The governing equations of a mathematical model describe how the values of the unknown variables (i.e. the dependent variables) change when one or more of the known (i.e. independent) variables change.
Physical systems can be modeled phenomenologically at various levels of sophistication, with each level capturing a different degree of detail about the system. A governing equation represents the most detailed and fundamental phenomenological model currently available for a given system.
For example, at the coarsest level, a beam is just a 1D curve whose torque is a function of local curvature. At a more refined level, the beam is a 2D body whose stress-tensor is a function of local strain-tensor, and strain-tensor is a function of its deformation. The equations are then a PDE system. Note that both levels of sophistication are phenomenological, but one is deeper than the other. As another example, in fluid dynamics, the Navier-Stokes equations are more refined than Euler equations.
As the field progresses and our understanding of the underlying mechanisms deepens, governing equations may be replaced or refined by new, more accurate models that better represent the system's behavior. These new governing equations can then be considered the deepest level of phenomenological model at that point in time.
A mass balance, also called a material balance, is an application of conservation of mass to the analysis of physical systems. It is the simplest governing equation, and it is simply a budget (balance calculation) over the quantity in question:
Differential equationedit
Physicsedit
The governing equations[1][2] in classical physics that are lectured[3][4][5][6] at universities are listed below.
The basic equations in classical continuum mechanics are all balance equations, and as such each of them contains a time-derivative term which calculates how much the dependent variable change with time. For an isolated, frictionless / inviscid system the first four equations are the familiar conservation equations in classical mechanics.
Darcy's law of groundwater flow has the form of a volumetric flux caused by a pressure gradient. A flux in classical mechanics is normally not a governing equation, but usually a defining equation for transport properties. Darcy's law was originally established as an empirical equation, but is later shown to be derivable as an approximation of Navier-Stokes equation combined with an empirical composite friction force term. This explains the duality in Darcy's law as a governing equation and a defining equation for absolute permeability.
The non-linearity of the material derivative in balance equations in general, and the complexities of Cauchy's momentum equation and Navier-Stokes equation makes the basic equations in classical mechanics exposed to establishing of simpler approximations.
Some examples of governing differential equations in classical continuum mechanics are
A governing equation may also be a state equation, 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 an oil production plant which on the average operates in a steady state mode. 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.
^Fletcher, Clive A.J. (1991). Computational Techniques for Fluid Dynamics 2; Chapter 1; Fluid Dynamics: The Governing Equations. Vol. 2. Berlin / Heidelberg, Germany: Springer Berlin Heidelberg. pp. 1–46. ISBN978-3-642-58239-4.
^Kline, S.J. (2012). Similitude and Approximation Theory (2012 ed.). Berlin / Heidelberg, Germany: Springer Science & Business Media. ISBN9783642616389.
^Nakariakov, Prof. Valery (2015). Lecture PX392 Plasma Electrodynamics (Lecture PX392 2015-2016 ed.). Coventry, England, UK: Department of Physics, University of Warwick.[1]
^Tryggvason, Viola D. Hank Professor Gretar (2011). Lecture 28 Computational Fluid Dynamics - CFD Course from B. Daly (1969) Numerical methods (Lecture 28 CFD Course 2011 ed.). Notre Dame, Indiana, US: Department of Aerospace and Mechanical Engineering, University of Notre Dame.[2]
^Münchow, Physical Oceanographer Ph.D. Andreas (2012). Lecture MAST-806 Geophysical Fluid Dynamics (Lecture MAST-806 2012 ed.). Newark, Delaware, US: University of Delaware.[3]
^Brenner, Glover Prof. Michael P. (2000). The dynamics of thin sheets of fluid Part 1 Water bells by G.I. Taylor (MIT course number 18.325 Spring 2000 ed.). Cambridge, Massachusetts, US: Harvard University.[4]
November 07, 2023
