Hydrodynamics continuity equation

This is merely the hydrodynamic continuity equation p = — div pv written for an arbitrary number of dimensions.If one now lets y(t) run over all realizations of Y(t), with their appropriate probabilities, equation (5.2) becomes a linear stochastic differential equation for p(w, t). 

In this form there is a close correspondence with the equations of hydrodynamics, or vortex-free flow of a fluid under the influence of conservative forces. Equation (12) resembles a hydrodynamic continuity equation if a2 is considered to be a density and if the stream velocity v = V(j). As V x v = 0,... 

Combining eq.(5.62) with the hydrodynamic continuity equation for each ion ... 

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)... 

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]. 

Equation (8.64) allows the shape of the velocity profile to be calculated (e.g., substitute ytr = constant and see what happens), but the magnitude of the velocity depends on the yet unknown value for dPjdz. As is often the case in hydrodynamic calculations, pressure drops are determined through the use of a continuity equation. Here, the continuity equation takes the form of a constant mass flow rate down the tube ... 

Trinh et al. [399] derived a number of similar expressions for mobility and diffusion coefficients in a similar unit cell. The cases considered by Trinh et al. were (1) electrophoretic transport with the same uniform electric field in the large pore and in the constriction, (2) hindered electrophoretic transport in the pore with uniform electric fields, (3) hydrodynamic flow in the pore, where the velocity in the second pore was related to the velocity in the first pore by the overall mass continuity equation, and (4) hindered hydrodynamic flow. All of these four cases were investigated with two different boundary condi-... 

Equation (7) is an exact analogue of the continuity equation (1.7) of hydrodynamics, and this allows definition of a probability current density... 

It has been shown that for practical calculation of the density quantities p(r, t) and j(r, t), one can have several schemes of which we discuss only two. In the first scheme, one has to solve the hydrodynamical equations, i.e., the continuity equation... 

This formulation results very insightful according to Equation 8.30, particles move under the action of an effective force — We , i.e., the nonlocal action of the quantum potential here is seen as the effect of a (nonlocal) quantum force. From a computational viewpoint, these formulation results are very interesting in connection to quantum hydrodynamics [21,27]. Thus, Equations 8.27 can be reexpressed in terms of a continuity equation and a generalized Euler equation. As happens with classical fluids, here also two important concepts that come into play the quantum pressure and the quantum vortices [28], which occur at nodal regions where the velocity field is rotational. 

Understanding the order of the hydrodynamics equations, continuity and momentum, can be somewhat confusing and possibly not the same from problem to problem. The continuity and momentum equations must be viewed as a closely coupled system. Again, it is clear that the momentum equations are second order in velocity and first order in pressure. The continuity equation is first order in density. However, an equation of state requires that density be a function of pressure, and vice versa. Density and pressure must be dependent on each other through an algebraic equation. Therefore a substitution could be done to eliminate either pressure or density. As a result the coupled system is third order, which can present some practical issues for boundary-condition assignment. The first-order behavior must carry information from some portions of the boundary into the domain, but it does not communicate information back. Therefore, over some portions of a problem... 

In the 1960s, the start of application of computers to the practice of marine research gave a pulse to the development of numerical diagnostic hydrodynamic models [33]. In them, the SLE (or the integral stream function) field is calculated from the three-dimensional density field in the equation of potential vorticity balance over the entire water column from the surface to the bottom. The iterative computational procedure is repeated until a stationary condition of the SLE (or the integral stream function) is reached at the specified fixed density field. Then, from equations of momentum balance, horizontal components of the current vector are obtained, while the continuity equation provides the calculations of the vertical component. The advantage of this approach is related to the absence of the problem of the choice of the zero surface and to the account for the coupled effect of the baroclinicity of... 

Many hydrodynamic systems have been studied theoretically7-11. The solution to (5.45) proceeds through analysis of the velocity profile, derived from the momentum continuity equation and which is, for an incompressible fluid,... 

The exact solution of the convection-diffusion equations is very complicated, since the theoretical treatments involve solving a hydrodynamic problem, i.e., the determination of the solution flow velocity profile by using the continuity equation or -> Navier-Stokes equation. For the calculation of a velocity profile the solution viscosity, densities, rotation rate or stirring rate, as well as the shape of the electrode should be considered. 

As the flow was presumed to be incompressible, as a result of the continuity equation, dwjdxj = 0 holds. The pressure satisfies the potential equation. Equations (3.101) and (3.102) are the basis of the hydrodynamic theory of lubrication, which encompasses, among others, the oil flow in bearings. 

The representation (1.1.17) of the hydrodynamic parameters of turbulent flow as the sum of the average and fluctuating components followed by the averaging process made it possible, based on the continuity equation (1.1.3) and the Navier-Stokes equations (1.1.4), to obtain (under some assumptions) the Reynolds equations... 

Averaging the pore scale transport process over the REV and assigning the average properties to the centroid of the REV results in continuous functions in space of the hydrodynamic properties and state variables. As for the flow equation (1), differential calculus can be applied to establish mass and momentum balance equations for infinitesimal small soil volume and time increments. For the case of inert solute transport in a macroscopic homogeneous soil, the general continuity equation applies ... 

Although it is beyond the scope of this chapter to treat hydrodynamics in any depth, a brief discussion of some of the concepts, terms, and equations is included to provide some feeling for the approach and the results that follow. For an incompressible fluid (i.e., a fluid whose density is constant in time and space), the velocity profile is obtained by two differential equations with appropriate boundary conditions. The continuity equation. 

In essence, the Fokker-Planck equation is a continuous equation, of temporary evolution of the electronic flux, compelled to an external potential V(x) with the drift factor and one of diffusion (stochastic noise). This thing can be easily notice if the Fokker-Planck equation is rewritten (5.250), as example, in a hydrodynamic form ... 

In polymer solutions, it is usually not the displacement of the mass that is measured, and consequently not but the change in the concentration dc with time. During diffusion, the law of conservation of mass must be maintained, i.e., the hydrodynamic continuity condition established by inserting equation (7-12), bearing in mind the opposing signs of flow and change in concentration ... 

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