Fluid motion equations

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] ... 

In this case, the first fluid motion equation (3.33) and the boundary conditions (3.34), (3.35) are equivalent to the conjugation boundary-value problem (3.60) if A = Ah and Pr = 1, so that the profiles of U(z) and T(z) coincide. This is the so-called Reynolds analogy between fluid mechanics and heat (mass) fields. 

These quantities satisfy compressible fluid motion equations, whose kinematic viscosity v is related to the relaxation parameter X in the BGK model (20.2) through V = [5]. 

Next it is necessary to determine the deformation of vessel walls and valve leaflets being influenced by of fluid flow, and also the distribution of body forces / in the fluid motion equation based on this deformation. It is possible to calculate the deformation of vessel walls and valve leaflets under this particular fluid pressure and the resistance forces by using the equations (12) - (13), which are numerically integrated by using any of quadrature formulas, and equations (10) - (11). Afterwards, the body forces / are recalculated, and it is possible to move to the next time step. 

To calculate Eq. 4.23, the flow velocity distribution within the package U (u, v, w) must be defined. This involves the fluid motion equation in porous media. As discussed in Chapter 3, Darcy s law and Brinkman equation can be used to characterise the flow behavioirr within the package. 

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 ... 

The fomulation of Equation (8.68) gives the fully developed velocity profile, Fz(r), which corresponds to the local values of ix(r) and p(r) without regard to upstream or downstream conditions. Changes in Fz(r) must be gradual enough that the adjustment from one axial velocity profile to another requires only small velocities in the radial direction. We have assumed Vy to be small enough that it does not affect the equation of motion for V. This does not mean that Vr is zero. Instead, it can be calculated from the fluid continuity equation,... 

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 ... 

This corresponds to a Hamiltonian system which is characterized by a weak oscillatory perturbation of the SHV streamfunction T r, ) —> Tfr, Q + HP, (r, ( ) x sin(fEt). The equations of fluid motion (4.4.4) are used to compute the inertial and viscous forces on particles placed in the flow. Newton s law of motion is then... 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

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). 

Note that the hydrodynamic boundary layer depends on the diffusion coefficient. Introducing the proportionality constant K° results in an equation valid for any desired hydrodynamic system based on relative fluid motion as proposed in Ref. 10 ... 

For a spherical particle in a fluid, the equation of motion for the Stokes law region is ... 

An example of this work is that of Farrell and co-workers [34], They present a rather complex model to attempt to account for the effects of fluid motion and turbulence in three different levels of scale, relative to the plume. They begin with classical equations of motion, but by breaking their particle velocity vector into three components related to the three scales of interest, they are able to introduce appropriate statistical descriptions for the components. The result is a model that retains both the diffusive and the filamentary nature of the plume. 

In the recent years, the advance of computer power has allowed numerical solutions for the differential equations that describe fluid motion. The use of computational fluid dynamics (CFD) is beginning to give a better understanding of the strongly swirling turbulent flow inside hydrocyclones and, consequently, of their performance [46-50]. 

With this simplification, the equations governing incompressible fluid motion are Eq. (1-33) and the continuity equation, Eq. (1-9). Several important consequences follow from inspection of these equations. The fluid density does not appear in either equation. Both equations are reversible in the sense that they are still satisfied if u is replaced by — u, whereas the nonlinearity of the Navier-Stokes equations prevents such reversibility. If we take the divergence of Eq. (1-33) and apply Eq. (1-9), we obtain... 

Prediction of fluid motion, drag, and transfer rates becomes much more complex when the motion is unsteady. Dimensional analysis gives an indication of the problems. A rigid sphere moving with steady velocity in a gravitational field is governed by an equation of the general form... 

The general equation for fluid motion in a mixing system contains no less than 13 terms. Of these terms, nine define geometric boundary conditions. If these can be fixed, and strict geometric similarity be adhered to, the equation can be simplified and written as... 

It is impossible to read much of the literature on viscosity without coming across some reference to the equation of motion. In the area of fluid mechanics, this equation occupies a place like that of the Schrodinger equation in quantum mechanics. Like its counterpart, the equation of motion is a complicated partial differential equation, the analysis of which is a matter for fluid dynamicists. Our purpose in this section is not to solve the equation of motion for any problem, but merely to introduce the physics of the relationship. Actually, both the concentric-cylinder and the capillary viscometers that we have already discussed are analyzed by the equation of motion, so we have already worked with this result without explicitly recognizing it. The equation of motion does in a general way what we did in a concrete way in the discussions above, namely, describe the velocity of a fluid element within a flowing fluid as a function of location in the fluid. The equation of motion allows this to be considered as a function of both location and time and is thus useful in nonstationary-state problems as well. 

This chapter established three important concepts that are essential for the derivation of the conservation equations governing fluid flow. First, the Reynolds transport theorem was developed to relate a system to an Eulerian control volume. The substantial derivative that emerges from the Reynolds transport theorem can be thought of as a generalized time derivative that accommodates local fluid motion. For example, the fluid acceleration vector... 

When motion of the fluid consists of only small fluctuations about a state of near-rest, Lhe continuum equations are linearized by neglecting nonlinear terms and they become lhe equalions of acoustics. A large variety of fluid motions are described as sound waves when the small-motion or acoustic description can be used, the principle of superposition is valid. This powerful principle allows addition of simple simultaneous motions to represent a more complex motion, such as the sound reaching lhe audience from the instruments of a symphony orchestra. The superposition principle does not apply to large-scale (nonacoustical) motions, and the subject of fluid dynamics (in distinction from acoustics) treats nonlinear flows. i.e.. those that cannot be described as superpositions of other flows. 

Conservation Laws. The fundamental conservation laws of physics can be used to obtain the basic equations of fluid motion, the equations of continuity (mass conservation), of flow (momentum conservation), of... 

From die Navier-Stokes equations it follows that three criteria of fluid motions (i.e., small arrows in Figure 3) are to be kept constant to obtain geometrically similar agitation patterns for both reactor sizes. 

Melting model using a Newtonian fluid with temperature independent viscosity p. For a Newtonian fluid the equation of motion reduces to... 

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