Non-slip boundary condition

Clearly then, the continuum approach as outlined above is faulty. Furthermore, since our erroneous result depends only on the non-slip boundary condition for V together with the Navier-Stokes equation (4.3), one or... 

The basis for the familiar non-slip boundary condition is a kinetic theory argument originally presented by Maxwell [23]. For a pure gas Maxwell showed that the tangential velocity v and its derivative nornial to a plane solid surface should be related by... 

Velocity images and profiles at several selected heights are shown in Figure 4.3.6, where the noisy points in the images indicate the air space where a liquid signal was not detected. When the fluid is inside the glass pipette, the velocity profile is nearly Poiseuille and a non-slip boundary condition is almost achieved. This is consistent with one of the early tube flow reports that the 0.5% w/v solution of... 

The formation of a surface wave in Fig. 1.8a requires the lateral displacement of liquid. Assuming a non-slip boundary condition at the substrate surface (lateral velocity v(z) = 0 at the surface (z = 0)), and the absence of normal stresses at the liquid surface, this implies a parabolic velocity profile (half-Poiseuille profile) in the film... 

Also note that the same profile will result if one of the non-slip boundary conditions is replaced by a symmetry condition at y = 0, namely duz/dy = 0. The mean velocity in the channel is obtained integrating the above equation,... 

The response of the QCM at the solid-liquid interface can be found by matching the stress and the velocity fields in the medium in contact. It is usually assumed that the relative velocity at the boundary between the liquid and the solid is zero. This corresponds to the non-slip boundary condition. Strong experimental evidence supports this assumption on the macroscopic scales [42,43]. hi this case the frequency shift, A, and the half-width of the... 

Although the non-slip boundary condition has been remarkably successful in reproducing the characteristics of liquid flow on the macroscopic scale, its application for a description of Hquid dynamics in microscopic liquid layers is questionable. A number of experimental [45-52] and theoretical [53,54] studies suggest the possibility of slippage at soUd-Hquid interfaces. Recent reviews [55-57] summarize the results of these works. Here we focus on the effect of slippage on the QCM response. 

The non-slip boundary condition is discussed in an excellent paper by Huh and Scriven They take note of the fact that, previous workers seem not to have been well informed by fluid mechanics , in aUuding to the essentially surface chemical analyses of spreading dynamics. Another point they address is that except for very smooth surfaces and non-adsorbing hquids the advancing or receding of the contact line proceeds in a slip-stick and discontinuous fashion a fact which is the focus of attention in current analyses of contact angle hysteresis using the theory of random fluctuations... 

Continuity of particle displacement across the solld/llquld interface requires that the surface displacement of the APM generate motion in the liquid. Solution of the Navler-Stokes equation in the liquid, subject to this non-slip boundary condition at the solid/li-quld interface, indicates that the liquid undergoes a shear motion which decays rapidly with distance from the surface (11). For an angular frequency of oscillation the velocity field decay length S of the liquid entrained by the plate mode is approximated by ... 

At the walls of the furnace the conventional non-slip boundary condition was used for momentum transfer, whereas constant heat fluxes, based on plant data, were prescribed at the walls for heat transfer. Details of the furnace boundary conditions for power and flow computations are provided in the following sub-sections. 

Since there was no current flow in the refractory walls, the magnetic vector potential. A, was set equal to zero. For heat transfer, a constant heat flux boundary condition, equal to the measured heat loss flux, was specified for this wall. The heat loss fluxes at the side wall for the water-cooled copper panels in Ae bullion and slag were 31.3 kW/m and 1.75 kW/m respectively. The conventional non-slip boundary condition was used for momentum transfer. 

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... 

The principal dimensions of the reactor are shown in Figure 6.1. The numerical values used here are D = 0.48 m //(= 1.02 m H, = 0.34 0 = 0.33 m s = 0.01 m r = 0.044125 m d = 0.0625 m d = 0.04 m and q = 0.07 m. Non-slip boundary conditions are assumed on the vessel wall. Both radial and axial velocities are set to zero on the shaft and impeller disk and the angular velocity is determined by the speed of rotation. On the free surface of the liquid, the axial component of velocity is zero with the other two components of velocity being stress free. Along the central line, below the impeller, the axial component of velocity is stress free and the other two components are zero. The temperature of the jacket at the vessel walls is fixed at 10 °C. Heat is lost by convection and at the free surface and there is an axis of symmetry along the centreline with no flux at the shaft and impeller boundaries. The flow is... 

As in the previous situations, we are considering hard anchoring and non-slip boundary conditions at the cylinders... 

We initiate our studies by deriving an axial momentum equation for the segment of the artery shown in Fig. 1. As in previous studies [8]-[10], we take the blood and the elastic vessel as a combined system and assume the non-slip boundary condition hence the complicated interaction between the blood and the elastic vessel can be treated as internal forces and need not be considered. Only forces acting on the surfaces in contact with the outside systems and the viscous force inside the blood will be included. 

Inamuro T, Yoshino M, Ogino F (1995) A non-slip boundary condition for lattice Boltzmann simulations. Phys Fluids 7(12) 2928-2930... 

One of the first approaches employed to impose a non-slip boundary condition at an external wall or at a moving object in a MFC solvent was to use ghost or wall particles [36,81]. In other mesoscale methods such as LB, no-slip conditions are modeled using the bounce-back rule the velocity of the particle is inverted from v to -V when it intersects a wall. For planar walls which coincide with the boundaries of the collision cells, the same procedure can be used in MFC. However, the walls will generally not coincide with, or even be parallel to, the cell walls. Furthermore, for small mean free paths, where a shift of the cell lattice is required to guarantee Galilean invariance, partially occupied boundary cells are unavoidable, even in the simplest flow geometries. 

---