Inertia finite

In several respects the finite inertia of the system of Eq. (1.7) produces effects similar to those of a weak additive stochastic force. In particular, as a consequence of finite inertia, the escape from the well is also possible in the absence of the additive stochastic noise. These effects are not taken into... 

As a second example, we consider the kinetic equation (KE) for monodisperse, isothermal solid particles suspended in a constant-density gas phase. For clarity, we assume that the particle material density is significantly larger than that of the gas so that only the fluid drag and buoyancy terms are needed to account for momentum exchange between the two phases (Maxey Riley, 1983). In this example, the particles are large enough to have finite inertia and thus they evolve with a velocity that can be quite different than that of the gas phase. 

Shafranov and Frolenkov who showed that the unstable spectrum either moves to high wave numbers where MHD theory is not applicable, or that the finite inertia of the plasma decreases... 

There is a contradiction between the low-shear limit of the Mercier criterion for a helical magnetic field and the closed--line criterion of a purely poloidal field. This apparent paradox was solved by Grad and Shafranov and Frolenkov who showed that the unstable spectrum either moves to high wave numbers where MHD theory does not apply, or that finite inertia decreases the growth rate to zero. The closed-line criterion predicts MHD stability under rather general conditions, in some cases even without shear and conducting walls. 

Drossinos, Y. Reeks, M. W. (2005). Brownian motion of finite-inertia particles in a simple shear flow, Phys. Rev. E 71 031113. 

Fladdadi, H. and Morris, J. F. 2014. Microstructure and rheology of finite inertia neutrally buoyant suspensions. J. Fluid Mech. 749, 431-459. 

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

Vertical CVD Reactors. Models of vertical reactors fall into two broad groups. In the first group, the flow field is assumed to be described by the one-dimensional similarity solution to one of the classical axisymmetric flows rotating-disk flow, impinging-jet flow, or stagnation point flow (222). A detailed chemical mechanism is included in the model. In the second category, the finite dimension of the susceptor and the presence of the reactor walls are included in a detailed treatment of axisymmetric flow phenomena, including inertia- and buoyancy-driven recirculations, whereas the chemical mechanism is simplified to a few surface and gas-phase reactions. 

In the high crack velocity regime three different values of Kid can be assigned to one rate of crack propagation depending on the state of crack acceleration. This behaviour was ascribed to inertia effects associated with crack acceleration and deceleration. Such a hypothesis is corroborated by the computed K data (also shown in Fig. 9), which were obtained from a finite element model, taking into consideration the mentioned transient dynamic linear elastic effects [35]. 

A first attack to Painleve s conjecture was done by von Zeipel [26]. von Zeipel considered how the size of an A-body system evolves in time. He chose the moment of inertia I = m,jq, p as the size of the A-body system. He has shown that a necessary condition for a solution having noncolhsional singularity is that the motion of the system becomes unbounded in finite time. 

The results illustrated on the right-hand side of Fig. 4 show that in this region the increase of k is much more sensitive to the increases in than it is in the high-friction region, thereby corroborating our statements about the role of inertia. This trend is especially emphasized in the limit y 0 and is better seen in Fig. 5. As remarked above, the reaction rate stays finite in this zero-friction limit, counter to Kramers prediction. 

In ref. 39 a computer calculation of the rate of escape was done based on the CFP of Chapter III. The agreement between this calculation and Eq. (4.2S) is good, as shown in Fig. 11. Note that the combination of weak inertia with multiplicative noise results in a finite rate of escape which is rigorously forbidden by the AEP when no additive noise is present. 

When a voltage Is applied to an "Ideal" dielectric, polarization of the dipoles occurs Instantaneously, and there Is no lag between the orientation of the molecules.and the variation of the electric field. Under these conditions the current has a displacement of tt/2 relative to the e.m.f., and no loss of electrical energy occurs. This is Illustrated in (a). For a real dielectric the situation Is quite different. Firstly, there Is an Inertia associated with the orientation polarization of a dipole, and a finite time Is required for the dipole to become oriented, and secondly, the dipoles tend to undergo a relaxation to the original... 

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... 

The real-time temperature trace at the axis. Fig. 6.6, shows the variations in temperature until extinction at 1230 K, similar to reported values of 1250 K [12], after which the thermocouple measured an exponential temperature decay slowed by its finite thermal inertia. The temperatures were close to a previously calculated value of 1200 K and provide support for the chemical extinction mechanism that was proposed [13]. It is apparent from Fig. 6.6 that the amplitude of high-... 

Figure 10.3 The influence of the finite dimension of particles in inertia-free flotation on their trajectory in the vicinity of a floating bubble. The liquid flow lines corresponding to target distances b(a,) and are indicated by dashed lines. The continuous lines are characteristic of the deviation of the trajectory of particles from the liquid flow lines under the influence of short-range hydrodynamic interaction...

Levin (1961) has shown that inertia deposition of particles below a critical size, which corresponds to a critical Stokes number St = 1/12, is impossible. Regarding a finite size of particles, the collision is characterised by Sutherland s formula (10.11). Comparison of the results obtained from Sutherland s relation and by Levin enables to conclude that in the region of small St < St the approximation of the material point, accepted by Levin and useful at fairly big St, becomes unsuitable for Stsmall Stokes numbers were studied by Dukhin (1982 1983b) for particles of finite size. Under these conditions inertia forces retard microflotation. 

It is worth noticing that negative effects of inertia forces appear at subcritical values of Stokes numbers when a positive effect is practically absent (cf Section 10.1). The inertia-free approach of a particle and a bubble is caused by the radial particle velocity when its centre is located at a distance from the bubble surface approximately equal to a. When the particle radius tends to zero, this velocity also tends to zero and deposition depends on the finite size of the particle. 

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