Free surface boundary conditions

The velocity potential satisfies the Laplace equation and the free surface boundary condition on cf> is obtained by differentiating Eq. (10.4.14) with respect to t and eliminating d ldt from Eq. (10.4.16). For waves on deep water the second boundary condition on the potential is supplied by the requirement that there are no disturbances deep in the water, or that = constant as... 

The observed stabilizing effect of surfactants toward convection induced by surface tension has been confirmed theoretically in a recent paper by Berg and Acrivos (B13), in which the stability analysis technique and the physical model were the same as Pearson s except that the free-surface boundary condition [(iii) of Table III] took into account the presence of surface active agents. Critical values for the Thompson number were computed as functions of two dimensionless parameters, one embodying the surface viscosity and the other the surface elasticity. ... 

Figure 18. Surface coverage in units of the complete -s/s monolayer for N2 on graphite near the melting transition from Monte Carlo simulations. Triangles 256 molecules forming an isolated patch subject to free surface boundary conditions. Circles (squares) 16 (64) molecules in the NPTensemble with deformable periodic boundary conditions. Inverted triangles same as circles but obtained from a cooling run starting in the fluid phase. (Adapted from Fig. 1 of Ref. 301.)...

Following the fluid flow analogy, the most obvious choice for a free surface boundary condition is to allow the shear stresses to vanish and thereby force the shear rate to go to zero (i.e., du/dy = 0). Although this condition has been used in chute flow calculations (Savage, 1979 Campbell and Brenen, 1985 etc.) it forces the velocity profile to a... 

Combined Kinematic and D mamic Free Surface Boundary Condition (CKDFSBC) ... 

Domain mapping of the WMBVP follows the theory by Joseph. The physical fluid domain shown in Fig. 2.14 for the fully nonlinear WMBVP is mapped to a fixed computational fluid domain, and the discretized coupled free-surface boundary conditions are computed by an implicit Crank Nicholson (C N) method. s each iteration of the C-N method, the potential field is computed by the conjugate gradient method. The wavemaker motion E y/h,t) is assumed to be periodic with period T — 2tt/u , and the WMBVP with the surface tension f is given by... 

The velocity potential ip is the field variable, and the free surface y may be expressed in terms of p> by the free surface boundary conditions and the contact line conditions at the vertical sidewalls.h The velocity potential ip and the free surface displacement r] are assumed to be linear combinations of the progressive wavemaker wave and the parametrically excited cross wave given by the following ... 

Assuming incompressible fluid and irrotational flow motion, the velocity potential exists, which satisfies the Laplace equation. Linearizing the free-surface boundary conditions, the following boundary value problem for the velocity potential x,z,t) is obtained ... 

Linearizing the free-surface boundary conditions, the following boundary value... 

Irregular (nonplanar) free-surface boundary conditions are far more difficult to implement in FD methods. Typically, such methods require a much finer sampling of the wave field for accurate results (e.g., Robertsson 1996 Ohminato and Chouet 1997). 

The solution domain is shown in Figure 2. Velocity is fully specified at the inflow and outflow boundaries, Fi and Fo, respectively. Symmetry boundary conditions are imposed along the centerline, labeled Fj. Free surface boundary conditions are imposed on the edge of the domain, denoted Ff. The specific boundary conditions are as follows. 

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