Creeping motion

The first part of Eq. (89), proportional to the inverse viscosity r] of the liquid film, describes a creeping motion of a thin film flow on the surface. In the (almost) dry area the contributions of both terms to the total flow and evaporation of material can basically be neglected. Inside the wet area we can, to lowest order, linearize h = hoo[ + u x,y)], where u is now a small deviation from the asymptotic equilibrium value for h p) in the liquid. Since Vh (p) = 0 the only surviving terms are linear in u and its spatial derivatives Vw and Au. Therefore, inside the wet area, the evolution equation for the variable part u of the height variable h becomes... 

X. Feng, E. A. Brener, D. E. Temkin, Y. Saito, H. Miiller-Krumbhaar, Creep motion of a solidification front in a two-dimensional binary alloy. Phys Rev E (to appear) X. Feng, E. A. Brener, D. E. Temkin, Y. Saito, H. Muller-... 

Fig. 19a-c. Schematic representation of a reptating chain in different time regimes a Short-time unrestricted Rouse motion b equilibration of density fluctuations along tha chain c creep motion of a chain out of its tube. 

De Gennes [61] and Doi and Edwards [6] have formulated a tractable analytic expression for the dynamic structure factor. They neglected the initial Rouse regime, i.e. the derived expression is valid for t>r once confinement effects become important. The dynamic structure factor is composed from two contributions S ° (Q,t) and S (Q,t) reflecting local reptation and escape processes (creep motion) from the tube ... 

Structure factor for creep motion within the reptation model... 

Values from tables of friction coefficients always have to be used with caution, since the experimental results not only depend on the materials but also on surface preparation, which is often not well characterized. In the case of plastic deformation, the static coefficient of friction may depend on contact time. Creeping motion due to thermally activated processes leads to an increase in the true contact area and hence the friction coefficient with time. This can often be described by a logarithmic time dependence... 

A. Nir and A. Acrivos, On the Creeping Motion of Two Arbitrary-sized Touching Spheres in a Linear Shear Field, J. Fluid Mech., 59, 209-223 (1973). 

H. Westborg and O. Hassager, Creeping Motion of Long Bubbles and Drops in Capillary Tubes, J. Colloid Interface Sci., 133, 135-147 (1989). 

In the case of creeping motion, it is necessary a) to transform the mass-related heat capacity Cp in n3 into a volume-related one (pCp) and b) g and p can only occur as gravity gp. These two requirements have been fulfilled when Gr has been multiplied with Pr and II x has been combined with n3 (= Pr-1) to give the Fourier number Fo ... 

To this point we have limited onr consideration to mass diffitsion in a station aiy medium, and thus the only ntotion involved was the creeping motion of molecules in the direction of decreasing concentration, and there was no motion of the mixture as a whole. Many practical problems, such as the evaporation of water from a lake under the iiifliience of the wind or the mixing of two fluids as they flow in a pipe, involve diffusion in a moving medium where the hoik motion i.s caused by an external force. Mass diffusion in such c.nses is complicated by the fact that chemical species are transported both by diffusion and by the bulk motion of the medium (i.e., convection). The velocities and mass flow rates of species in a moving medium consist of two components one due to molecular diffusion and one due to convection (Fig. 14-29). 

The observation that a macromolecular brush gets stretched as the side chains get adsorbed on a flat surface provides a means to stimulate molecular motility by desorption of the brush molecule or a segment of it. If the molecule is in a subsequent period allowed to relax to the adsorbed stretched state it will eventually do a step forward. This is depicted schematically in Figure 28 as a sort of a creep motion. Here, the desorbed state might be characterized as an excited state whose formation requires input of energy. In the case that the structure of the surface and of the molecule favor relaxation into a distinct direction, i.e., in the case of an asymmetric potential, the motion of the molecule can be become directed. 

Figure 28. Creep motion by contraction upon desorption and subsequent stretching upon adsorption of an abstract molecular object.

The same equation can be derived within a simple Kramers picture [55,80,86] for the escape from a well (locked state), assuming that the pulling force produces a small constant potential bias that reduces a height of a potential barrier. The progressive increase of the force results in a corresponding increase of the escape rate that leads to a creep motion of the atom. However, this behavior is different from what occurs when the atom (or an AFM tip) is driven across the surface and the potential bias is continuously ramped up as the support is moved [87,88]. The consequences of this effect will be discussed in more detail in Section III.B. 2. 

