Boundary conditions interface deformation

Fig. 12.22 Velocity of the polymer in the vicinity of the wall as it makes the transition from inside to outside the capillary die. For x < 0 (inside the die), measurements were made of slow-moving particles, that is, those nearest the wall. Forx > 0 (outside the die), the measurements were made at the air-polymer interface. The flow rate at which the onset of sharkskin is observed is noted, (a) Without the polymer process additive. For the higher flow rates, we show the velocity of both the core and the surface regions, (b) With the polymer process additive. [Reprinted by permission from K. B. Migler, Extensional Deformation, Cohesive Failure, and Boundary Conditions during Sharkskin Melt Fracture, J. Rheol., 46, 383-400 (2002).]...

A model of deformed evaporating liquid film, which is moved hy co-current vapor flow and gravity, is developed. Shear stress from co-current vapor is included as a boundary condition on interface. Intermolecular forces are taking into account as disjoining pressure component and surface roughness is considered also. 

Combining the general equation of films with deformable interfaces (Equation 5.255), the mass balance (Equations 5.276 and 5.277), and the boundary condition for the interfacial stresses (Equation 5.281), we can derive ... 

Equation (6-148), plus the boundary conditions (6 142) and the integral constraint (6 143), is sufficient to determine h(x). We should note that we do not necessarily expect Eq. (6-148) to hold all the way to the end walls atx = 0 andx = 1, for it was derived by means of the governing equation, (6-119), (6-120) and (6-137), and these are valid only for the core region of the shallow cavity. Nevertheless, we will at least temporarily ignore this fact and integrate (6-148) over the whole domain, with the promise to return to this issue later. Qualitatively, we can see that the interface deformation is determined by a balance between the nonuniform pressure associated with the flow in the cavity, e g., Eq. (6 145), which tends to deform the interface, and the effects of capillary and gravitational forces, both of which tend to maintain the interface in its flat, undeformed state, i.e., h = 1. 

Closely related to the Taylor problem is the situation sketched in Fig. 7-6, when a flat plate is drawn into a viscous fluid through a free surface (that is, an interface). In reality, of course, the interface will tend to deform as a result of the motion of the plate, but we assume here that the interface remains flat. Then the problem is identical to the previous Taylor problem except for the boundary conditions, which now become... 

Derive dimensionless equations and boundary conditions whose solution would be sufficient to determine the drop velocity (and shape) to 0(8). Use the method of domain perturbations to express all boundary conditions at the deformed drop interface in terms of equivalent conditions at the spherical surface of the undeformed drop. Show that 5 = Ca. 

Finally, we have to apply the boundary conditions at the interface. These conditions are strictly applied at the deformed interface z = eh. However, the domain perturbation argument from the Rayleigh-Taylor section shows that the boundary conditions for the linearized disturbance problem can equally well be applied at the unperturbed surface, z = 0. 

This boundary condition, given by (8-220), reveals that A = Vapproach- The non-deformable nature of the bubble requires that Vr must vanish at the gas-liquid interface. This boundary condition atr = R translates into... 

In order for this approach to be effective, it is necessary to ensure that the compressive and/or shear strains to which the simulation cell is subjected are transmitted to the atoms within the cell. Indeed, it is possible in principle to deform the cell in arbitrary ways without altering the positions of the atoms at all and stiU obtain a perfectly suitable periodically repeated system. To ensure the atoms in the cell move in conjunction with the lattice vectors, it is common practice to represent the atomic positions in fractional coordinates. This approach also ensures that the Lees-Edwards boundary conditions [124] are satisfied to ensure that artificial slip planes are not introduced at the interface between each periodically repeated cell. 

For the low-deformation bubble simulations where the boundary-fitted method is most widely used, it is assumed that Reynolds number (Re = pUDIIfi) is low, the gas flow inside bubbles is neglected, and a constant gas pressure assumed. The following boundary conditions are applied at the 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... 

The underlying idea behind the stability analysis is that all possible initial perturbations z(x) consistent with system boundary conditions may occur. If all of these perturbations diminish in amplitude with time, the system is stable. But if any permrbation grows, instability occurs and the interface never remms to its initial flat configuration. It is clear from Equation 5.2 that if the system is stable with respect to all initial interfacial deformations having the forms Zia and Z2a of individual Fourier components, it is stable with respect to a general deformation. But if it is unstable with respect to even one Fourier component, interfacial deformation can be expected to increase continuously with time, and there is no return to the initial state. 

Taylor and Acrivos (1964) made the solution satisfy the boundary conditions on the surface r = a, but satisfied the normal stress boundary condition on the surface of a slightly deformed sphere a(l + < )) by using a Taylor series expansion to express any functiony(r) at the interface as j a) + Since... 

At high temperature, creep is the main mechanism of plastic deformation. It involves the diffusion of vacancies in the oxide or in the substrate. The vacancies preferentially coalesce at grain boundaries or at the oxide-metal interface. As a result of coalescence, microscopic cavities form that cause a weakening of the cohesion of the oxide at the grain boundaries or of the adhesion at the oxide-metal interface. This in turn facilitates cracking or spalling. Under these conditions, plastic deformation itself can thus be a source of deterioration of oxide films. 

Note that the term [1 — MaC(0 — C)] tends to avoid the vertical displacement (deformation) of the interface. For the temperature field, the boundary condition at the free surface, eq. (23), becomes... 

The total number of elements (Quad4, MSC Software Corp.) [29] used for each model was 400. The symmetry boundary condition was applied to the vertical and horizontal hnes, and fixed boundary condition to the outside edge. As shown in Rgure 9.10, each IPMC diaphragm consists of the IPMC part and a Nation part. Therefore, when a voltage is apphed on an IPMC part, the vertical interface between IPMC and Nation can rotate easily to produce large bending deformation, since Nafion has a low elastic modulus. 

---