Boundary conditions surface force balances

To close the system of equations for the fluid motion the tangential stress boundary condition and the force balance equation are used. The boundary condition for the balance of the surface excess linear momentum, see equations (8) and (9), takes into account the influence of the surface tension gradient, surface viscosity, and the electric part of the bulk pressure stress tensor. In the lubrication approximation the tangential stress boundary condition at the interface, using Eqs. (17) and (18), is simplified to... 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

The boundary conditions at the z=0 surface arise from the mechanical equilibrium, which implies that both the nonnal and tangential forces are balanced there. This leads to... 

The Smoluchowski-Levich approach discounts the effect of the hydrodynamic interactions and the London-van der Waals forces. This was done under the pretense that the increase in hydrodynamic drag when a particle approaches a surface, is exactly balanced by the attractive dispersion forces. Smoluchowski also assumed that particles are irreversibly captured when they approach the collector sufficiently close (the primary minimum distance 5m). This assumption leads to the perfect sink boundary condition at the collector surface i.e. cp 0 at h Sm. In the perfect sink model, the surface immobilizing reaction is assumed infinitely fast, and the primary minimum potential well is infinitely deep. 

Liquid-liquid interface At the interface between two immiscible liquids, the boundary conditions that must be satisfied are (a) a continuity of both the tangential and the normal velocities (this implies a no-slip condition at the interface), (b) a continuity of the shear stress, and (c) the balance of the difference in normal stress across the interface by the interfacial (surface) force. Thus the normal stresses are not continuous at the interface, but differ by an amount given in the following expression ... 

At the inlet to the finite element domain, the flow is parallel so the equations of the lubrication approximation are used to specify the inlet velocity profile. These equations are integrated from -oo to the inlet, generating an equation relating the flow rate to the inlet pressure. The remaining boundary conditions are as shown in Figure 3. The only complexity here is that the fluid traction, n T, at the free surface has to be specified as a boundary condition on the momentum equation. A force balance there gives, in dimensionless form,... 

To express the force equilibrium condition in a mathematical form, we can now consider a force balance on an arbitrary surface element of a fluid interface, which we denote as A. A sketch of this surface element is shown in Fig. 2-14, as seen when viewed along an axis that is normal to the interface at some arbitrary point within A. We do not imply that the interface is flat (though it could be) - indeed, we shall see that curvature of an interface almost always plays a critical role in the dynamics of two-fluid systems. We denote the unit normal to the interface at any point in A as n (to be definite, we may suppose that n is positive when pointing upward from the page in Fig. 2-14) and let t be the unit vector that is normal to the boundary curve C and tangent to the interface at each point (see... 

But the simple no-flow picture of equation 4 can no longer hold in view of equations 7a and 7b. At the liquid-vapor boundary, the viscous shear force must balance the force imposed by surface tension gradients, rjdu/dz = da/dx (z = h). This boundary condition leads to a linear flow profile toward the drying line,... 

We conclude this discussion by alerting the reader to the concept of the dynamic contact angle (and line), which appears in the literature of flows governed by surface tension (Dussan V. 1979). In a flow field where the contact line moves, it is necessary to know the contact angle as a boundary condition for determining the meniscus shape. If this angle is a function of the speed of the contact line relative to the solid surface, then the force balance inherent in... 

Clearly, then, we must know the concentration, temperature, and charge distributions at the interface in order to define the surface tension variation required to solve the hydrodynamic problem. However, these distributions are themselves coupled to the equations of conservation of mass, energy, and charge through the appropriate interfacial boundary conditions. The boundary conditions are obtained from the requirement that the forces at the interface must balance. This implies that the tangential shear stress must be continuous across the interface, and the net normal force component must balance the interfacial pressure difference due to surface tension. 

Thus the shear stress depends on the local surface tension gradient, in the absence of which Eq. (10.5.3) simply reduces to the usual fluid dynamic boundary condition that the tangential viscous stress is continuous at the interface of two different fluids. The normal force balance simply gives the scalar equation... 

The last boundary condition on the velocity comes from the balance between the change in surface tension due to temperature variations along the surface with the tractive force induced at the free surface. Here, as in Section 10.5, the rate of change of surface tension is taken to be linear with temperature... 

Thus, in order to solve the hydrodynamic problem of liquid motion in view of the change of 2 at the interface, we should first And out the distribution of substance concentration, temperature and electric charge over the surface. These distributions, in turn, are influenced by the distribution of hydrodynamic parameters. Therefore the solution of this problem requires utilization of conservation laws - the equations of mass, momentum, energy, and electric charge conservation with the appropriate boundary conditions that represent the balance of forces at the interface the equality of tangential forces and the jump in normal forces which equals the capillary pressure. In the case of Boussinesq model, it is necessary to know the surface viscosity of the layer. From now on, we are going to neglect the surface viscosity. 

Free surfaces are often encountered in extmsion processes, e.g. when the shape of the extrudate leaving the die is of interest. UnUke the boundaries discussed so far, the shape of a free surface boundary is not know a priori since it evolves as part of the solution. Thus two boundary conditions are necessary a kinematic condition that signifies that the particles at the free surface move with the local fluid velocity and a dynamic condition that assures force balance at the surface. The kinematic condition is expressed as... 

In the problems with free surfaces and interfaces, two different boundary ccardi-tions are implemented at the interface. One is the stress balance and the other is the kinematic condition. The stress balance at the interface between the liquid and its surrounding fluid is one of the main factors in the evolution of the liquid surface shape. This stress is governed by both the surface tension forces and the viscous forces. 

In the presence of an electric field, the surface charge density and polarization force density at the interface influence the interfacial force balance boundary condition, and the electric field can affect fluid flow. The interfacial stress balance boundary condition in the i direction on a surface whose normal, n, is in the j direction is... 

To derive the corresponding dynamic boundary condition, suppose that the surface patch S depicted in Fig. 2 separates two fluids (called fluid 1 and fluid 2) at some instant of time, with the unit vector n pointing into fluid 2. The balance of forces on the surface patch S thus takes the form ... 

Velocimetry. - An analytic model for the velocity field within a tubeless siphon (Fano flow) was presented. The model was based on a simple differential equation in which extensional, shear and gravitational pressure gradient forces are balanced. The role of surface tension in determining boundary conditions for the flow is considered. The analysis is applied to NMR velocimetry data (Xia and Callaghan, J. Magn. Reson., 2003, 16, 365) on a... 

The Smoluchowski equation applies when the double layer is thin enough or R is large enough such that the motion of the diffuse part of the double layer can be considered to be nniform and parallel to a flat surface. The flow is taken to be laminar i.e. infinitesimal layers of liquid flow past each other. Within each layer, the electrical and viscous forces are balanced. By balancing these forces and nsing Poisson s equation (Eq. 3.6) with suitable boundary conditions, it can be shown that the mobility has a form similar to the Hiickel equation, althongh the numerical prefactor is different ... 

The mobility tensor can be derived from Stokes-flow hydrodynamics. Consider a set of spherical particles, located at positions r, with radius a, surrounded by a fluid with shear viscosity rj. Each of the particles has a velocity v which, as a result of stick boundary conditions, is identical to the local fluid velocity on the particle surface. The resulting fluid motions generate hydrodynamic drag forces Ff, which at steady state are balanced by the conservative forces, Ff + F- = 0. The commonly used approximation scheme is a systematic multipole expansion, similar to the analogous expansion in electtostatics [17-21]. For details, we refer the reader to the original literature [17], where the contributions from rotational motion of the beads are also considered. As a result of the linearity of Stokes flow, the particle velocities and drag forces are linearly related,... 

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