The Equations of Continuity

This equation expresses the condition that, provided matter is neither created nor destroyed, a liquid as it flows does not pull apart and that different rates of flow into and out of a given region must result in changes in density of the liquid contained therein. 

Consider again the liquid contained in the element of volume shown in Fig. 5.2. It will be assumed that the components of the liquid velocity u 3ieu, Uy, parallel to theand z directions, 

The mass of liquid flowing into the element across face A in the X direction in a time 6r is  

Taking the liquid flow in the and z directions into account, the net mass inflow into the volume 8x 6z in a time St is  

Equating the two expressions for the change in mass of the element in a time St gives  

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Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

The aim of this section is to show how the modulus-phase formulation, which is the keytone of our chapter, leads very directly to the equation of continuity and to the Hamilton-Jacobi equation. These equations have formed the basic building blocks in Bohm s formulation of non-relativistic quantum mechanics [318]. We begin with the nonrelativistic case, for which the simplicity of the derivation has... 

As already discussed, in general, polymer flow models consist of the equations of continuity, motion, constitutive and energy. The constitutive equation in generalized Newtonian models is incorporated into the equation of motion and only in the modelling of viscoelastic flows is a separate scheme for its solution reqixired. 

Application of the weighted residual method to the solution of incompressible non-Newtonian equations of continuity and motion can be based on a variety of different schemes. Tn what follows general outlines and the formulation of the working equations of these schemes are explained. In these formulations Cauchy s equation of motion, which includes the extra stress derivatives (Equation (1.4)), is used to preseiwe the generality of the derivations. However, velocity and pressure are the only field unknowns which are obtainable from the solution of the equations of continuity and motion. The extra stress in Cauchy s equation of motion is either substituted in terms of velocity gradients or calculated via a viscoelastic constitutive equation in a separate step. 

In the absence of body force the equations of continuity and motion representing Stokes flow in a two-dimensional Cartesian system are written, on the basis of Equations (1.1) and (1.4), as... 

The majority of polymer flow processes involve significant heat dissipation and should be regarded as nou-isothermal regimes. Therefore in the finite element modelling of polymeric flow, in conjunction with the equations of continuity... 

Non-dimensionalization of the stress is achieved via the components of the rate of deformation tensor which depend on the defined non-dimensional velocity and length variables. The selected scaling for the pressure is such that the pressure gradient balances the viscous shear stre.ss. After substitution of the non-dimensional variables into the equation of continuity it can be divided through by ieLr U). Note that in the following for simplicity of writing the broken over bar on tire non-dimensional variables is dropped. 

General Equation of Motion. Neglecting relativistic effects, the rate of accumulation of mass within a Cartesian volume element dx-dy-dz must equal the sum of the rates of inflow minus outflow. This is expressed by the equation of continuity ... 

Abramovich was the first to study axisymmetric confined jets analytically. He suggested the method based on utilizing the equations of continuity and momentum conservation. He also assumed that the width of the layer of a jet mixing with a counterflow equals the width of a free jet with a velocity distribution according to Schlichting s formula ... 

Fluid flow is additionally described by a conservation of flow equation, i.e., the equation of continuity... 

We now describe the conditions that correspond to the interface surface. Eor stationary capillarity flow, these conditions can be expressed by the equations of continuity of mass, thermal fluxes on the interface surface and the equilibrium of all acting forces (Landau and Lifshitz 1959). Eor a capillary with evaporative meniscus the balance equations have the following form ... 

Where F is the flux of diffusing species and dc/dx is its gradient of concentration c in the x direction. From the equation of continuity,... 

Based on this configuration, the reformer and combustor are modeled with partial differential equations. Since the thickness of the plates is relatively small, only the flow direction is considered. Using the equation of continuity, the component mass balances are constructed and the energy balance considering with heat loss and momentum balance are established as follows. 

A very general relation, that is known as the equation of continuity, has applications in many brandies of physics and chemistry. It can be derived by taking the divergence of Gq. (66). Then, from Eq. (62) the relation... 

It leads directly to the equation of continuity for the charge density in a closed volume, viz. 

As with acid-base reactions, the equation of continuity used to describe the mass transport of a solid in a fluid is... 

Transport of component i in a binary system is described by the equation of continuity [2], which is an expression for mass conservation of the subject component in the system, i.e.,... 

The choice of vx is a matter of convenience for the system of interest. Table 1 summarizes the various definitions of vx and corresponding, /Y, commonly in use [3], The various diffusion coefficients listed in Table 1 are interconvertible, and formulas have been derived. For polymer-solvent systems, the volume average velocity, vv, is generally used, resulting in the simplest form of Jx,i- Assuming that this vv = 0, implying that the volume of the system does not change, the equation of continuity reduces to the common form of Fick s second law. In one dimension, this is... 

Motion of the virtual ensemble in phase space may be likened to fluid flow by introducing a 2n-dimcnsional vector of phase velocity v with components qi,pi(i = 1,2,..., n). Since the systems of the virtual ensemble can neither be created nor destroyed in the flow, the equation of continuity... 

The Manning Equation and the Equation of Continuity are the basic hydraulic tools for description of the wastewater flow in a pipe at stationary, uniform flow ... 

The three-dimensional transport equation for inert pollutant dispersion results from timesmoothing the equation of continuity of the emitted substance. In Cartesian coordinates the distribution of a pullutant is given by the partial differential equation of second order for the concentration C(x, y, z, t) 111 ... 

If we substitute Eqs. (2.12) and (2.13) into Eq. (2.2) and perform the averaging operation indicated by the overbar, using the equation of continuity for an incompressible fluid,... 

The x-direction velocity component is perpendicular to the flight edge. It is assumed that the flow in the x direction is fully developed, forcing dl /dx to be zero. Moreover, from the equation of continuity (Eq. A7.4), both dVy/dy and Vy are zero for this simplified analysis. The x-component equation of motion (Eq. A7.5) reduces to ... 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

From Eq. 10.1, together with the equation of continuity, we can derive Pick s second law, otherwise known simply as the diffusion equation in one-dimensional form, this is given by... 

At the steady state the first term drops out and substituting Eq. (6) into the above equation, the equation of continuity becomes ... 

Detonation, Rayleigh (or Mikhel son) Line and Transformation in. (Called here Rayleigh-MikheTson Line) The Chapman-Jouguet theory deals with adiabatic transformations in steady, non-viscous, onedimensional flows in stream tubes or ducts of constant cross-section. Such transformations can be called Rayleigh transformations. From the equation of continuity valid for flow of constant cross-section and from the momentum equation (Ref 1, p 117 Ref 2, p 99), with use of the formula c = y-Pv for sonic velocity in an ideal gas, can be derived the relationship ... 

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