Equation motion

Standard Galerkin procedure - to discretize the circumferential component of the equation of motion, Equation (5.23), for the calculation of vs. 

Continuous penalty method - to discretize the continuity and (r, z) components of the equation of motion, Equations (5.22) and (5.24), for the calculation of r,. and v. Pressure is computed via the variational recovery procedure (Chapter 3, Section 4). 

A variety of techniques have been detailed for handling Newton s equations of motion, equation 5 (86—90). Integration techniques yield atomic... 

These three terms represent contributions to the flux from migration, diffusion, and convection, respectively. The bulk fluid velocity is determined from the equations of motion. Equation 25, with the convection term neglected, is frequently referred to as the Nemst-Planck equation. In systems containing charged species, ions experience a force from the electric field. This effect is called migration. The charge number of the ion is Eis Faraday s constant, is the ionic mobiUty, and O is the electric potential. The ionic mobiUty and the diffusion coefficient are related ... 

This equation is ealled the motion equation for the system, and it ean be rewritten as follows ... 

The externaiiy appiied periodic force has a frequency lu, which can vary independentiy of the system parameters. The motion equation for this system may be obtained by any of the previousiy stated methods. The Newtonian approach wiii be used here because of its conceptuai simpiicity. The freebody diagram of the mass m is shown in Figure 5-ii. 

Substituting the previous relationships into motion equation (5-17), the following relationship is obtained ... 

If we ignore inertia force and follow the conventional assumptions in liquid lubrication, motion equation (here we consider the incompressible fluids in the absence of volume force and volume momentum) is ... 

Note that depending on the manner in which the drag force and the buoyancy force are accounted for in the decomposition of the total fluid particle interactive force, different forms of the particle motion equation may result (Jackson, 2000). In Eq. (36), the total fluid-particle interaction force is considered to be decomposed into two parts a drag force (fd) and a fluid stress gradient force (see Eq. (2.29) in Jackson, 2000)). The drag force can be related to that expressed by the Wen-Yu equation, /wen Yu> by... 

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

Let us examine the droplet motion (Equation (12.37)) along with its evaporation transport (Equation (12.32)) and the conservation of mass for a single drop ... 

The size, shape and charge of the solute, the size and shape of the organism, the position of the organism with respect to other cells (plankton, floes, biofilms), and the nature of the flow regime, are all important factors when describing solute fluxes in the presence of fluid motion. Unfortunately, the resolution of most hydrodynamics problems is extremely involved, and typically bioavailability problems under environmental conditions are in the range of problems for which analytical solutions are not available. For this reason, the mass transfer equation in the presence of fluid motion (equation (17), cf. equation (14)) is often simplified as [48] ... 

We have vibrationally averaged the CAS /daug-cc-pVQZ dipole and quadmpole polarizability tensor radial functions (equation (14)) with two different sets of vibrational wavefunctions j(i )). One was obtained by solving the one-dimensional Schrodinger equation for nuclear motion (equation (16)) with the CAS /daug-cc-pVQZ PEC and the other with an experimental RKR curve [70]. Both potentials provide identical vibrational... 

In short, the distributivity of the transformation f/t implies that retains the reducibility of the Liouville equation into a pair of Schrodinger equations. Furthermore, this transformation retains the time-reversal invariance of these equations, since the free-motion equations [Eqs. (15)] are time-reversal invariant. 

Since there are three dimensions, two dimensionless groups, e.g., and defined in Chapter 5, suffice to describe the motion. If the motion is unsteady, it is necessary to introduce the particle density explicitly, since it determines the particle inertia as well as the net gravity force. Also, since L/r varies with time and position, a further parameter must be introduced. This may be the distance x moved since the start of the motion. Equation (11-1) is then replaced by... 

The discussion of Kapral s kinetic theory analysis of chemical reaction has been considered in some detail because it provides an alternative and intrinsically more satisfactory route by which to describe molecular scale reactions in solution than using phenomenological Brownian motion equations. Detailed though this analysis is, there are still many other factors which should be incorporated. Some of the more notable are to consider the case of a reversible reaction, geminate pair recombination [286], inter-reactant pair potential [454], soft forces between solvent molecules and with the reactants, and the effect of hydrodynamic repulsion [456b, 544]. Kapral and co-workers have considered some of the points and these are discussed very briefly below [37, 285, 286, 454, 538]. 

By far the largest contribution to a gas-phase species entropy comes from translational motion. Equation 8.98 provided a means to calculate this contribution ... 

To describe the geminate kinetics, let us start with the motion equations for the probability density W(fX, rB, i), where W(fX, rB, )dfXdrB gives the probability to find at the moment t particle A at volume dfX centred at fX and particle B at drB centred at fie. The relevant motion equation reads [64]... 

Differentiating equation (3.2.5) and making use of the motion equation (3.2.4), we arrive at the well-known Smoluchowski equation [65-70]... 

In the case of particle interaction described by their drift in potential, the black sphere model again allows us to obtain a simple solution of the kinetics sought for. Imposing the boundary conditions, equation (3.2.17) and w(go, t) = 0, as well as the initial condition (3.2.14) on the motion equation... 

In these last two examples of equations of motion, the objective is to determine functions of the form h = /(/) or x=g(t), respectively, which satisfy the appropriate differential equation. For example, the solution of the classical harmonic motion equation is an oscillatory function, x=g t), where g(f) = cos a>t, and a> defines the frequency of oscillation. This function is represented schematically in Figure 7.1 (see also Worked Problem 4.4). 

The last formulae can be obtained from the motion equation written in the Lamb form... 

What is the size of the overlapping volume The complete independence of the constant in Equation 1 from the degree of polymerization shows that the overlapping volume always consists of the same portion of the volume of the polymer coil (12). This can be easily understood by assuming that two polymer coils are able to migrate nearly unhindered through each other. Then the mean depth of permeation and, therefore, the time of overlapping is determined only by the statistics of the free Brownian motion. Equation 1 is based on this assumption. 

---