Generic scalar transport equation

Generic scalar transport equation

The generic scalar transport equation is a general partial differential equation that describes transport phenomena such as heat transfer, mass transfer, fluid dynamics (momentum transfer), etc. A general form of the equation is

: frac{partial phi}{partial t } + abla cdot f(t,x,phi, ablaphi) = g(t,x,phi)

where f is called the flux, and g 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 by convection 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 and diffusion.

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 a concentration of a substance c_i is

: frac{partial{ ho c_i{partial t } + abla cdot ( ho vec u c_i + vec J) = R_i

where vec u denotes the velocity field, vec J denotes the diffusion flux of the chemical species, and R_i denotes the rate of generation of R_i caused by the chemical reaction.

The "x-momentum" equation for a Newtonian fluid can be written as

: frac{partial{ ho u{partial t} + abla cdot ( ho vec u u ) = abla cdot (mu abla u ) - frac {partial p}{partial x} + B_x + V_x

where B_x is the body force in the "x"-direction and V_x includes the viscous terms that are not expressed by abla cdot (mu abla u ).

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 phi, the generic differential equation is

: underbrace{ frac{partial{ ho phi{partial t_{ ext{Transient term + underbrace{ abla cdot ( ho vec u phi )}_{ ext{Convection term =underbrace { abla cdot (Gamma abla phi )}_{ ext{Diffusion term + underbrace {S_{phi_{ ext{Source term

where Gamma is the diffusion coefficient, or diffusivity.

* The "transient term", frac{partial{ ho phi{partial t} , accounts for the accumulation of phi in the concerned control volume.
* The "convection term", abla cdot ( ho vec u phi ), accounts for the transport of phi due to the existence of the velocity field (note the velocity vec u multiplying phi ).
* The "diffusion term", abla cdot (Gamma abla phi ), accounts for the transport of phi due to its gradients.
* The "source term", S_{phi} , accounts for any sources or sinks that either create or destroy phi . 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 in financial markets can be also categorized as generic scalar transport equations. Examples include the Black-Scholes equation.

ee also

* Continuity equation
* Buckley–Leverett equation

External links

*CFDWiki|name=Generic_scalar_transport_equation


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