The Continuity Equations

In the real world, matter must be conserved. Let us relate the rate of variation of the mass contained in an arbitrary volume 12 to the flux across the surface I 

Once mass flux is expressed as a function of density and velocity, it becomes 

This equation holds true for any arbitrary volume, and so must be true for the volume element dV. We can therefore drop the integration sign and term dV to obtain 

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The earliest appearance of the nonrelativistic continuity equation is due to Schrodinger himself [2,319], obtained from his time-dependent wave equation. A relativistic continuity equation (appropriate to a scalar field and formulated in terms of the field amplitudes) was found by Gordon [320]. The continuity equation for an electron in the relativistic Dirac theory [134,321] has the well-known form [322] ... 

Again, the summation convention is used, unless we state otherwise. As will appear below, the same strategy can be used upon tbe Dirac Lagrangean density to obtain the continuity equation and Hamilton-Jacobi equation in the modulus-phase representation. 

The variational derivative of this with respect to ((> yields the continuity equation... 

When one takes its vaiiational derivative with respect to the phases < >, one obtains the continuity equation in the form... 

The result of interest in the expressions shown in Eqs. (160) and (162) is that, although one has obtained expressions that include corrections to the nonrelativistic case, given in Eqs. (141) and (142), still both the continuity equations and the Hamilton-Jacobi equations involve each spinor component separately. To the present approximation, there is no mixing between the components. 

The terms before the square brackets give the nonrelativistic part of the Hamilton-Jacobi equation and the continuity equation shown in Eqs. (142) and (141), while the term with the squaie brackets contribute relativistic corrections. All terms from are of the nonmixing type between components. There are further relativistic terms, to which we now turn. 

In Eq. (168), the first, magnetic-field term admixes different components of the spinors both in the continuity equation and in the Hamilton-Jacobi equation. However, with the z axis chosen as the direction of H, the magnetic-field temi does not contain phases and does not mix component amplitudes. Therefore, there is no contribution from this term in the continuity equations and no amplitude mixing in the Hamilton-Jacobi equations. The second, electric-field term is nondiagonal between the large and small spinor components, which fact reduces its magnitude by a further small factor of 0 particle velocityjc). This term is therefore of the same small order 0(l/c ), as those terms in the second line in Eqs. (164) and (166) that refer to the upper components. 

The continuity equation is the expression of the law of conservation of mass. This equation is written as... 

As already explained the necessity to satisfy the BB stability condition restricts the types of available elements in the modelling of incompressible flow problems by the U-V P method. To eliminate this restriction the continuity equation representing the incompressible flow is replaced by an equation corresponding to slightly compressible fluids, given as... 

The penalty method is based on the expression of pressure in terms of the incompressibility condition (i.e. the continuity equation) as... 

In a fixed two-dimensional Cartesian coordinate system, the continuity equation for a free boundary is expressed as... 

In Equation (4.12) the discretization of velocity and pressure is based on different shape functions (i.e. NjJ = l,n and Mil= l,m where, in general, mweight function used in the continuity equation is selected as -Mi to retain the symmetry of the discretized equations. After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and the pressure terms (to maintain the consistency of the formulation) and algebraic manipulations the working equations of the U-V-P scheme are obtained as... 

As described in Chapter 3, Section 5.1 the application of the VOF scheme in an Eulerian framework depends on the solution of the continuity equation for the free boundary (Equation (3.69)) with the model equations. The developed algorithm for the solution of the described model equations and updating of the free surface boundaries is as follows ... 

Wc now obtain the integral of the continuity equation for incompressible fluids with respect to the local gap height hr this flow domain... 

Substituting from Equations (5.73a) and (5.73b) into the continuity equation (5.65) yields... 

This equation is also known as the continuity equation. 

Simplified forms of Eq. (6-8) apply to special cases frequently found in prac tice. For a control volume fixed in space with one inlet of area Ai through which an incompressible fluid enters the control volume at an average velocity Vi, and one outlet of area Ao through which fluid leaves at an average velocity V9, as shown in Fig. 6-4, the continuity equation becomes... 

Mass Balance, Continuity Equation The continuity equation, expressing consei vation of mass, is written in cartesian coordinates as... 

The substantial derivative, also called the material derivative, is the rate of change in a Lagrangian reference frame, that is, following a material particle. In vector notation the continuity equation may oe expressed as... 

Selecting the inlet and outlet surfaces 1 and 2 as shown, the continuity equation Eq. (6-9) can he used to find the exit velocity V2 = ViAi/A2. The mass flow rate is obtained by m = pViAi. 

The continuity equation gives V2 = V AJa, and Vj = Q/A. The pressure drop measured by the manometer is pi —p2= (p — p)gA . Substituting these relations into the energy balance and rearranging, the desired expression for the flow rate is found. 

Since = 0 at y = ff/2, the continuity equation integrates to = 0. This is a direct result of the assumption of fuUy developed flow. 

The chapters presented by different experts in the field have been structured to develop an intuition for the basic principles by discussing the kinematics of shock compression, first from an extremely fundamental level. These principles include the basic concepts of x-t diagrams, shock-wave interactions, and the continuity equations, which allow the synthesis of material-property data from the measurement of the kinematic properties of shock compression. A good understanding of these principles is prerequisite... 

This model v/as used by Atwood et al (1989) to compare the performance of 12 m and 1.2 m long tubular reactors using the UCKRON test problem. Although it was obvious that axial conduction of matter and heat can be expected in the short tube and not in the long tube, the second derivative conduction terms were included in the model so that no difference can be blamed on differences in the models. The continuity equations for the compounds was presented as ... 

The quantity of gaseous effluent leaving a process is usually calculated from the continuity equation, which for this use is written as... 

The continuity equation is a mathematical formulation of the law of conservation of mass of a gas that is a continuum. The law of conservation of mass states that the mass of a volume moving with the fluid remains unchanged... 

F(r) and G(T) are the discrete transition and controi matrices and are obtained by converting the matrices in the continuous equation (9.49) into discrete form using equations (8.78) and (8.80). 

The mass flow is found using the continuity equation riv= punrd /4 and the Reynolds number formula Re = dm/firp dv)-. 

Equations (4.281)-(4.283) have to be solved at the same time as the continuity equation (4.278). The following dimensionless vatiables are used ... 

It is assumed in the above-mentioned methods that the influence of confined space on the supplied jet can be described by the reduction of the axial component and the value as for jet development in the counterflow. The value of i/,. is assumed to be the same throughout each cross-section but variable along the jet length. The value of can be found from the continuity equation, which in the case of jet distribution in a space of cylindrical shape can be presented as... 

The governing equations for mass flow, energy flow, and contaminant flow in a room will be the continuity equation, Navier-Stokes equations (one in each coordinate direction), the energy equation, and the mass transport equation, respectively. 

The continuity equation for an incompressible flow is given by the following expression ... 

In this section we show how the fundamental equations of hydrodynamics — namely, the continuity equation (equation 9.3), Euler s equation (equation 9.7) and the Navier-Stokes equation (equation 9.16) - can all be recovered from the Boltzman equation by exploiting the fact that in any microscopic collision there are dynamical quantities that are always conserved namely (for spinless particles), mass, momentum and energy. The derivations in this section follow mostly [huangk63]. 

Since the continuity equation (equation 9.3) is essentially a restatement of the principle of conservation of mass, wc should not be surprised to learn that it is easily recovered from Boltzmaii s equation by setting k — rn in equation 9.52 ... 

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