- Generic scalar transport equation
The generic scalar transport equation is a general
partial differential equation that describestransport phenomena such asheat transfer ,mass transfer ,fluid dynamics (momentum transfer), etc. A general form of the equation is:
where is called the flux, and is called the source.
All the transfer processes express a certain conservation principle. In this respect, any differential equation addresses a certain quantity as its dependent variable and thus expresses the balance between the phenomena affecting the evolution of this quantity. For example, the
temperature of a fluid in a heated pipe is affected byconvection due to the solid-fluid interface, and due to the fluid-fluid interaction. Furthermore, temperature is also diffused inside the fluid. For a steady-state problem, with the absence of sources, a differential equation governing the temperature will express a balance between convection anddiffusion .A brief inspection of the equations governing various transport phenomena reveal that all of these equations can be put into a generic form thus allowing a systematic approach for a
computer simulation . For example, the conservation equation of aconcentration of a substance is:
where denotes the velocity field, denotes the diffusion flux of the chemical species, and denotes the rate of generation of caused by the chemical reaction.
The "x-momentum" equation for a
Newtonian fluid can be written as:
where is the body force in the "x"-direction and includes the viscous terms that are not expressed by
Upon inspection of the above equations, it can be inferred that all the dependent variables seem to obey a generalized conservation principle. If the dependent variable (scalar or vector) is denoted by , the generic differential equation is
:
where is the diffusion coefficient, or diffusivity.
* The "transient term", , accounts for the accumulation of in the concerned
control volume .
* The "convection term", , accounts for the transport of due to the existence of the velocity field (note the velocity multiplying ).
* The "diffusion term", , accounts for the transport of due to its gradients.
* The "source term", , accounts for any sources or sinks that either create or destroy . Any extra terms that cannot be cast into the convection or diffusion terms are considered as source terms.The objective of all
discretization techniques (finite difference , finite element, finite volume, boundary element, etc.) is to devise a mathematical formulation to transform each of these terms into an algebraic equation. Once applied to all control volumes in a given mesh, we obtain a full linear system of equations that needs to be solved.calar transport equation in financial mathematics
Some equations that governs the dynamics of
financial derivatives infinancial markets can be also categorized as generic scalar transport equations. Examples include theBlack-Scholes equation .ee also
*
Continuity equation
*Buckley–Leverett equation External links
*CFDWiki|name=Generic_scalar_transport_equation
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