Boundary conditions dynamic

Zaika Yu.V. (1996) The solvability of the equations for a model of gas transfer through membranes with dynamic boundary conditions, Computer Mathematics and Math. Physics 36(12), 1731-1741. 

Zaika Yu.V. (2004) Identification of a hydrogen transfer model with dynamical boundary conditions, Int. J. of Mathematics and Math. Sciences 4, 195-216. 

Zaika Yu.V. (2001) Parametric regularization of hydrogen permeability model with dynamic boundary conditions, Mathematical modeling, 13(11), 69-87 (in Russian). 

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-... 

In summary, we have so far seen that there are two types of boundary conditions that apply at any solid surface or fluid interface the kinematic condition, (2-117), deriving from mass conservation and the dynamic boundary condition, normally in the form of (2-122), but sometimes also in the form of a Navier-slip condition, (2-124) or (2-125). When the boundary surface is a solid wall, then u is known and the conditions (2-117) and (2-122) provide a sufficient number of boundary conditions, along with conditions at other boundaries, to completely determine a solution to the equations of motion and continuity when the fluid can be treated as Newtonian. 

When a bounding surface is a fluid fluid interface instead of the surface of a solid, the kinematic and dynamic boundary conditions can be seen, from (2 112) and (2-122), to provide either two (or three) independent relationships between the unknown velocity vectors, u and u. However, there are a total of either four or six unknown components of u and u (the number depending on whether the flow is 2D or frilly 3D), and thus additional conditions must be imposed at an interface to completely specify the solutions of the Navier-Stokes and continuity equations. In this section, we assume that there is no phase change at the interface. 

The fluid dynamical boundary conditions are similar to those applied in the previous problem, with two notable exceptions. First, u = 0 at z = 0 (i.e., the lower boundary is stationary). Second, the tangential-stress condition is modified to account for the presence of Marangoni stresses that are due to gradients of the interfacial tension at the fluid interface,... 

The linearized dynamic boundary condition is physically the same as given by Eq. (10.4.7) for the plane surface wave, which with allowance for the cylindrical symmetry of the jet problem may, from the Young-Laplace equation, be written... 

The excess pressure is then found from the dynamic boundary condition Eq. (10.4.26). Using the unsteady Bernoulli equation + d(f)ldt - 0, we can eliminate the unknown constant A to give the eigenvalue relation... 

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... 

Unlike creeping flow about a solid sphere, the r9 component of the rate-of-strain tensor vanishes at the gas-liquid interface, as expected for zero shear, but the simple velocity gradient (dvg/dr)r R is not zero. The fluid dynamics boundary conditions require that [(Sy/dt)rg]r=R = 0- The leading term in the polynomial expansion for vg, given by (11-126), is most important for flow around a bubble, but this term vanishes for a no-slip interface when the solid sphere is stationary. For creeping flow around a gas bubble, the tangential velocity component within the mass transfer boundary layer is approximated as... 

The presence of droplets also introduces new kinematic and dynamic boundary conditions on the fluid flow. Since the immiscible fluids cannot cross the interface, boundary condition states that the local normal component of the velocities in each fluid must be equal to the interface velocity, the velocity tangents to the interface must be also equal inside and outside the droplet, and the tangential shear stresses must be balanced at the interface when it is clean of surfactants. 

Dynamic boundary condition needed to solve (5.16) is given by the balance of normal forces,... 

The kinematics and dynamics boundary conditions at the interfaces close the hydrodynamic problem (l)-(2). On the solid-liquid boundary the non-slip boundary conditions are applied -the liquid velocity close to the particle boundary is equal to the velocity of particle motion. In the case of pure liquid phases the non-slip boundary condition is replaced by the dynamic boundary condition. The tangential hydrodynamic forces of the contiguous bulk phases, nx(P+Pb) n, are equal from both sides of the interface, where n is the unit normal of the mathematical dividing surface. The capillary pressure compensates the difference between the... 

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. 

To derive the corresponding dynamic boundary condition, suppose that the surface patch 5 depicted in Fig. 2 separates two fluids (called fluid 1 and fluid 2) at some instant of time, with the... 

The covariance of the interface can be related to the other quantities in the mixture by using the dynamic boundary condition on the interface. This condition gives for the Fourier-Stieltjes component of the fluctuation in the pressure of the mixture,... 

The dynamic boundary condition that governs the forces balance at the interface, is ... 

The Maxwell boundary condition [Eq. (122)] also gives rise to hydro-dynamic boundary conditions of the form (125) with o>, t, and x roughly prop)ortional to So, as long as a is of order unity (more specifically a IJ/h) the corrections to stick boundary conditions remain small, and only if a becomes of order boundary conditions differing appreciably from stick are obtained.This is an indication why stick boundary conditions are for most purposes a very good approximation in hydrodynamic theory a reflection mechanism that is almost specular is not very likely to occur in nature, due to irregularities in surface structures and thermal motion of the surface molecules. 

One can use the dynamic boundary conditions of Eqs. (12)-(13) without considering the influence of the osmotic pressure and the Gibbs adsorption (the case of significant contribution of these values to the interfacial balance of forces is analyzed in Refs. [4,19]) ... 

Molecular Dynamics Boundary Conditions for Regular Crystal Lattices. 

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