manifolds have also been developed [192] but have not yet been applied in combustion systems. 

Another consequence of the integral theorem (8-111) is that we can calculate inertial and non-Newtonian corrections to the force on a body directly from the creeping-flow solution. Let us begin by considering inertial corrections for a Newtonian fluid. In particular, let us recall that the creeping-flow equations are an approximation to the full Navier-Stokes equations we obtained by taking the limit Re -> 0. Thus, if we start with the ftdl equations of motion for a steady flow in the form... 

One of the more beautiful techniques in fluid mechanics is the dimensional analysis, that provides - via simple calculations - a reliable judgment of the relative importance of the various forces that can drive or retard the flow of fluids. In short the procedure rests on constmction of non-dimensional groups of parameters that constitute the so-called non-dimensional munbers. The most important non-dimensional munber in fluid mechanics is the Re molds number that judges the relative importance of the inertial and viscous effects. At low values of the Reynolds number - a situation that is common to microsystems - the Navier Stokes equation can be well approximated by the Stokes equation, that, in the absence of body forces, reads ... 

Particles of finite size or with a density different from that of the surrounding fluid (e.g. liquid droplets or dust particles suspended in a fluid), due to their inertia and non-vanishing size, have an instantaneous velocity that is somewhat different from the local velocity of the fluid. Therefore such inertial effects can have a significant influence on the distribution of suspended particles. If the Reynolds number based on the size of the particle and its velocity relative to the fluid is small, the flow around the particle can be approximated... 

---