Boundary Slip

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... 

Figure 5.14 (a) The predicted velocity field corresponding to no-slip wall boundary conditions, (b) Tlie predicted velocity field corresponding to partial slip boundary conditions... 

E. Bansch, B. Hdhn. Numerical treatment of the Navier-Stokes equations with slip-boundary condition. Preprint 9-98, Mathematische Fakultat Freiburg. SIAM J Sci Comput (submitted). 

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]. 

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 friction coefficient of a large B particle with radius ct in a fluid with viscosity r is well known and is given by the Stokes law, Q, = 67tT CT for stick boundary conditions or ( = 4jit ct for slip boundary conditions. For smaller particles, kinetic and mode coupling theories, as well as considerations based on microscopic boundary layers, show that the friction coefficient can be written approximately in terms of microscopic and hydrodynamic contributions as ( 1 = (,(H 1 + (,/( 1. The physical basis of this form can be understood as follows for a B particle with radius ct a hydrodynamic description of the solvent should... 

Figure 9b shows the friction constant as a function of a. For large a the friction coefficient varies linearly with ct in accord with the prediction of the Stokes formula. The figure also shows a plot of (slip boundary conditions) versus ct. It lies close to the simulation value for large ct but overestimates the friction for small ct. For small ct, microscopic contributions dominate the friction coefficient as can be seen in the plot of (m. The approximate expression 1 = + 071 interpolates between the two limiting forms. Cluster friction... 

Simulations are then performed for gas bubbles emerging from a single nozzle with 0.4 cm I.D. at an average nozzle velocity of lOcm/s. The experimental measurements of inlet gas injection velocity in the nozzle using an FMA3306 gas flow meter reveals an inlet velocity fluctuation of 3-15% of the mean inlet velocity. A fluctuation of 10% is imposed on the gas velocity for the nozzle to represent the fluctuating nature of the inlet gas velocities. The initial velocity of the liquid is set as zero. An inflow condition and an outflow condition are assumed for the bottom wall and the top walls, respectively, with the free-slip boundary condition for the side walls. 

Subsequently, simulations are performed for the air Paratherm solid fluidized bed system with solid particles of 0.08 cm in diameter and 0.896 g/cm3 in density. The solid particle density is very close to the liquid density (0.868 g/ cm3). The boundary condition for the gas phase is inflow and outflow for the bottom and the top walls, respectively. Particles are initially distributed in the liquid medium in which no flows for the liquid and particles are allowed through the bottom and top walls. Free slip boundary conditions are imposed on the four side walls. Specific simulation conditions for the particles are given as follows Case (b) 2,000 particles randomly placed in a 4 x 4 x 8 cm3 column Case (c) 8,000 particles randomly placed in a 4 x 4 x 8 cm3 column and Case (d) 8,000 particles randomly placed in the lower half of the 4x4x8 cm3 column. The solids volume fractions are 0.42, 1.68, and 3.35%, respectively for Cases (b), (c), and (d). 

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... 

Prescribed pressure cell, free-slip boundaries... 

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. 

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