Slip-wall boundary conditions

Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as 

Equations (3.59) and (3.60) are recast in terms of their components and solved together. After algebraic manipulations and making use of relations (3.61) slip-wall velocity components are found as 

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On no-slip walls zero velocity components can be readily imposed as the required boundary conditions (v = v, = 0 on F3 in the domain shown in Figure 3.3). Details of the imposition of slip-wall boundary conditions are explained later in Section 4.2. 

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... 

After the imposition of no-slip wall boundary conditions the last term in Equation (5.64) vanishes. Therefore... 

The Hagen-Poiseuille equation reported above assumes a no-slip condition at the capillary wall, that is, the molecules in the fluid layer in contact with the capillary wall have zero velocity. It should be noted that this is an approximation and there is no reason why particles at the wall should not have a finite velocity. The nature and very existence of slip is an intensely debated topic and is covered in detail elsewhere in this book. Several fluid flow experiments in nanometer channels showing huge slip lengths will be described later in this chapter. In these experiments, the slip length X has been calculated via the Hagen-Poiseuille equation with a slip wall boundary condition ... 

In RANS models, the solid wall boundary conditions have traditionally been modeled using wall functions. Wall functions use empirical profiles to replace the actual boundary conditions, such as no-slip (zero velocity) condition at solid surfaces. An example of an empirical law is the logarithmic velocity profile ... 

For solid walls, no-penetration and no-slip are typically applied to the momentum equation. Boundary conditions such as velocity, pressure and temperature at the inlet are usually known and specified, whereas their counterparts at the outlet are derived from assumptions of no-stress or fully developed and simulated flow. The thermal wall boundary conditions influence the flow significantly. Simple assumptions of constant wall temperature, insulated side walls and constant wall heat transfer flux have been used extensively for simple applications. More specifically, the following assumptions are normally made for a retort as shown in Figure 6.29 (the directions of the velocity in the following description refer to a cylindrical coordinate system in this figure). 

The governing equations are elliptic so boundary conditions are required at all boundaries. The normal velocity components for both phases are set to zero at the vertical boundaries. The wall boundary conditions for the vertical velocity component, k and e are specified in accordance with the standard wall function approach. The particle phase is allowed to slip along the wall following the boundary condition given by (4.99). 

Ad slip parameter for wall boundary condition in granular theory (m) Af t) Eulerian longitudinal integral length scale (m)... 

Step 4 Set the boundary conditions as follows. The centerline, inlet velocity, and exit velocity/pres sure are set as in the laminar case slip/symmetry, v = 2, Normal flow/ Pressure, p = 0. The wall boundary condition, though, is set to the Logarithmic wall function. This is an analytic formula for the velocity, turbulent kinetic energy, and rate of dissipation, as determined by experiment (Deen, 1998, pp. 527-528). 

It is shown that fluid flow and heat transfer at microscale differ greatly from those at macroscale. At macroscale, classical conservation equations are successfully coupled with the corresponding wall boundary conditions, usual no-slip for the hydrodynamic boundary condition and no-temperature-jump for the thermal boundary condition. These two boimdary conditions are valid only if the fluid flow adjacent to the surface is in thermal equilibrium. However, they are not valid for gas flow at microscale. For this case, the gas no longer reaches the velocity or the temperature of the surface and therefore a slip condition for the velocity and a jump condition for the temperature should be adopted. 

For the gas velocity u, the wall boundary conditions can be of free slip, no-slip, or turbulent law-of-the-wall type. The free-slip condition demands that the normal velocity components of the fluid and the wall coincide, while the tangential components of the stress tensor satisfy o = slip boundary conditions require the fluid velocity to coincide with the wall velocity. No-slip boundary conditions impose large velocity gradients, which, because of lack of sufficient computational resources, cannot be resolved. Therefore, wall functions are used. The following discussion reflects the exhibition given by Amsden et al. [4]. 

Periodic boundary conditions may be implemented in DPD just as in MD. There are different approaches taken to implement wall boundary conditions in the solution of the DPD equations. Some of these will now be discussed. Two common methods are adopted to implement no-slip boundary conditions. In the first one, a layer of frozen particles is placed at the boundary and interaction of fluid particles with this layer obeys certain rules. In the second one, an extra layer of particles reflected from the DPD particles in the fluid is used for achieving no-slip. 

The steady-state heat convection between two parallel plates and in circular, rectangular, and annular channels with viscous heat generation for both thermally developing and fully developed conditions is solved. Both constant wall temperature and constant heat flux boundary conditions are crmsidered. The velocity and the temperature distributions are derived from the momentum and energy equations, and the proper slip-flow boundary conditions are considered. 

Figure 1 Schematic representation of a no slip (la) or a slip (lb) boundary condition for the fluid velocity at the solid wall...

The walls are assumed to be solid and impermeable so that flow velocities are zero. No slip conditions are employed at the boundary walls. These wall boundary conditions are implemented using the near-wall approach of Launder and Spalding [54] for the momentum and scalar transport equations. The method assumes that near the wall, Couette flow prevails and the velocity profile obeys the universal logarithmic law. Therefore, the log-law is generally employed to calculate the velocity component parallel to the wall thus... 

The above-mentioned velocity solution indicates not only a velocity slip on the wall but also a correction to the slope of the profile (see Figure 3.13) for the Couette flow with slip flow boundary condition. The velocity profile as a function of Knudsen number and momentum accommodation coefficient has been shown in Figure 3.14. There is an increase in slip velocity... 

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