Fluid velocity field

By using the relationship between the fluid current and its veloeity field, J = Pv, a quantum fluid velocity field of... 

At Che opposite limit, where Che density Is high enough for mean free paths to be short con ared with pore diameters, the problem can be treated by continuum mechanics. In the simplest ease of a straight tube of circular cross-section, the fluid velocity field can easily be obtained by Integrating Che Nsvler-Stokes equations If an appropriate boundary condition at Che... 

Let us examine the critical dynamics near the bulk spinodal point in isotropic gels, where K + in = A(T — Ts) is very small, Ts being the so-called spinodal temperature [4,51,83-85]. Here, the linear theory indicates that the conventional diffusion constant D = (K + / )/ is proportional to T — Ts. Tanaka proposed that the density fluctuations should be collectively convected by the fluid velocity field as in near-critical binary mixtures and are governed by the renormalized diffusion constant (Kawasaki s formula) [84],... 

The proposed model consists of a biphasic mechanical description of the tissue engineered construct. The resulting fluid velocity and displacement fields are used for evaluating solute transport. Solute concentrations determine biosynthetic behavior. A finite deformation biphasic displacement-velocity-pressure (u-v-p) formulation is implemented [12, 7], Compared to the more standard u-p element the mixed treatment of the Darcy problem enables an increased accuracy for the fluid velocity field which is of primary interest here. The system to be solved increases however considerably and for multidimensional flow the use of either stabilized methods or Raviart-Thomas type elements is required [15, 10]. To model solute transport the input features of a standard convection-diffusion element for compressible flows are employed [20], For flexibility (non-linear) solute uptake is included using Strang operator splitting, decoupling the transport equations [9],... 

As before, let P be the local stress tensor, and denote by an overbar the statistical average of any quantity. The definition of the fluid-velocity field may be analytically extended to the solid-particle interiors and the pressure therein assumed to vanish. As such, taking the statistical average of the... 

The mean vorticity of the fluid phase is the curl of the mean fluid velocity field, coc = "X Uf, and is a measure of the mean rotation of the continuous phase. 

Except when natural convection is considered, the analysis of mass transfer can be determined after the flow field is obtained. Here, we thus assume that fluid velocity fields are known. Since Schmidt numbers in aqueous electrolytes are typically on the order of 1000 and can be much larger, the accurate resolution of concentration fields may require much finer meshes than those for the flow fields. It thus may be advantageous to develop methodologies that permit the use of different grids for the concentration fields. 

We assume that the medium density and viscosity are independent of concentration and temperature, and hence, the concentration and temperature distributions do not affect the flow field. This allows one to analyze the hydrodynamic problem about the fluid motion and the diffusion-heat problem of finding the concentration and temperature fields independently. (More complicated problems in which the flow field substantially depends on diffusion-heat factors will be considered later in Chapter 5.) It is assumed that the information about the fluid velocity field necessary for the solution of the diffusion-heat problem is known. We also assume that the diffusion and thermal conductivity coefficients are independent of concentration and temperature. For simplicity, we restrict our consideration to the case of two-component solutions. 

For each w satisfying condition (4.4.27), the leading terms of the asymptotic expansions for Eqs. (4.4.26) and (4.4.28) with the same boundary conditions coincide in the inner and outer regions. Therefore, as Pe - 0, in the diffusion equation one can replace the actual fluid velocity field v by w. This fact allows one to use the results presented later on in Section 4.11. Namely, as w we take the velocity field for the potential flow of ideal fluid past the cylinder. This approximation yields an error of the order of Pe in the inner expansion. By retaining only the leading terms in (4.11.15), we obtain the dimensionless diffusion flux at small Peclet numbers in the form... 

Mass transfer inside a drop is described by Eq. (4.12.1) and the first two conditions (4.12.2). The fluid velocity field v = (vr,vg) at low Reynolds numbers is given by the Hadamard-Rybczynski stream function and, in the dimensionless variables, has the form... 