Due to its great importance in reactor simulations, a brief survey of the main steps involved in Stokes solution of the Navier-Stokes equation for creeping motion about a smooth immersed rigid sphere is provided. The details of the derivation is not repeated in this book as this task is explained very well in many textbooks [169, 14, 103, 15]. 

Dagan, Z., Weinbaum, S., and Pfeffer, R., An infinite-series solution for the creeping motion through an orifice of finite length. J. Fluid Meeh. 1982 115, 505-523. 

However, this is just the Reynolds number that, according to the creeping-motion assumption, is arbitrarily small. In addition, in this case, S = 1 and Re/S = Re 1. It follows that the velocity and pressure fields adjust instantaneously relative to the rate at which the geometry of the flow domain changes and therefore always appear to be at steady state with respect to the present configuration. Thus time appears in a creeping-flow solution only as a parameter that characterizes the instantaneous boundary velocity, or boundary geometry, either of which may depend on time. 

We have seen that the Navier-Stokes and continuity equations reduce, in the creeping-motion limit, to a set of coupled but linear, PDEs for the velocity and pressure, u andp. Because of the linearity of these equations, a number of the classical solution methods can be utilized. In the next three sections we consider the general class of 2D and axisymmetric creeping flows. For this class of flows, it is possible to achieve a considerable simplification of the mathematical problem by combining the creeping-flow and continuity equations to produce a single higher-order DE. 

We saw, in the previous section, that problems of creeping-motion in two dimensions can be reduced to the solution of the biharmonic equation, (7 46), subject to appropriate boundary conditions. To actually obtain a solution, it is convenient to express (7 46) as a coupled pair of second-order PDEs ... 

The general solution (7-71) can be applied to examine 2D flows in the region between two plane boundaries that intersect at a sharp corner. This class of creeping motion problems was considered in a classic paper by Moffatt,11 and our discussion is similar to that given by Moffatt. A typical configuration is shown in Fig. 7-5 for the case in which one boundary at 6 = 0 is moving with constant velocity U in its own plane and the other at 6 = a is stationary. 

We saw in Section C that the creeping-motion and continuity equations for axisymmetric, incompressible flow can be reduced to the single fourth-order PDE for the streamfunction,... 

In this section, we begin by considering the buoyancy-driven motion of a single gas bubble or drop through an otherwise stationary viscous fluid under the assumption that the bubble or drop shape is nearly spherical. We denote the viscosities and densities of the two fluids as /x, /x, p, and p with the variables with carets corresponding to the fluid inside the drop. In this section, we also assume that the interfacial tension, which we denote as y, is uniform at the drop surface, and that the Reynolds numbers for both the interior and exterior flows are sufficiently small that the creeping-motion approximation can be applied for both fluids. Under these circumstances, experimental evidence shows (and we will assume) that the drop or bubble will translate with a constant velocity U. In addition, though we must consider the shape of the drop to be unknown, we may also anticipate that it will be axisymmetric about an axis that is collinear with the velocity vector U. 

S. Wakiya, Application of bipolar co-ordinates to the two-dimensional creeping-motion of a liquid. I. Flow over a projection or a depression on a wall, J. Phys. Soc. Jpn. 39, 111 3-20 (1975) M. E. O Neill, On the separation of a slow linear shear flow from a cylindrical ridge or trough in a plane, Z. Angew. Math. Phys. 28,438-48 (1977) A. M. J. Davis and M. E. O Neill, Separation in a Stokes flow past a phase with a cylindrical ridge or trough, Q. J. Mech. Appl. Math. 30, 355-68 (1977). 

Problem 7-9. Motion of a Force- and Torque-Free Axisymmetric Particle in a General Linear Flow. We consider a force- and torque-free axisymmetric particle whose geometry can be characterized by a single vector d immersed in a general linear flow, which takes the form far from the particle y°°(r) = U00 + r A fl00 + r E00, where U°°, il, and Ex are constants. Note that E00 is the symmetric rate-of-strain tensor and il is the vorticity vector, both defined in terms of the undisturbed flow. The Reynolds number for the particle motion is small so that the creeping-motion approximation can be applied. 

Problem 7-12. Bubble in an Axisymmetric Flow. A gas bubble is immersed in a viscous Newtonian fluid that is undergoing an axisymmetric extensional flow. The fluid is viscous enough that the relevant Reynolds number is small so that the creeping-motion approximation can be applied. The capillary number based on the extension rate, E, and the surface tension, a, is small, i.e.,... 

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