Boundary conditions, fluid interface

In spite of this, it is perhaps useful to briefly consider the conditions at solid boundaries and fluid interfaces for complex/non-Newtonian fluids. One reason for doing this is that it provides additional emphasis to the idea from the proceeding paragraphs that there will be conditions when the commonly applied no-slip condition breaks down. It should be stated, at the outset, that the question of slip or no-slip is still a matter of current research interest for complex fluids. Nevertheless, the occurrence of sbp is generally accepted to be much more common for complex/non-Newtonian fluids than for Newtonian/small molecule liquids. In the latter case, we have seen that slip generally involves either extreme shear stresses or solid walls that exhibit extremely weak attractive interactions with the hquids, and the issue is primarily one of basic scientific interest. Polymer melts, on the other hand, commonly exhibit apparent manifestations of slip that play a critical role in the success or failure of certain types of commercial processing applications.43... 

The disadvantage of this method is that the two phases have to coexist in the same simulation cell, including two interfaces (Due to the periodic boundary conditions, the interfaces are oriented parallel to a surface of the cubic simulation box, generating the fluid in a slab-like configuration. Hence, one faces the problem of slow equilibration of interface fluctuations, and to distinguish bulk properties from those of the interface region [97, 98, 99,100]). 

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... 

In the previous sections, we have seen how computer simulations have contributed to our understanding of the microscopic structure of liquid crystals. By applying periodic boundary conditions preferably at constant pressure, a bulk fluid can be simulated free from any surface interactions. However, the surface properties of liquid crystals are significant in technological applications such as electro-optic displays. Liquid crystals also show a number of interesting features at surfaces which are not seen in the bulk phase and are of fundamental interest. In this final section, we describe recent simulations designed to study the interfacial properties of liquid crystals at various types of interface. First, however, it is appropriate to introduce some necessary terminology. 

The mapping (7) introduces the unknown interface shape explicitly into the equation set and fixes the boundary shapes. The shape function h(x,t) is viewed as an auxiliary function determined by an added condition at the melt/crystal interface. The Gibbs-Thomson condition is distinguished as this condition. This approach is similar to methods used for liquid/fluid interface problems that include interfacial tension (30) and preserves the inherent accuracy of the finite element approximation to the field equation (27)... 

In order to describe correctly the dynamic evolution of a fluid/fluid interface, a number of boundary conditions have to be implemented into the computational models. 

Condition (2) is also quite common. For instance, in crystals it results in a reduced sound velocity, v q) when q approaches a boundary of the Brillouin zone [93,96], a direct result of the periodicity of a crystal lattice. In addition, interaction between modes can lead to creation of soft mode with qi O and corresponding structural transitions [97,98]. The importance of nonlocality at fluid interfaces and the corresponding softening of surface modes has been demonstrated recently, both theoretically [99] and experimentally [100]. 

In the IBM, the presence of the solid boundary (fixed or moving) in the fluid can be represented by a virtual body force field -rp( ) applied on the computational grid at the vicinity of solid-flow interface. Considering the stability and efficiency in a 3-D simulation, the direct forcing scheme is adopted in this model. Details of this scheme are introduced in Section II.B. In this study, a new velocity interpolation method is developed based on the particle level-set function (p), which is shown in Fig. 20. At each time step of the simulation, the fluid-particle boundary condition (no-slip or free-slip) is imposed on the computational cells located in a small band across the particle surface. The thickness of this band can be chosen to be equal to 3A, where A is the mesh size (assuming a uniform mesh is used). If a grid point (like p and q in Fig. 20), where the velocity components of the control volume are defined, falls into this band, that is... 

Recent times have seen much discussion of the choice of hydrodynamic boundary conditions that can be employed in a description of the solid-liquid interface. For some time, the no-slip approximation was deemed acceptable and has constituted something of a dogma in many fields concerned with fluid mechanics. This assumption is based on observations made at a macroscopic level, where the mean free path of the hquid being considered is much smaller... 

Lu et al. [7] extended the mass-spring model of the interface to include a dashpot, modeling the interface as viscoelastic, as shown in Fig. 3. The continuous boundary conditions for displacement and shear stress were replaced by the equations of motion of contacting molecules. The interaction forces between the contacting molecules are modeled as a viscoelastic fluid, which results in a complex shear modulus for the interface, G = G + mG", where G is the storage modulus and G" is the loss modulus. G is a continuum molecular interaction between liquid and surface particles, representing the force between particles for a unit shear displacement. The authors also determined a relationship for the slip parameter Eq. (18) in terms of bulk and molecular parameters [7, 43] ... 

For solid particles a sufficient set of boundary conditions is provided by the no slip condition, the requirement of no flow across the particle surface, and the flow field remote from the particle. For fluid particles, additional boundary conditions are required since Eqs. (1-1) and (1-9) apply simultaneously to both phases. Two additional boundary conditions are provided by Newton s third law which requires that normal and shearing stresses be balanced at the interface separating the two fluids. 

The boundary conditions are the same as for steady motion considered in Chapters 1, 3, and 4, i.e., uniform flow remote from the particle, no slip and no normal flow at the particle boundary, and, for fluid particles, continuity of tangential stress at the interface. For a sphere the normal stress condition at the interface is again formally redundant, but indicates whether a fluid particle will remain spherical. 

A number of authors [46 to 48] employ the single sphere model in which the packed bed is considered as a set of equal spheres that are under the same state of extraction, and the fluid flowing around them is solute-free. That is, equation (3.4-90) would be valid, but without the generation term [46], The transport at the solid-fluid interface obeys the boundary condition (Eqn. 3.4-94) with C = 0 (fluid-flows at a large velocity). Under these assumptions, there is an analytical solution to the above problem (without axial dispersion) in terms of the Biot number (Bi = k, R/De), included in the following equation ... 

The definition of a solid state reaction implies that the reaction product is a solid. If, for example, one of the reactants is a fluid, no deviatoric stresses are transmitted across the common interface. This situation simplifies the mechanical boundary condition significantly and explains why studies on boundary morphology are often performed with solid/fluid systems. 

The same Navier dynamic boundary condition Eq. (1) and the subsequent expression Eq. 3 for the extrapolation length b can also be written down for non-Newtonian and polymeric fluids, where r is the shear viscosity and 11 is the local viscosity at the interface. The expression Eq. (2b) for 3 is equally valid for poly-... 

The other set of boundary condition is applied at the interface, which is the no-fluid through the interface condition i.e. 

An important class of fluid flow problems involves free surface flows where only the flow phenomena in a single phase are of importance despite the fact that two or more phases are present in the system. Such situations are encountered, for example, in gas-liquid two-phase flows where, due to the large density difference, from a hydrodynamic point of view the flow can be treated as a liquid flow in a vacuum. The analysis of such free surface flows is rather complex due to the fact that in addition to the fluid flow problem the position of the interface and enforcement of appropriate boundary conditions have to be dealt with. 

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