Inertial regime

There is a natural limitation to Washburn s law for very short times (i 0). The problem is that the impregnation velocity (which varies as l/ /f) then diverges, which is physically impossible. What restores sanity, to the situation is the inertia of the liquid, which was neglected in Washburn s model. The tube (or the porous medium) is connected to a vessel containing a liquid at rest, which resists sudden movements. Since this early phase of the process occurs for short times as well as for small heights (2 — 0), we may neglect in equation (5.39) both the viscous friction force Frf and the weight W. We are then left with 

The impregnation velocity defined by this equation is about 30 cm/s for 

We conclude this analysis with three remarks  

The constant velocity solution ultimately catches up to Washburn s diffusive solution. This defines a time r below which inertia prevails over viscosity. This time is given dimensionally by pR /ry and can approach 1 s for a millimeter-size tube and for water. The quantity r can also be interpreted as the time required for a viscous flow to establish itself in the tube. This happens as soon as the liquid enters the tube by way of diffusion of the viscous boundary layer (of thickness S). The progress of this layer obeys Prandtl s law S V Vp)- It therefore requires a time T for the layer to make its way from the wall to the center of the tube (S R). 

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That the most likely coarse velocity is equal to the most likely terminal velocity can only be true in two circumstances either the system began in the steady state and the most likely instantaneous velocity was constant throughout the interval, or else the system was initially in a dynamically disordered state, and x was large enough that the initial inertial regime was relatively negligible. These equations are evidently untrue for x —> 0, since in this limit the most... 

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. 

It can be seen from the tables and from Fig. 2.a. that the addition of small amount of acetonitrile in benzene speeds up the initial decay of the solvent response, usually assigned to inertial motions [6], On the other hand, the solvent response at times longer than 1 ps is at first only slightly modified by the presence of acetonitrile. Only large amount of acetonitrile lead to a faster decay of the solvent response function in this non-inertial regime. 

Low-frequency acquisition of the curves corresponds to a non-inertial regime wherein the mass of the cantilever does not play any role and the system can be treated as two springs in series. The in-phase and out-of-phase mechanical response of the cantilever in FMM-SFM was interpreted in terms of stiffness and damping properties of the sample, respectively [125,126]. This interpretation works rather good for compliant materials, but can be problematic for stiff samples. Assuming low damping, the cantilever response (Eqs. 9 and 10) below the resonance frequency (O0 for the case of is given by... 

Inertial regime. For low-viscosity liquids like molten metals, the use of the viscous friction model is questionable because inertial forces can become very important. For a sessile drop of mass ma, the inertial force is given by ... 

The dependence of the relative pressure drop on the porosity at otherwise equal conditions in the viscous and inertial flow regimes are shown in Figure 8.5, where AP0A is the pressure drop at eb = 0.4. All other variables in the equation remaining constant a change in the void fraction from 0.4 to 0.5 reduces the pressure drop more than 2.8 times in viscous flow regime and more than 2.3 times in the inertial regime. 

Figure 18. Phase diagram for stick slip motion with M = 1.2 kg. Stick slip occurs in region 1 and smooth sliding in region 2. The h ansition between the two is continuous in the creep regime (CR) and discontinuous in the inertial regime (Ini. With permission from Ref, 53. Phys. Rev. E 49, 4973 (1994).

One of them is much slower than the others, and the fastest one becomes Hbrational (i.e., it acquires a detectable imaginary part) in the inertial regime (wi = 0.5). Note, however, that since the second rank potential coupling provides two potential minima in which the solute can reorient (with the possibility of jump motions), the nature of the slow mode in... 

Also note that up to Pe = 1, the effective molecular diffusion dominates. Therefore, the very low Pe results given in Table 9.1 do not make any significant contribution to the total dif-fusivity. With this in mind, and for Pr characteristics of gases (<1), the significant dispersion contribution will be associated with Re > 1. This, in principle, requires inclusion of the inertial term in the momentum equation. However, as is shown later, the experimental results indicate that even for the inertial regime, Pe remains the only parameter on which D depends. 

Previous experimental studies of the dispersive stress used simple shear devices rather than turbulent pipe flows. Bagnold identified two regions of behavior the macroviscous and the inertial regimes, distinguished by a dimensionless group B, defined as... 

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