Boundary conditions at a fluid interface

This is not to say that there are no unresolved issues in formulating the basic principals for a continuum description of fluid motions. Effective descriptions of the constitutive behavior of almost all complex, viscoelastic fluids are still an important fundamental research problem. The same is true of the boundary conditions at a fluid interface in the presence of surfactants, and effective methods to make the transition from a pure continuum description to one which takes account of the molecular character of the fluid in regions of very small scale is still largely an open problem. 

N. The Role of Surfactants in the Boundary Conditions at a Fluid Interface... 

N. THE ROLE OF SURFACTANTS IN THE BOUNDARY CONDITIONS AT A FLUID INTERFACE... 

Inside the drop, we require that the velocity and pressure fields be bounded at the origin [which is a singular point for the spherical coordinate system that we will use to solve (7 199)]. Finally, at the drop surface, we must apply the general boundary conditions at a fluid interface from Section L of Chap. 2. However, a complication in using these boundary conditions is that the drop shape is actually unknown (and, thus, so too are the unit normal and tangent vectors n and t and the interface curvature V n). As already noted, we can expect to solve this problem analytically only in circumstances when the shape of the drop is approximately (or exactly) spherical, and, in this case, we can use the method of domain perturbations that was first introduced in Chap. 4. In this procedure, we assume that the shape is nearly spherical, and develop an asymptotic solution that has the solution for a sphere as the first approximation. An obvious question in this case is this When may we expect the shape to actually be approximately spherical ... 

In this contribution, first a number of fundamental concepts that are central to interface capturing are presented, including definitions of level set functions and unit normal and curvature at an interface. This is followed by consideration of kinematic and dynamic boundary conditions at a sharp interface separating two immiscible fluids and various ways of incorporating those conditions into a continuum, whole-domain formulation of the equations of motion. Next, the volume-of-fluid (VOE) and level set methods are presented, followed by a brief outlook on future directions of research and other interface capturing/tracking methods such as the diffuse interface model and front tracking. 

Ross, S.M. Theoretical model of the boundary condition at a fluid-porous interface. AIChE J. 1983, 29, 840-846. 

J. L. Barrat and L. Bocquet, Influence of wetting properties on hydrodynamic boundary conditions at a fluid/solid interface, Faraday Discuss., 112,119 [1999],... 

A similar solution for creeping flow past a spherical droplet of fluid were derived independently by Hadamard [55] and Rybczynski [124], In this case the fluid stream has a velocity V at infinity and viscosity p/, while the droplet has a viscosity /j,p and a fixed interface. The boundary conditions at the droplet interface are (1) zero radial velocities and (2) equality of surface shear and... 

An arbitrary disturbance form in the x and y directions could be expressed as a sum of the Fourier modes of wave number a x and a y, but because the governing equations are linear with coefficients that are independent of x, y, it is enough to consider the stability of these disturbance quantities one mode at a time, for arbitrary values of a x and a y. The fimctions of z must be chosen to satisfy boundary conditions on the fluid interface. The stability is determined by the sign of the real part of a. The reader is reminded that the primes on all of the symbols mean that they are dimensional. 

Summary of Results for Creeping Viscous Flow Around a Gas Bubble. The shortcut method described above and boundary conditions at a gas-liquid interface are useful to analyze creeping flow of an incompressible Newtonian fluid... 

The solution for n = 1 must be discarded because the fluid is stagnant at large r. Hence, A = 0. The boundary condition at the fluid-sohd interface yields B = QR. The creeping viscous flow solution is... 

The dimensionless coefficient, accounts for the change in the hydrodynamic friction between the fluid and the particles (created by the hydrodynamic interactions between the particles). The dimensionless surface mobility coefficient, takes into account the variation of the friction of a molecule in the adsorption layer. The diffusion problem, Eqs. (4) and (5), is connected with the hydrodynamic problem, Eqs. (1) and (2), through the boundary conditions at the material interface. 

Pulsatile flow in an elastic vessel is very complex, since the tube is able to undergo local deformations in both longitudinal and circumferential directions. The unsteady component of the pulsatile flow is assumed to be induced by propagation of small waves in a pressurized elastic tube. The mathematical approach is based on the classical model for the fluid-structure interaction problem, which describes the dynamic equilibrium between the fluid and the tube thin wall (Womersley, 1955b Atabek and Lew, 1966). The dynamic equilibrium is expressed by the hydrodynamic equations (Navier-Stokes) for the incompressible fluid flow and the equations of motion for the wall of an elastic tube, which are coupled together by the boundary conditions at the fluid-wall interface. The motion of the liquid is described in a fixed laboratory coordinate system (f , 6, f), and the dynamic... 

In the continuum model (Kn<0.001), the following assumptions can be made (1) a linear relation between stress and strain, (2) no-slip boundary condition at the fluid-soUd interface, (3) linear relation between heat flux and temperature and (4) no-jump condition of temperature at the fluid-solid interface. If the mean free path is not much smaller than the characteristic length, the flow is not near equilibrium and the above assumptions are no longer valid. 

Jager, W. and Mikelic, A., On the boundary condition at the contact interface between a porous medium and free fluid, 1995, to appear... 

It is well known that the continuum theory in the Navier-Stokes equations only validates when the mean free path of the molecules is smaller than the characteristic length scale of the gas flow. Otherwise, the fluid will no longer be in thermodynamic equilibrium and the linear relationship between the shear stress and rate of shear strain cannot be applied. The commonly used no-slip boundary condition at the fluid-solid interface is not fully valid, and a slip length has to be introduced. 

Chandesris, M., Jamet, D., 2006. Boundary conditions at a planar fluid-porous interface for a Poiseuille flow. Int J. Heat Mass Transf. 49,2137 2150. 

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... 

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