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The '''governing equations''' of a [[mathematical model]] describe how the values of the unknown variables (i.e. the [[dependent variable]]s) change when one or more of the known (i.e. [[independent variable|independent]]) variables change.


Physical systems can be modeled [[Phenomenological model|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 [[Euler–Bernoulli beam theory|beam]] is just a 1D curve whose torque is a function of local curvature. At a [[Timoshenko–Ehrenfest beam theory|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 (fluid dynamics)|Euler equations]].
The '''governing equations''' of a [[mathematical model]] describe how the values of the unknown variables (i.e. the [[dependent variable]]s) will change. The change of the value of a variable with respect to time may be explicit, in that a governing equation includes a derivative with respect to time, or implicit, such as when a governing equation has velocity or flux as an unknown variable.


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.
==Examples==

⚫
The classic governing equations<ref name="Fletcher1991">{{cite book|last1=Fletcher|first1=Clive A.J.|year=1991|title=Computational Techniques for Fluid Dynamics 2; Chapter 1; Fluid Dynamics: The Governing Equations |pages= 1-46|volume=2|publisher=Springer Berlin Heidelberg|location=Berlin / Heidelberg, Germany|isbn=978-3-642-58239-4}}</ref><ref
== Mass balance ==
⚫
name="Kline2012">{{cite book|last1=Kline|first1=S.J.|year=2012|title=Similitude and Approximation Theory|edition=2012|publisher=Springer Science & Business Media|location=Berlin / Heidelberg, Germany|isbn=9783642616389}}</ref> in physics that are
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:
⚫
lectured<ref name="Nakariakov2015">{{cite book|last1=Nakariakov|first1=Prof. Valery|year=2015|title=Lecture PX392 Plasma Electrodynamics |edition= Lecture PX392 2015-2016|publisher=Department of Physics, University of Warwick|location= Coventry, England, UK}}[https://www2.warwick.ac.uk/fac/sci/physics/research/cfsa/people/valery/teaching/px420/addres/mhd_int1.pdf]</ref><ref name="Tryggvason2011">{{cite book|last1=Tryggvason|first1=Viola D. Hank Professor Gretar|year=2011|title=Lecture 28 Computational Fluid Dynamics - CFD Course from B. Daly (1969) Numerical methods|edition= Lecture 28 CFD Course 2011|publisher=Department of Aerospace and Mechanical Engineering, University of Notre Dame|location= Notre Dame, Indiana, US}}[http://www3.nd.edu/~gtryggva/CFD-Course/2011-Lecture-28.pdf]</ref><ref

name="Münchow2012">{{cite book|last1=Münchow |first1=Physical Oceanographer Ph.D. Andreas|year=2012|title=Lecture MAST-806 Geophysical Fluid Dynamics |edition=Lecture MAST-806 2012|publisher=University of Delaware|location=Newark, Delaware, US}}[http://muenchow.cms.udel.edu/html/classes/gfd/book/IntroGFDChapt3.pdf]</ref><ref
<div align="center"><math> \text{Input} + \text{Generation} = \text{Output} + \text{Accumulation} \ + \text{Consumption} </math></div>
name="Brenner2000">{{cite book|last1=Brenner|first1=Glover Prof. Michael P.|year=2000|title=The dynamics of thin sheets of fluid Part 1 Water bells by G.I. Taylor|edition=MIT course number 18.325 Spring 2000 |publisher=Harvard University|location=Cambridge, Massachusetts, US}}[http://www.seas.harvard.edu/brenner/taylor/handouts/waterbell/node2.html]</ref>

==Differential equation==
===Physics===
⚫
The governing equations<ref name="Fletcher1991">{{cite book|last1=Fletcher|first1=Clive A.J.|year=1991|title=Computational Techniques for Fluid Dynamics 2; Chapter 1; Fluid Dynamics: The Governing Equations |pages= 1–46|volume=2|publisher=Springer Berlin Heidelberg|location=Berlin / Heidelberg, Germany|isbn=978-3-642-58239-4}}</ref><ref
⚫
name="Kline2012">{{cite book|last1=Kline|first1=S.J.|year=2012|title=Similitude and Approximation Theory|edition=2012|publisher=Springer Science & Business Media|location=Berlin / Heidelberg, Germany|isbn=9783642616389}}</ref> in classical physics that are
⚫
lectured<ref name="Nakariakov2015">{{cite book |last1=Nakariakov |first1=Prof. Valery |title=Lecture PX392 Plasma Electrodynamics |publisher=Department of Physics, University of Warwick |year=2015 |edition=Lecture PX392 2015-2016 |location=Coventry, England, UK}}[https://www2.warwick.ac.uk/fac/sci/physics/research/cfsa/people/valery/teaching/px420/addres/mhd_int1.pdf]</ref><ref name="Tryggvason2011">{{cite book |last1=Tryggvason |first1=Viola D. Hank Professor Gretar |title=Lecture 28 Computational Fluid Dynamics - CFD Course from B. Daly (1969) Numerical methods |publisher=Department of Aerospace and Mechanical Engineering, University of Notre Dame |year=2011 |edition=Lecture 28 CFD Course 2011 |location=Notre Dame, Indiana, US}}[http://www3.nd.edu/~gtryggva/CFD-Course/2011-Lecture-28.pdf]</ref><ref name="Münchow2012">{{cite book |last1=Münchow |first1=Physical Oceanographer Ph.D. Andreas |title=Lecture MAST-806 Geophysical Fluid Dynamics |publisher=University of Delaware |year=2012 |edition=Lecture MAST-806 2012 |location=Newark, Delaware, US}}[http://muenchow.cms.udel.edu/html/classes/gfd/book/IntroGFDChapt3.pdf]</ref><ref name="Brenner2000">{{cite book |last1=Brenner |first1=Glover Prof. Michael P. |title=The dynamics of thin sheets of fluid Part 1 Water bells by G.I. Taylor |publisher=Harvard University |year=2000 |edition=MIT course number 18.325 Spring 2000 |location=Cambridge, Massachusetts, US}}[http://www.seas.harvard.edu/brenner/taylor/handouts/waterbell/node2.html]</ref>
at universities are listed below.
at universities are listed below.


