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 Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...

) variables change.

Mass balance

mass balance In physics, a mass balance, also called a material balance, is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have b ...

, also called a material balance, is an application of

conservation of mass In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass can ...

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:

\text + \text = \text + \text \ + \text

Differential equation

Physics

The governing equations in classical physics that are lectured
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mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...

* balance of (linear) momentum * balance of

angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...

* balance of

energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...

* balance of

entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...

* Maxwell-Faraday equation for induced electric field * Ampére-Maxwell equation for induced magnetic field * Gauss equation for electric flux * Gauss equation for magnetic flux

Classical continuum mechanics

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 Darcy's law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of ...

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.

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 continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material der ...

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 Hele-Shaw flow is defined as Stokes flow between two parallel flat plates separated by an infinitesimally small gap, named after Henry Selby Hele-Shaw, who studied the problem in 1898. Various problems in fluid mechanics can be approximated to Hele ...

Plate theory In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions ...

Kirchhoff–Love plate theory The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler-Bernoulli beam theory and ...

Mindlin–Reissner plate theory The Uflyand-Mindlin theory of vibrating plates is an extension of Kirchhoff–Love plate theory that takes into account shear deformations through-the-thickness of a plate. The theory was proposed in 1948 by Yakov Solomonovich UflyandUflyand, Y ...

* Vortex shedding * Annular fin *

Astronautics Astronautics (or cosmonautics) is the theory and practice of travel beyond Earth's atmosphere into outer space. Spaceflight is one of its main applications and space science its overarching field. The term ''astronautics'' (originally ''astron ...

Finite volume method for unsteady flow Unsteady flows are characterized as flows in which the properties of the fluid are time dependent. It gets reflected in the governing equations as the time derivative of the properties are absent. For Studying Finite-volume method for unsteady flow ...

* Acoustic theory *

Precipitation hardening Precipitation hardening, also called age hardening or particle hardening, is a heat treatment technique used to increase the yield strength of malleable materials, including most structural alloys of aluminium, magnesium, nickel, titanium, and ...

Kelvin's circulation theorem In fluid mechanics, Kelvin's circulation theorem (named after William Thomson, 1st Baron Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and ...

* Kernel function for solving integral equation of surface radiation exchanges * Nonlinear acoustics *

Large eddy simulation Large eddy simulation (LES) is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, and first explored by Deardorff (1970). LES is c ...

Föppl–von Kármán equations The Föppl–von Kármán equations, named after August Föppl and Theodore von Kármán, are a set of nonlinear partial differential equations describing the large deflections of thin flat plates. With applications ranging from the design of sub ...

Timoshenko beam theory Tymoshenko ( uk, Тимошенко, translit=Tymošenko), Timoshenko (russian: Тимошенко), or Tsimashenka/Cimašenka ( be, Цімашэнка) is a surname of Ukrainian origin. It derives from the Christian name Timothy, and its Ukrainian ...

Biology

A famous example of governing differential equations within biology is * Lotka-Volterra equations are prey-predator equations

Sequence of states

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 An oil production plant is a facility which processes production fluids from oil wells in order to separate out key components and prepare them for export. Typical oil well production fluids are a mixture of oil, gas and produced water. An oil p ...

which on the average operates in a

steady state In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...

mode. Results from one

thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...

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.

References

{{Reflist Equations