Gradient of the divergence

Two other possibilities for successive operation of the del operator are the curl of the gradient and the gradient of the divergence. The curl of the gradient of any differentiable scalar function always vanishes. 

The mathematics is completed by one additional theorem relating the divergence of the gradient of the electrical potential at a given point to the charge density at that point through Poisson s equation... 

If there are no reactions, the conservation of the total quantity of each species dictates that the time dependence of is given by minus the divergence of the flux ps vs), where (vs) is the drift velocity of the species s. The latter is proportional to the average force acting locally on species s, which is the thermodynamic force, equal to minus the gradient of the thermodynamic potential. In the local coupling approximation the mobility appears as a proportionality constant M. For spontaneous processes near equilibrium it is important that a noise term T] t) is retained [146]. Thus dynamic equations of the form... 

A completely different approach has also been proposed to compute dQJdx [14,15] instead of finding the derivatives of Equation (3.7), one can differentiate the basic PCM electrostatic equations and then find the solutions to the new equations. By the repeated application of the divergence theorem, this procedure leads to the following expression for the free energy gradients ... 

The implementation of the algorithm outlined above is somewhat delicate due to our use of the polar representation of the complex Hermite polynomials that project onto the final states (3 or / . When the complex polynomial is zero, the phase is ill-defined. This is reflected in the expression of the force in (44) by the apparent singularity in the off-diagonal terms. The existence of a divergence in the force, however, depends on the behavior of the gradients of the off-diagonal terms of the electronic Hamiltonian. As they usually are, or go to, zero very rapidly in regions of zero population of the final state, it is... 

The Laplacian operator on the LHS of the pressure equation is the product of the divergence operator originating from the continuity equation and the gradient operator that comes from the momentum equations. The RHS of the pressure equation consists of a sum of derivatives of the convective terms in the three components of the momentum equation. In all these terms, the outer derivative stems from the continuity equation while the inner derivative arises from the momentum equation. In a numerical approximation, it is essential that the consistency of these operators is maintained. The approximations of the terms in the Poisson equations must be defined as the product of the divergence and gradient approximations used in the basic equations. Violation of this constraint may lead to convergence problems as the continuity equation is not appropriately satisfied. 

If we take the divergence of the gradient of the scalar function ip, as is done for the pressure field formulating an equation for the pressure, we obtain ... 

Evaluation of the Curl of the Divergence of the Velocity Gradient. Begin by expressing the velocity gradient tensor using summation notation ... 

The product rule for the divergence operator is applied to both terms on the right-hand side of equation (9-27). In any coordinate system, the divergence of the product of a scalar and a vector is expanded as a product of the scalar and the divergence of the vector plus the scalar (i.e., dot) product of the vector and the gradient of the scalar. This vector identity was employed in equation (9-14). The pseudo-binary mass transfer equation for component i is... 

For the sake of simplification, the treatment of the PFTR shall be limited to steady state operations. For a more detailed reaction engineering discussion of the PFTR operated under unsteady conditions it is referred to the literature earlier recommended [44], Furthermore it shall be assumed that the input and output volume flow rates are constant and that radial gradients in concentration shall be negligible. The first assumption allows the use of the mean residence time as the characteristic time. The assumption that no axial concentration gradients are present allows the restriction to a one-dimensional analysis of the divergence in the axial direction z of the reactor. With these boundary conditions one obtains ... 

There are also other reasons that truncate the order parameter divergence such as spatial inhomogeneities or external fields. For example, to describe a spatial inhomogeneous system, a term quadratic in the gradient of the order parameter G(Vri) must be added to the density of free energy and all the Landau expansion should be integrated over the system volume ... 

According to (24) and (25), eddy accelerations are due to buoyancy forces (represented by the gradient of the geopotential) and to the Coriolis force. It turns out that the latter can be neglected when the horizontal scale of the motion is sufficiently small (a few hnndred kilometers). Then, differentiation of (24) with respect to x and of (25) with respect to y yields an equation for the divergence, ( i + v y) ... 

The system of Eqs. [18], [54], and [55] is usually solved iteratively, with each iteration defined by the successive solution of the three equations. An initial guess is first supplied for the force field P in Eq. [54], which is then solved on a discrete grid to provide the components of the cnrrent density /. The divergence of J is then computed with the steady-state continuity equation (Eq. [55]) to obtain the charge distribution that, in turn, is used in the forcing function of Poisson s equation. From the gradient of the computed potential, one derives a new (better) approximation to the force P that is used to start... 

The ion distribution within the diffuse layer is given by the solution of the Poisson equation which relates the divergence of the gradient of the electric potential to the charge density p at that point (see for instance [23, 24]) ... 

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