Boundary no-slip

Wu and Cheng (2003) measured the friction factor of laminar flow of de-ionized water in smooth silicon micro-channels of trapezoidal cross-section with hydraulic diameters in the range of 25.9 to 291.0 pm. The experimental data were found to be in agreement within 11% with an existing theoretical solution for an incompressible, fully developed, laminar flow in trapezoidal channels under the no-slip boundary condition. It is confirmed that Navier-Stokes equations are still valid for the laminar flow of de-ionized water in smooth micro-channels having hydraulic diameter as small as 25.9 pm. For smooth channels with larger hydraulic diameters of 103.4-103.4-291.0pm, transition from laminar to turbulent flow occurred at Re = 1,500-2,000. 

If the range of the channel height is limited to be above 10 pm, then the no-slip boundary condition can be adopted. Furthermore, with the assumptions of uniform inlet velocity, pressure, density, and specified pressure Pout at the outlet, the boundary conditions can be expressed as follows ... 

The azimuthal component of the fluid velocity, v, is identical to vx and the local fluid angular velocity is co = v /q. This azimuthal velocity is y sin0oo(0) = qoo(0) and the local shear rate, y, is -sin0 which, for no slip boundary conditions, is Q/a for small a. Under uniform shear with no slip, it may be shown that dvx/dy 0 and dvx/dy y[2, 17]. 

Considering a surface temperature which is higher than the Leidenfrost temperature of the liquid in this study, it is assumed that there exists a microscale vapor layer which prevents a direct contact of the droplet and the surface. Similar to Fujimoto and Hatta (1996), the no-slip boundary condition is adopted at the solid surface during the droplet-spreading process and the free-slip... 

Applying Immersed or Embedded Boundary Methods (Mittal and Iaccarino, 2005) circumvents the whole issue of the friction between the more or less steady overall flow in the bulk of the vessel and the strongly transient character of the flow in the zone of the impeller. These methods are introduced below. In the context of a LES, Derksen and Van den Akker (1999) introduced a forcing technique for both the stationary vessel wall and the revolving impeller. They imposed no-slip boundary conditions at the revolving impeller and at the stationary tank wall (including baffles). To this purpose, they developed a specific control algorithm. 

A no-slip boundary condition is used on all impermeable solid surfaces, but the choice of boundary conditions for the inlet and outlet of the model is not so... 

There are two constants of integration in equation 1.56 so two boundary conditions are required. The first is the no-slip condition at r = r, and the second is that the velocity gradient is zero at r = 0. Using the latter condition in equation 1.55 shows that A2 = 0 so that equation 1.56 becomes identical to equation 1.53. The no-slip boundary condition gives the value of B as before. 

For potential flow, ie incompressible, irrotational flow, the velocity field can be found by solving Laplace s equation for the velocity potential then differentiating the potential to find the velocity components. Use of Bernoulli s equation then allows the pressure distribution to be determined. It should be noted that the no-slip boundary condition cannot be imposed for potential flow. 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

This type of boundary condition has been explored extensively in the acoustics literature. Ferrante et al. [41] modeled the interface between a surface oscillating in the shear direction and a semi-infinite liquid as a spring connecting two masses (Fig. 2). The no-slip boundary condition for the displacements at the interface is replaced by a complex-valued ratio of the upper and lower displacements,... 

We have rewritten the contribution due to viscous dissipation and chemical reaction in terms of their intrinsic phase averages because they are a more appropriate average for these quantities. For a stationary porous media the first surface integral on the left-hand side does not come into play and the second integral is zero due to the no slip boundary condition. For a moving porous media the two integrals cancel each other. 

Couette and Poiseuille flows are in a class of flows called parallel flow, which means that only one velocity component is nonzero. That velocity component, however, can have spatial variation. Couette flow is a simple shearing flow, usually set up by one flat plate moving parallel to another fixed plate. For infinitely long plates, there is only one velocity component, which is in the direction of the plate motion. In steady state, assuming constant viscosity, the velocity is found to vary linearly between the plates, with no-slip boundary conditions requiring that the fluid velocity equals the plate velocity at each plate. There... 

Apply no-slip boundary conditions at the walls (f,- and r0) to determine the constants of integration. 

After substituting the relationship between the friction factor and the nondimensional pressure gradient, solve the nondimensional differential equation to develop an expression for the circumferential velocity profile w(r). The product Re/ should appear as a parameter in the differential equation. Assume no-slip boundary conditions at the channel walls. 

For the situation illustrated in Fig. 5.7, no-slip boundary conditions are applied at the lower and upper walls,... 

Application of the no-slip boundary conditions determines the constants Ci and C2, with the solution becoming... 

In nonreacting, continuum fluid mechanics the fluid velocity normal to a solid wall is zero, which is a no-slip boundary condition. However, if there are chemical reactions at the wall, then the velocity can be nonzero. The so-called Stefan flow velocity occurs... 

Figure 1. No slip boundary condition steady laminar velocity profile for fluid sheared between two plates.

Fixed pressure boundary conditions are implemented by assigning the equilibrium distribution functions, computed with zero velocity and specified density at the reservoirs, to the distribution functions. Periodic boundary conditions are applied in the span-wise directions. A particle distribution function bounce-back scheme57 is used at the walls to obtain no-slip boundary condition. 

For the velocity field the situation is a little more complex We assume no-slip boundary conditions, i.e., the velocity of the fluid and the velocity of the plate are the same at the surface of the plate. It is convenient to split the velocity field in to two parts the shear field Vo which satisfies the governing equations and the no-slip boundary condition and the correction vi to this shear field. The boundary condition for v now reads... 

To find a solution to this problem using BEM, we must solve the Stokes system of equations with their corresponding equivalent integral formulation eqn. (10.82) with traction boundary conditions at the entrance and end of the tube and with no-slip boundary conditions at the tube walls. We start by creating the surface mesh and by selecting the position of the internal points where we are seeking the solution. Figure 10.19 shows a typical BEM mesh with 8-noded quadratic elements. 

Figure 13 plots an example of the processed PIV frame. The turbulent velocity field and its boundaries, solid wall, and liquid-free surface are simultaneously shown in Figure 13. The turbulence structures such as the coherent vortical structure near the bottom wall and its modification after release from the no-slip boundary condition near the free surface of the open-channel flow, and the evolvement of the free-surface wave can be seen in Figure 13. This simultaneous measurement technique for free-surface level and velocity field of the liquid phase using PIV has been successfully applied to the investigation of wave-turbulence interaction of a low-speed plane liquid wall-jet flow (Li et al., 2005d), and the characteristics of a swirling flow of viscoelastic fluid with deformed free surface in a cylindrical container driven by the constantly rotating bottom wall (Li et al., 2006c).

A detailed derivation of Eq. (1.6a) using the no-slip boundary condition is provided in Section A.2 of the Appendix. If we were to generalize the analysis above with the partial-slip boundary condition, that is, Sv/Sz = pv (p = slip parameter) instead of the no-slip condition in Eq. (A.2) at the lubricant/solid boundary with q (z) = p, we would obtain... 

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