Equation continuity

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

We first derive the so-called continuity equation, which is a direct consequence of the conservation of mass. If p is the density, or mass per unit volume, then the total mass of a fluid contained in F is equal to M = fj p dF. Letting dS — fi dS be an element of the surface, with n a unit vector perpendicular to the surface, the mass flow per unit time through the surface element is pv dS. The total fluid flow out of the volume F is then given by 

Since a continuity of flow implies that the fluid flow through the surface must equal the loss of fluid in the enclosed volume, we have that 

Given a deterministic thermostatting method, the density will evolve under the continuity equation, 

We divide this into four parts p = Wi + W2 + W3 + W4, where 

We seek perturbations of Hamiltonian dynamics with separable equilibria, i.e., such that p = p (//(z))p( ) is a stationary solution. 

Taking this as an ansatz, the first term can be simplified as follows  

Using the volume preservation of Hamiltonian systems, the second term satisfies 

The flow of materials is accounted for with two balances conservation of mass and conservation of momentum transfer. The most important is a momentum balance, which is also called the equation of motion. The mass balance (also called the continuity equation) makes sure that mass is conserved. 

3 Charge Carrier Management (Thermal Noise Engineering) 

We assume that the concentration of ionized dopants (the difference between the concentrations of ionized donors and acceptors) are not time dependent, i.e., we neglect slow states. This is not fully justified, since these states are actually responsible for flicker noise. 

By introducing (3.17) into the divergence of (3.16) and simultaneously adding and subtracting net generation-recombination rate we obtain two continuity equations 

---

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

The phenomenology of model B, where (j) is conserved, can also be outlined simply. Since (j) is conserved, it obeys a conservation law (continuity equation) ... 

For a general dimension d, the cluster size distribution fiinction n(R, x) is defined such that n(R, x)dR equals the number of clusters per unit volume with a radius between andi + dR. Assuming no nucleation of new clusters and no coalescence, n(R, x) satisfies a continuity equation... 

If the state-specific rate constants are assumed continuous, equation (A3.12.65) can be written as [103]... 

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 U-V-P scheme based on the slightly compressible continuity equation... 

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... 

The momentum and continuity equations give rise to a 22 x 22 elemental stiffness matrix as is shown by Equation (3.31). In Equation (3.31) the subscripts I and / represent the nodes in the bi-quadratic element for velocity and K and L the four corner nodes of the corresponding bi-linear interpolation for the pressure. The weight functions. Nr and Mf, are bi-qiiadratic and bi-linear, respectively. The y th component of velocity at node J is shown as iPj. Summation convention on repeated indices is assumed. The discretization of the continuity and momentum equations is hence based on the U--V- P scheme in conjunction with a Taylor-Hood element to satisfy the BB condition. 

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

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

Therefore the continuity equation for an incompressible axisymmetric flow is written 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... 

The described continuous penaltyf) time-stepping scheme may yield unstable results in some problems. Therefore we consider an alternative scheme which provides better numerical stability under a wide range of conditions. This scheme is based on the U-V-P method for the slightly compressible continuity equation, described in Chapter 3, Section 1.2, in conjunction with the Taylor-Galerkin time-stepping (see Chapter 2, Section 2.5). The governing equations used in this scheme are as follows... 

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... 

Continuity equation corresponding to the first-order terms... 

Integration of the first-order continuity equation between the limits of the thin layer gives... 

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

When electrons are injected as minority carriers into a -type semiconductor they may diffuse, drift, or disappear. That is, their electrical behavior is determined by diffusion in concentration gradients, drift in electric fields (potential gradients), or disappearance through recombination with majority carrier holes. Thus, the transport behavior of minority carriers can be described by a continuity equation. To derive the p—n junction equation, steady-state is assumed, so that = 0, and a neutral region outside the depletion region is assumed, so that the electric field is zero. Under these circumstances,... 

Mechanism. The thermal cracking of hydrocarbons proceeds via a free-radical mechanism (20). Siace that discovery, many reaction schemes have been proposed for various hydrocarbon feeds (21—24). Siace radicals are neutral species with a short life, their concentrations under reaction conditions are extremely small. Therefore, the iategration of continuity equations involving radical and molecular species requires special iategration algorithms (25). An approximate method known as pseudo steady-state approximation has been used ia chemical kinetics for many years (26,27). The errors associated with various approximations ia predicting the product distribution have been given (28). 

With the introduction of Gear s algorithm (25) for integration of stiff differential equations, the complete set of continuity equations describing the evolution of radical and molecular species can be solved even with a personal computer. Many models incorporating radical reactions have been pubHshed. 

Material Balances Whenever mass-transfer applications involve equipment of specific dimensions, flux equations alone are inadequate to assess results. A material balance or continuity equation must also be used. When the geometiy is simple, macroscopic balances suffice. The following equation is an overall mass balance for such a unit having bulk-flow ports and ports or interfaces through which diffusive flux can occur ... 

TABLE 5-12 Continuity Equation in Various Coordinate Systems... 

This equation is also known as the continuity equation. 

---