governing, equation, governing, equations, mathematical, model, describe, values, unknown, variables, dependent, variables, change, when, more, known, independent, variables, change, physical, systems, modeled, phenomenologically, various, levels, sophisticati. The governing equations of a mathematical model describe how the values of the unknown variables i e the dependent variables change when one or more of the known i e independent variables change Physical systems can be modeled phenomenologically at various levels of sophistication with each level capturing a different degree of detail about the system A governing equation represents the most detailed and fundamental phenomenological model currently available for a given system For example at the coarsest level a beam is just a 1D curve whose torque is a function of local curvature At a more refined level the beam is a 2D body whose stress tensor is a function of local strain tensor and strain tensor is a function of its deformation The equations are then a PDE system Note that both levels of sophistication are phenomenological but one is deeper than the other As another example in fluid dynamics the Navier Stokes equations are more refined than Euler equations As the field progresses and our understanding of the underlying mechanisms deepens governing equations may be replaced or refined by new more accurate models that better represent the system s behavior These new governing equations can then be considered the deepest level of phenomenological model at that point in time Contents 1 Mass balance 2 Differential equation 2 1 Physics 2 2 Classical continuum mechanics 2 3 Biology 3 Sequence of states 4 See also 5 ReferencesMass balance editA mass balance also called a material balance is an application of conservation of mass to the analysis of physical systems It is the simplest governing equation and it is simply a budget balance calculation over the quantity in question Input Generation Output Accumulation Consumption displaystyle text Input text Generation text Output text Accumulation text Consumption nbsp Differential equation editPhysics edit The governing equations 1 2 in classical physics that are lectured 3 4 5 6 at universities are listed below balance of mass balance of linear momentum balance of angular momentum balance of energy balance of entropy Maxwell Faraday equation for induced electric field Ampere Maxwell equation for induced magnetic field Gauss equation for electric flux Gauss equation for magnetic flux Classical continuum mechanics edit The basic equations in classical continuum mechanics are all balance equations and as such each of them contains a time derivative term which calculates how much the dependent variable change with time For an isolated frictionless inviscid system the first four equations are the familiar conservation equations in classical mechanics Darcy s law of groundwater flow has the form of a volumetric flux caused by a pressure gradient A flux in classical mechanics is normally not a governing equation but usually a defining equation for transport properties Darcy s law was originally established as an empirical equation but is later shown to be derivable as an approximation of Navier Stokes equation combined with an empirical composite friction force term This explains the duality in Darcy s law as a governing equation and a defining equation for absolute permeability The non linearity of the material derivative in balance equations in general and the complexities of Cauchy s momentum equation and Navier Stokes equation makes the basic equations in classical mechanics exposed to establishing of simpler approximations Some examples of governing differential equations in classical continuum mechanics are Hele Shaw flow Plate theory Kirchhoff Love plate theory Mindlin Reissner plate theory 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 Foppl von Karman equations Timoshenko beam theory Biology edit A famous example of governing differential equations within biology is Lotka Volterra equations are prey predator equationsSequence of states editA governing equation may also be a state equation 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 an oil production plant which on the average operates in a steady state mode 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 See also editConstitutive equation Mass balance Master equation Mathematical model Primitive equationsReferences edit Fletcher Clive A J 1991 Computational Techniques for Fluid Dynamics 2 Chapter 1 Fluid Dynamics The Governing Equations Vol 2 Berlin Heidelberg Germany Springer Berlin Heidelberg pp 1 46 ISBN 978 3 642 58239 4 Kline S J 2012 Similitude and Approximation Theory 2012 ed Berlin Heidelberg Germany Springer Science amp Business Media ISBN 9783642616389 Nakariakov Prof Valery 2015 Lecture PX392 Plasma Electrodynamics Lecture PX392 2015 2016 ed Coventry England UK Department of Physics University of Warwick 1 Tryggvason Viola D Hank Professor Gretar 2011 Lecture 28 Computational Fluid Dynamics CFD Course from B Daly 1969 Numerical methods Lecture 28 CFD Course 2011 ed Notre Dame Indiana US Department of Aerospace and Mechanical Engineering University of Notre Dame 2 Munchow Physical Oceanographer Ph D Andreas 2012 Lecture MAST 806 Geophysical Fluid Dynamics Lecture MAST 806 2012 ed Newark Delaware US University of Delaware 3 Brenner Glover Prof Michael P 2000 The dynamics of thin sheets of fluid Part 1 Water bells by G I Taylor MIT course number 18 325 Spring 2000 ed Cambridge Massachusetts US Harvard University 4 Retrieved from https en wikipedia org w index php title Governing equation amp oldid 1179415469, wikipedia, wiki, book, books, library,
