User:Frode54/sandboxgoverning

Elements of a mathematical model
The traditional mathematical model contains four major elements. These are
 * 1) Governing equations
 * 2) Constituent equations
 * 3) Constraints
 * 4) Kinematic equations

The governing equations describes how the unknown variables (i.e. the dependent variables) will change. The change of variables w.r.t. time may be explicit (i.e. a governing equation includes derivative with respect to time) or implicit (e.g. a governing equation has velocity or flux as unknown variable). The classic governing equations in physics are


 * Balance of mass
 * Balance of (linear) momentum
 * Balance of angular momentum
 * Balanse of energy
 * Balance of entropy

For isolated systems the upper four equations are the familiar conservation equations. A governing equation may also take the form of a flux equation like the diffusion equation or the heat conduction equation. In these cases the flux itself is a variable describing change of the unknown variable or property (e.g. mole concentration or internal energy or temperature). A governing equation may also be an approximation and adaption of the above basic equations to the situation or model in question. A governing equation may also be derived directly from experimental results and therefore be an empiric equation. These last two considerations are the main the reasons for introducing the concept of governing equations as an alternative to the term balance equation used for the most famous and fundamental equations. A governing equation may also be an equation describing the state of the system, and thus actually be a constituent equation that has "risen through the ranks" because the model in question is not meant to include a time-dependent term in the equation. This is the case with thermodynamic equilibrium calculations where the results of one equation, or one calculation, are input data to the next equilibrium calculation together with some new state parameters and so on. In this case the algorithm and sequence of input data form a chain of actions, or calculations, that describes a change of states from the first state (based solely on input data) to the last state that finally comes out of the calculation sequence. Some examples are


 * Hele-Shaw flow
 * Bending of Kirchhoff-Love plates
 * Bending of thick Mindlin plates
 * Bending of Reissner-Stein cantilever plates
 * Vortex shedding
 * Annular fin
 * Astronautics
 * Finite volume method for unsteady flow
 * Acoustic theory
 * Precipitation hardening
 * Kelvin's circulation theorem
 * Kernel function for solving integral equation of surface radiation exchanges
 * Nonlinear acoustics
 * Large eddy simulation
 * Föppl–von Kármán equations
 * Timoshenko beam theory

The constituent equations describes how some unknown variables are related to each other or how a parameter (often a material property) varies in space or as a function of unknown variables (e.g. temperature). Some examples are


 * Pcow = F(Sw) - describes how much water (relative to existing oil) the capillary pressure will lift to a height with a given capillary pressure value
 * P = F(T,V,n) - equation of state is used to describe the state of a fluid (pressure, temperature, volume and mole relationship) at the end of a dynamic time step
 * $$\mu_i$$(P,Vg,T,ng,y) = $$\mu_i$$ (P,Vl,T,nl,x) - a fluid mixture splits into equilibrium of a liquid in contact with a gas
 * D = F(T) - diffusion coefficient is a function of temperature
 * K = F(x,y,z) - Permeability values for a porous rock is being distributed in a 3D model of an oil reservoir

The constraints are specific statements saying how the unknown variables, and possibly their derivatives, must behave at the boundaries of the model, at the sinks and sources within the model, and/or initially. The specific statements may be equations (i.e. explicit or implicit functions), inequalities or algorithms in the form of subroutines or even inclusion of adapted programs.

The kinematic equations of Galileo Galilei, for pure translation or including rotational coordinate systems, is fundamental in both the classic mechanics of rigid bodies and the traditional continuum mechanics. In continuum mechanic for fluids and (elastic) solids, there is seldom need for change of coordinate system, especially to moving coordinate systems, so kinematic equations usually have more focus in very basic research than in applied continuum mechanics.