Governing equation

Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, game theoretic models, recurrence relations, an algorithm for calculation of a sequence of related states (e.g. equilibrium states) and possibly even more forms. The governing equations of a mathematical model 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 or an algorithm). The classic governing equations in continuum mechanics are

• Balance of mass
• Balance of (linear) momentum
• Balance of angular momentum
• Balance of energy
• Balance of entropy

For isolated systems the upper four equations are the familiar conservation equations in physics. 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. A governing equation may also be an equation describing the state of the system, and thus actually be a constitutive equation that has "stepped up the ranks" because the model in question was not meant to include a time-dependent term in the equation. This is the case for a model of a petroleum processing plant. Results from one thermodynamic equilibrium 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 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 using differential equations are

• Lotka-Volterra equations are predator-prey equations
• Hele-Shaw flow
• Plate theory
  • Kirchhoff–Love plate theory or Bending of Kirchhoff-Love plates
  • Mindlin–Reissner plate theory or Bending of thick Mindlin plates or 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

References

Prof. Valery Nakariakov, Department of Physics, University of Warwick; Lecture PX392 Plasma Electrodynamics (2015-2016)

Viola D. Hank Professor Gretar Tryggvason, Department of Aerospace and Mechanical Engineering, University of Notre Dame; 2011 lecture 28 Computational Fluid Dynamics - CFD Course from B. Daly (1969) Numerical methods

Physical Oceanographer Andreas Münchow Ph.D., University of Delaware, Lecture MAST-806 Geophysical Fluid Dynamics (2012)

S.J. Kline (2012) Similitude and Approximation Theory

Glover Professor of Applied Mathematics and Applied Physics Michael P. Brenner, Harvard University; MIT course number 18.325 Spring 2000 The dynamics of thin sheets of fluid Part 1 Water bells by G.I. Taylor