Let us consider a transient solute concentration field in a liquid outside and inside a spherical drop of radius a moving at a constant velocity U in an infinite fluid medium. We assume that the fluid velocity fields for the continuous and disperse phases are determined by the Hadamard-Rybczynski solution [177, 420], obtained for low Reynolds numbers. The concentration far from the drop is maintained constant and equal to C,. At the initial time f = 0, the concentration outside the drop is everywhere uniform and is equal to C inside the drop, it is also uniform, but is equal to Co-... 

Let us consider mass and heat transfer for a monodisperse system of spherical particles of radius a with volume density of the solid phase. We use the fluid velocity field obtained at low Reynolds numbers from the Happel cell model (see Section 2.9) to find the mean Sherwood number [74,76 ... 

Preliminary remarks. In the preceding chapters it was assumed that the fluid velocity field is independent of the temperature and concentration distributions. However, there are a few phenomena in which the influence of these factors on the hydrodynamics is critical. This influence arises from the fact that various physical parameters of fluids, such as density, surface tension, etc. are temperature or concentration dependent. 

Remark. The problem of mass transfer to a drop for the diffusion regime of reaction on its surface under the conditions of thermocapillary motion is stated in the same way as in its absence (see Section 4.4) taking into account the corresponding changes in the fluid velocity field. In [144], a more complicated problem is considered for the chemocapillary effect with the heat production, which was described in [147-149,419], It was assumed that a chemical reaction of finite rate occurs on the drop surface. 

Figure 7.16 is a diagram of an endcap. There are two distinct zones. A partial differential equation could be written for each zone, but it would depend upon radial position, axial position, and time. A problem arises with this approach. The equations would require information on heat-transfer coefficients in the fluid/wall, solid/wall, solid/solid, and fluid/solid interfaces. This would require knowledge of the fluid velocity field in the endcaps along with correlations for the heat-transfer coefficients. That information is not available. Therefore, writing complete, comprehensive energy balances for the endcaps would not be productive. Simplifications must be made. 

In order to account for the variable density and the resulting coupling to the fluid velocity field (buoyancy effects), the Lattice-Boltzmann method has to be extended by a source term characterizing the temporal density change. Hence, besides... 

In all equations of change obtained so far, as well as in the equation of continuity, the fluid velocity appears via v or i>. If the migration velocities of different species do not influence the fluid flow field or the bulk fluid motion, then the fluid velocity field may be obtained from the solution of what is known as the equation of motion- ... 

The fluid velocity field v is quite important in convective diffusion of particles occurring in filters and scruhhers used for gas cleaning. Knowledge of v in the separator is essential in predicting particle separation. 

Fig. 6 Snapshots of the fluid velocity field (see colour scale) and particle cluster distribution with the poly-disperse size distribution at Rep = 1.0 and St = lA (Case R-P) at various instants of time fi m top to bottom tlhoi = 0.0 (a), 0.2 (b), 0.4 (c), 0.6 (d), 0.8 (e) and 1.0 (f). Spheres of the same colour have formed an agglomerate. Cut through the centre of the three-dimensional domain...

Basing on the statement that bone cells in vitro actually respond to fluid shear stresses about 0.2 - 6 Pa over their surface [5], it would be able to determine a fluid velocity bound valued in range 2.10 - 5.10 ps that correspond to the shear stress range (see [13] for the detail procedure to evaluate flow-induced shear stress in canaliculi from pore fluid velocity). As a result, the fluid velocity fields obtained in following tests will be compared with these values. 

We first assume that there are no micro-cracks in the domain. The distance between osteons is determinated with d = 0jjm. Thanks to the periodical condition, only one Representative Elementary Volume (REV) (see Fig. 2) is needed to be modeled. Fig. 2 depicts the response of fluid velocity field when applying a frequency of loading fo = IHz and a strain rate = 0.0035that were taken... 

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