Nonslip boundary condition

The surface boundary conditions are critical to modeling electrokinetic phenomena in nanofluidics. For the hydrodynamic boundary condition, we use the nonslip model at the silica surfaces. Although the slip boundaries have been adopted and have shown significant effects to improve the energy-conversirai efficiency, a careful molecular study showed that the hydro-dynamic boundary conditirm, slip or not, depended on the molecular interactions between fluid and solid and the channel size. For the dilute solution in silica nanochannels considered in this work(/x 2 nm), the nonslip boundary condition is still valid very well. 

The response of the QCM at the solid/liquid interface can be found by matching the stress and the velocity fields in the media 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 nonslip boundary condition. Strong experimental evidence supports this assumption on the macroscopic scales [56,57]. In this case the frequency shift. A/, and the width of the resonance, F/, can be written as follows [10,11] ... 

Although the nonslip boundary condition has been remarkably successful in reproducing the characteristics of liquid flow on the macroscopic scale, its application for a description of liquid dynamics in microscopic liquid layers is questionable. A number of experimental [62-64] and theoretical [65,66] studies suggest the possibiility of slippage at solid/liquid interfaces. 

A, is usually introduced [65,67,68]. The traditional nonslip boundary condition is replaced by... 

The prefactor in this equation is for the nonslip boundary condition, where the velocity of the particle is the same as that of the solvent at its surface. The basic input in the derivation of the above Stokes law is how the velocity field in the fluid is modified by a surface element of the particle and how this modified field gets further modified by other surface elements, while the surface elements remain structurally correlated in size and shape (Figure 6.4). It may be equivalently stated that the Stokes law is a consequence of full hydrodynamic coupling among all surface elements of the sphere arising Ifom velocity... 

In the case of basic hydrophobic surfaces, the water drop falls across the dirt particles and the dirt particles are mainly displaced to the sides of the droplet and re-deposited behind the droplet, this may be due to the nonslip boundary conditions. Hydrophobic particles especially tend to remain on such surfaces (Figure 17.12, left). In the case of water-repellent rough surfaces, the solid/water interface is minimized and water forms a spherical droplet that collects the particles from the surface (Figure 17.12, right). This is called as the lotus effect. ... 

Integration of Equation A6.4 and Equation A6.5 using (1) a nonslip boundary condition on the capillary surface at r = R, (2) a symmetry condition in the capillary center at r = 0, and (3) equality of velocities and tangential stresses at the liquid-liquid interface at r = / - /t, we can deduce expressions for axial velocities, in both liquids 1 and 2. Substitution of those expressions in Equation A6.8 and Equation A6.9 results in... 

The boundary conditions are a nonslip boundary condition on the fiber... 

The boundary conditions at x = 1 are replaced by those of the nonslip for the velocity and by the transport conditions at the electrically inhomogeneous surface for the electrolyte concentration and the electric potential of the form... 

The velocity boundary conditions are simple, due to the rigid, impermeable tube walls and to nonslip velocity at the wall. These conditions can be expressed as... 

Stokes law is based on the solution of equations of continuum fluid mechanics and therefore is applicable to the limit Kn —> 0. The nonslip condition used as a boundary condition is not applicable for high Kn values. When the particle diameter Dp approaches the same magnitude as the mean free path X of the suspending fluid (e.g., air), the drag force exerted by the fluid is smaller than that predicted by Stokes law. To account for... 

The boundary condition implementations play a very critical role in the accuracy of the numerical simulations. The hydrodynamic boundary conditions for the LBM have been smdied extensively. The conventional bounce-back rule is the most popular method used to treat the velocity boundary condition at the solid-fluid interface due to its easy implementation, where momentum from an incoming fluid particle is bounced back in the opposite direction as it hits the wall [20]. However, the conventional bounce-back rale has two main disadvantages. First, it requires the dimensionless relaxation time to be strictly within the range (0.5, 2) otherwise, the prediction will deviate from the correct result. Second, the nonslip boundary implemented by the conventional bounce-back rule is not located exactly on the boundary nodes, as mentioned before, which will lead to inconsistence when coupling with other partial differential equation (PDF) solvers on a same grid set [17]. 

Finally, the appropriate boundary conditions are the standard nonslip conditions at the channel walls and periodicity conditions along the homogeneous directions ... 

Imagine two parallel plates of area A between which is sandwiched a liquid of viscosity r). If a force F parallel to the x direction is applied to one of these plates, it will move in the x direction as shown in Figure 4.1. Our concern is the description of the velocity of the fluid enclosed between the two plates. In order to do this, it is convenient to visualize the fluid as consisting of a set of layers stacked parallel to the boundary plates. At the boundaries, those layers in contact with the plates are assumed to possess the same velocities as the plates themselves that is, v = 0 at the lower plate and equals the velocity of the moving plate at that surface. This is the nonslip condition that we described in Chapter 2, Section 2.3. Intervening layers have intermediate velocities. This condition is known as laminar flow and is limited to low velocities. At higher velocities, turbulence sets in, but we do not worry about this complication. 

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