Body forces defined

In this equation, the presence of the solid particle in the fluid is represented by a virtual boundary body force field, Fp(4>p), defined by the IBM which will be discussed in Section IV.C.2. Fvapor is vapor pressure force exerting on the droplet-particle contact area due to the effect of the evaporation, which will be discussed in vapor-layer model of Section IV.C.3. 

In the IBM, the presence of the solid boundary (fixed or moving) in the fluid can be represented by a virtual body force field -rp( ) applied on the computational grid at the vicinity of solid-flow interface. Considering the stability and efficiency in a 3-D simulation, the direct forcing scheme is adopted in this model. Details of this scheme are introduced in Section II.B. In this study, a new velocity interpolation method is developed based on the particle level-set function (p), which is shown in Fig. 20. At each time step of the simulation, the fluid-particle boundary condition (no-slip or free-slip) is imposed on the computational cells located in a small band across the particle surface. The thickness of this band can be chosen to be equal to 3A, where A is the mesh size (assuming a uniform mesh is used). If a grid point (like p and q in Fig. 20), where the velocity components of the control volume are defined, falls into this band, that is... 

If we assume that the body force is only due to gravity and use the definition r = —prL +1 where pr is the isotropic resin pressure, / is the unit tensor, and r is the deviatoric stress as well as assuming a constant density and define a new pressure Pr —pr + prgh, Equation 5.21 simplifies to,... 

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

In equations 5-8, the variables and symbols are defined as follows p0 is reference mass density, v is dimensional velocity field vector, p is dimensional pressure field vector, x is Newtonian viscosity of the melt, g is acceleration due to gravity, T is dimensional temperature, tT is the reference temperature, c is dimensional concentration, c0 is far-field level of concentration, e, is a unit vector in the direction of the z axis, Fb is a dimensional applied body force field, V is the gradient operator, v(x, t) is the velocity vector field, p(x, t) is the pressure field, jl is the fluid viscosity, am is the thermal diffiisivity of the melt, and D is the solute diffiisivity in the melt. The vector Fb is a body force imposed on the melt in addition to gravity. The body force caused by an imposed magnetic field B(x, t) is the Lorentz force, Fb = ac(v X v X B). The effect of this field on convection and segregation is discussed in a later section. 

Accidentally this relation agrees with Eq. (9.33), defining the tricritical region. This coincidence happens only in three dimensions. Indeed, our crossover diagrams ignore the three-body forces which induce an additional structure in the 0-region. This structure involves the strength of the three-body interaction as a new scale and therefore is not fixed relative to the curves shown here. 

The integral equation theory consists in obtaining the pair correlation function g(r) by solving the set of equations formed by (1) the Omstein-Zernike equation (OZ) (21) and (2) a closure relation [76, 80] that involves the effective pair potential weff(r). Once the pair correlation function is obtained, some thermodynamic properties then may be calculated. When the three-body forces are explicitly taken into account, the excess internal energy and the virial pressure, previously defined by Eqs. (4) and (5) have to be, extended respectively [112, 119] so that... 

The weight of a body is defined as the force exerted on the body as a result of the acceleration of gravity. Thus... 

It is proportional to the mass of the fluid element. The body force is defined by... 

In the case of a multicomponent mixture, the different effects of the body forces on the individual components must also be considered. The body force k j acting on component A will be defined by... 

For each balance law, the values of -0, J and 4> defines the transported quantity, the diffusion flux and the source term, respectively, v denotes the velocity vector, T the total stress tensor, gc the net external body force per unit of mass, e the internal energy per unit of mass, q the heat flux, s the entropy per unit mass, h the enthalpy per unit mass, u>s the mass fraction of species s, and T the temperature. 

The change of momentum for a particle in the disperse phase is typically due to body forces and fluid-particle interaction forces. Among body forces, gravity is probably the most important. However, because body forces act on each phase individually, they do not result in momentum transfer between phases. In contrast, fluid-particle forces result in momentum transfer between the continuous phase and the disperse phase. The most important of these are the buoyancy and drag forces, which, for reasons that will become clearer below, must be defined in a consistent manner. However, as detailed in the work of Maxey Riley (1983), additional forces affect the motion of a particle in the disperse phase, such as the added-mass or virtual-mass force (Auton et al., 1988), the Saffman lift force (Saffman, 1965), the Basset history term, and the Brownian and thermophoretic forces. All these forces will be discussed in the following sections, and the equations for their quantification will be presented and discussed. 

Most real systems, as distinct from idealized assemblies of point masses linked by two-body forces, show deviations from the simple behaviour discussed in the previous sections. One of these types of more complex behaviour is associated with the internal friction of the material which leads to the damping of free vibrations, even in the most perfectly isolated systems. We have seen that the most convenient way of determining elastic moduli is by examining the vibration frequencies of carefully shaped samples and, in the same way, internal friction or mechanical relaxation can be studied by examining the logarithmic decrement of these same vibrations when the external excitation is removed. The logarithmic decrement 8 is so defined that an oscillation of angular frequency (o is damped by a factor exp (- It is thus n times the loss... 

Here the body force, f = pg, is the gravitational force. With the reference system shown in Fig. 1.15, the component of g along the plane (which defines the Jc-direction) is g sincr so that the Navier-Stokes equation reads ... 

State defined by (3.222) (or by (3.220), (3.221)) the persistence of which is achieved by the zero body heating (3.231), the zero inertial and body forces (i = o, b = o) and the zero velocity v = o (3.223) everywhere. The body is in the uniform equilibrium state mentioned above and as may be seen, such a state may be realized in the isolatedhody in which no exchange of heat, work and mass with environment exists and the boundary of which is fixed. Denoting constant (throughout the body and time) equilibrium values of temperature T° density p° and therefore also specific volume v°, internal energy and entropy s° (cf. (3.191), (3.192), (3.199)) we can express the volume V°, energy E° and entropy S° of the body in such equilibrium by... 

---