{{div col|2}}
{{div col|colwidth=30em}}
* balance of [[mass]]
* balance of [[mass]]
* balance of (linear) [[momentum]]
* balance of (linear) [[momentum]]
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* balance of [[entropy]]
* balance of [[entropy]]
* [[Maxwell's equations#Formulation in SI units convention|Maxwell-Faraday equation for induced electric field]]
* [[Maxwell's equations#Formulation in SI units convention|Maxwell-Faraday equation for induced electric field]]
* Ampére-Maxwell equation for induced magnetic field
* [[Maxwell's equations#Formulation in SI units convention|Ampére-Maxwell equation for induced magnetic field]]
* Gauss equation for electric flux
* [[Maxwell's equations#Formulation in SI units convention|Gauss equation for electric flux]]
* Gauss equation for magnetic flux
* [[Maxwell's equations#Formulation in SI units convention|Gauss equation for magnetic flux]]
{{div col end}}
{{div col end}}


=== Classical continuum mechanics ===
The basic equations in [[Continuum mechanics|classical continuum mechanics]] are all [[Continuum mechanics#Governing equations|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#Transport fluxes|flux]] caused by a pressure gradient. A flux in classical mechanics is normally not a governing equation, but usually a defining equation for [[Transport phenomena|transport properties]]. [[Darcy's law#Derivation|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.
For isolated systems the first four equations are the familiar [[conservation equation]]s in [[classical mechanics]]. A governing equation may also take the form of a [[flux]] equation such as 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]]).


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.
⚫
A governing equation may also be an approximation and adaptation of the above basic equations to the situation or model in question. A governing equation may also be derived directly from [[experiment]]al results and therefore be an empirical equation. A governing equation may also be a [[state variable#Control systems engineering|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 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
Some examples of governing differential equations in classical continuum mechanics are


{{div col|2}}
{{div col|colwidth=30em}}
⚫
* [[Lotka-Volterra equations]] are predator-prey equations
* [[Hele-Shaw flow]]
* [[Hele-Shaw flow]]
* [[Plate theory]]
* [[Plate theory]]
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* [[Timoshenko beam theory]]
* [[Timoshenko beam theory]]
{{div col end}}
{{div col end}}

===Biology===
A famous example of governing differential equations within biology is

{{div col|colwidth=30em}}
⚫
* [[Lotka-Volterra equations]] are prey-predator equations
{{div col end}}

== Sequence of states ==

⚫
A governing equation may also be a [[state variable#Control systems engineering|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 ==
== See also ==
* [[Constitutive equation]]
* [[Mass balance]]
* [[Master equation]]
* [[Master equation]]
⚫
* [[Primitive equations]]
* [[Mathematical model]]
* [[Mathematical model]]
⚫
* [[Primitive equations]]


== References ==
== References ==

Latest revision as of 00:14, 10 October 2023

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.

Mass balance

[edit]

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 equation

[edit]

Physics

[edit]

The governing equations[1][2] in classical physics that are lectured[3][4][5][6] at universities are listed below.

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

Biology

[edit]

A famous example of governing differential equations within biology is

Sequence of states

[edit]

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.

See also

[edit]

References

[edit]
  1. ^ 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.
  2. ^ Kline, S.J. (2012). Similitude and Approximation Theory (2012 ed.). Berlin / Heidelberg, Germany: Springer Science & Business Media. ISBN 9783642616389.
  3. ^ Nakariakov, Prof. Valery (2015). Lecture PX392 Plasma Electrodynamics (Lecture PX392 2015-2016 ed.). Coventry, England, UK: Department of Physics, University of Warwick.[1]
  4. ^ 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]
  5. ^ 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]
  6. ^ 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]
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