Wall, boundary condition

Equations II-12 and 11-13 illustrate that the shape of a liquid surface obeying the Young-Laplace equation with a body force is governed by differential equations requiring boundary conditions. It is through these boundary conditions describing the interaction between the liquid and solid wall that the contact angle enters. 

It also has interesting historical roots and, as we shall see, it raises fundamental questions regarding the boundary conditions at the wall, which apparently are not widely understood. We shall therefore investigate what is involved in formulating and solving this problem. 

Here f denotes the fraction of molecules diffusely scattered at the surface and I is the mean free path. If distance is measured on a scale whose unit is comparable with the dimensions of the flow channel and is some suitable characteristic fluid velocity, such as the center-line velocity, then dv/dx v and f <<1. Provided a significant proportion of incident molecules are scattered diffusely at the wall, so that f is not too small, it then follows from (4.8) that G l, and hence from (4.7) that V v° at the wall. Consequently a good approximation to the correct boundary condition is obtained by setting v = 0 at the wall. ... 

Maxwell obtained equation (4.7) for a single component gas by a momentum transfer argument, which we will now extend essentially unchanged to the case of a multicomponent mixture to obtain a corresponding boundary condition. The flux of gas molecules of species r incident on unit area of a wall bounding a semi-infinite, gas filled region is given by at low pressures, where n is the number of molecules of type r per... 

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. 

In the finite element solution of the energy equation it is sometimes necessary to impose heat transfer across a section of the domain wall as a boundary condition in the process model. This type of convection (Robins) boundary condition is given as... 

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... 

Nuj. is the Nusselt number for uniform wall temperature boundary condition. 

Structurally, carbon nanotubes of small diameter are examples of a onedimensional periodic structure along the nanotube axis. In single wall carbon nanotubes, confinement of the stnreture in the radial direction is provided by the monolayer thickness of the nanotube in the radial direction. Circumferentially, the periodic boundary condition applies to the enlarged unit cell that is formed in real space. The application of this periodic boundary condition to the graphene electronic states leads to the prediction of a remarkable electronic structure for carbon nanotubes of small diameter. We first present... 

In this paper, we review progress in the experimental detection and theoretical modeling of the normal modes of vibration of carbon nanotubes. Insofar as the theoretical calculations are concerned, a carbon nanotube is assumed to be an infinitely long cylinder with a mono-layer of hexagonally ordered carbon atoms in the tube wall. A carbon nanotube is, therefore, a one-dimensional system in which the cyclic boundary condition around the tube wall, as well as the periodic structure along the tube axis, determine the degeneracies and symmetry classes of the one-dimensional vibrational branches [1-3] and the electronic energy bands[4-12]. 

When a coarse grid is used, wall functions are used for imposing boundary conditions near the walls (Section 11.2.3.3). The nondimensional wall distance should be 30 < y < ]Q0, where y = u,y/p. We cannot compute the friction velocity u. before doing the CFD simulation, because the friction velocity is dependent on the flow. However, we would like to have an estimation of y" to be able to locate the first grid node near the wall at 30 < y < 100. If we can estimate the maximum velocity in the boundary layer, the friction velocity can be estimated as n, — 0.04rj, . . After the computation has been carried out, we can verify that 30 nodes adjacent to the walls. 

The machines are very complicated, so they are simplified to a level which maintains the main features observed. In this case, the hall already existed, and the ventilation system needed to be improved. In the lower parr of the machines are rotating wheels and axes which lead to a net flow across the floor in one direction through all the machines. Such a flow was generated by adding moving walls in the lower part of the machine model (Fig. 11.24), and the size of the velocity was adjusted to fit the measured speed in the real hall. Periodic boundary conditions are attached to the walls to the left and right in Fig. 11.24. 

Under steady-state conditions, the temperature distribution in the wall is only spatial and not time dependent. This is the case, e.g., if the boundary conditions on both sides of the wall are kept constant over a longer time period. The time to achieve such a steady-state condition is dependent on the thickness, conductivity, and specific heat of the material. If this time is much shorter than the change in time of the boundary conditions on the wall surface, then this is termed a quasi-steady-state condition. On the contrary, if this time is longer, the temperature distribution and the heat fluxes in the wall are not constant in time, and therefore the dynamic heat transfer must be analyzed (Fig. 11.32). 

The highly resolved velocity profile can be mapped in the vicinity of solid boundaries such as the walls of a room and in the entire enclosure, providing relevant data for CFD boundary conditions. These data form a basis for verification of CFD results and for improvement of CFD codes. 

Subject to the boundary conditions that must be imposed at the axis, at the inlet, and on the wall boundaries of the cyclone, Bloor and Ingham found that the solution for 4 may be approximated by the expression... 

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