Advection equation

An example of a linear hyperbolic equation is the advection equation for flow of contaminants when the x and y velocity components are u and v, respectively. 

Even if the initial value of the level-set function (x,0) is set to be the distance function, the level set function r/j may not remain as a distance function at t >() when the advection equation, Eq. (3), is solved for . Thus, a redistance scheme is needed to enforce the condition of V0 = 1. An iterative procedure was designed (Sussman et al., 1998) to reinitialize the level-set function at each time step so that the level-set function remains as a distance function while maintaining the zero level set of the level-set function. This is achieved by solving for the steady-state solution of the equation (Sussman et al., 1994, 1998 Sussman and Fatemi, 1999) ... 

In this model, two level-set functions (d, p) are defined to represent the droplet interface (d) and the moving particle surface (p), respectively. The free surface of the droplet is taken as the zero in the droplet level-set function 0> and the advection equation (Eq. (3)) of the droplet level-set function (droplet surface. The particle level-set function (4>p) is defined as the signed distance from any given point x in the Eulerian system to the particle surface ... 

In the second model, the distribution and removal rates of tracers in the ocean are characterized through a one dimensional, (vertical) diffusion-advection equation. In this model, which ignores all horizontal processes, the equation governing the distribution of tracer in the soluble phase is [51,52,53,54] ... 

Consider the instantaneous release of a fixed mass of material, Q, into an infinite expanse of air (a ground surface will be added later). The coordinate system is fixed at the source. Assuming no reaction or molecular diffusion, the concentration C of material resulting from this release is given by the advection equation... 

Solving the purely advective equation or even introducing an advection term into the diffusion equation is a source of numerical difficulties. The simplest advection equation of a medium moving at velocity v in one dimension can be written... 

This study has shown that the application of EO can move TCE from a contaminated zone across an intact column of tight soil. The output from a ID diffusion-advection equation agrees well with the observed TCE concentration profiles, indicating that the transport equation is an appropriate model. 

Stute et al. (1992) made numerical calculation to study flow dynamics of the Great Hungarian Plain (GHP) aquifer system with the use of He concentration data. They solved the diffusion-advection equation for He transport for a two-dimensional case. 

Equation (3.47) is known as the advection equation. For one-dimensional fluid flow the advection equation reduces to... 

To develop the two-region Sangren-Sheppard model, consider a substance that traverses the endothelial cell clefts but does not enter endothelial cells (such as L-glucose, which is not taken up by cells.) This solute is assumed to exchange passively between the capillary and interstitial fluid (ISF) spaces. Applying a onedimensional approximation, the governing equation for solute concentration in the blood is the advection equation ... 

The volume of fluid (VOF) approach simulates the motion of all the phases rather than tracking the motion of the interface itself. The motion of the interface is inferred indirectly through the motion of different phases separated by an interface. Motion of the different phases is tracked by solving an advection equation of a marker function or of a phase volume fraction. Thus, when a control volume is not entirely occupied by one phase, mixture properties are used while solving governing Eqs (4.1) and (4.2). This avoids abrupt changes in properties across a very thin interface. The properties appearing in Eqs (4.1) and (4.2) are related to the volume fraction of the th phase as follows ... 

In the Hamiltonian formulation the Liouville equation can be seen as a continuity or advection equation for the probability distribution function. This theorem is fundamental to statistical mechanics and requires further attention. 

In this section we consider the translational terms in the context of a generalized advection equation [12, 77, 83, 40, 104, 39]. 

Equation (2.89) is a generalized advection equation stating that / remains unchanged if the observer moves along with the system points in phase space. 

A common feature for all the different formulations of the VOF model is that the location and orientation of the interface are defined through a volume fraction function. The evolution of this volume fraction function in time and space is determined by a scalar advection equation defined by ... 

The conservative form of the advection equation is obtained by use of the continuity equation, hence ... 

The truncation error associated with convection/advection schemes can be analyzed by using the modified equation method [205]. By use of Taylor series all the time derivatives except the 1. order one are replaced by space derivatives. When the modified equation is compared with the basic advection equation, the right-hand side can be recognized as the error. The presence of Ax in the leading error term indicate the order of accuracy of the scheme. The even-ordered derivatives in the error represent the diffusion error, while the odd-ordered derivatives represent the dispersion (or phase speed) error. 

Since v is known, this closed hyperbolic advection equation for the disperse-phase volume fraction is in conservative form. 

The idea underlying chaotic advection is the observation that a certain regular velocity field,u(x,i) can produce fluid pathlines,x(x, i), which uniformly fill the volume in an ergodic way. The motion of passive tracers is governed by the advection equation ... 

Since the vorticity is conserved within the fluid elements the equation of motion for the vortex centers in a system of point vortices is given by the advection equation r = v[x = t]. Each point vortex is passively advected just like any other fluid element by the superposition of the velocity fields generated by all the other vortices (excluding self-interaction). Thus the equation of motion of a system of point vortices is given by... 

This interpretation of the effective diffusion in terms of individual trajectories of an ensemble of particles advected by the flow and a superimposed random Brownian motion, as described by the stochastic advection equation (2.34), can be extended further. The characteristic time for molecular diffusion across the channel td L2/D gives the correlation time of the longitudinal velocity experienced by a particle. Thus the longitudinal motion can be described as a collection of independent longitudinal displacements of typical length Utd over time intervals td- Thus, for long times, t td, the effective diffusion coefficient of such random walk can be estimated as Deff (Utd)2/td U2L2/D that is consistent with (2.51) when Pe > 1. 

Consider the binary chemical reaction A + B —> C. The reaction-diffusion-advection equations read, in the case of equal diffusion co-... 

In the case of the fast binary reaction we could eliminate the reaction term from the reaction-diffusion-advection equation. But in general this is not possible. In this chapter we consider another class of chemical and biological activity for which some explicit analysis is still feasible. We consider the case in which the local-reaction dynamics has a unique stable steady state at every point in space. If this steady state concentration was the same everywhere, then it would be a trivial spatially uniform solution of the full reaction-diffusion-advection problem. However, when the local chemical equilibrium is not uniform in space, due to an imposed external inhomogeneity, the competition between the chemical and transport dynamics may lead to a complex spatial structure of the concentration field. As we will see in this chapter, for this class of chemical or biological systems the dominant processes that determine the main characteristics of the solutions are the advection and the reaction dynamics, while diffusion does not play a major role in the large Peclet number limit considered here. Thus diffusion can be neglected in a first approximation. 

To obtain the chemical concentration C(x, t) at a point x at time t one first needs to integrate the advection equation backwards in time to find the past trajectory of the fluid parcel, r(t — r) for 0 < r < t satisfying r(t) = x. Then the second equation in (6.25) for the chemical dynamics along this trajectory can be solved as (compare with (2.6))... 

In the previous chapters we discussed various aspects of chemical and biological activity in fluid flows presenting certain classes of dynamical behavior that can be described by reaction-diffusion-advection equations and analyzed using dynamical systems approaches. However, there are many research areas of chemical or biological processes taking place in fluid environments that were not covered in the previous chapters. Here we briefly discuss some of these areas and point the reader to the relevant literature for further reading. Apart from classical well-studied topics, here we also focus on more recent developments and active areas of current research. 

Movement and fate of radionuclides in groundwater follow the transport components represented by the basic diffusion/dispersion-advection equation. The following expression describes the basic equation for the advective and dispersive transport with radioactive decay and retardation for the radionuclide transport in the groundwater ... 

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