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.
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The governingequations of a mathematical model describe how the values of the unknown variables (i.e. the dependent variables) change when one or more...
y)=w_{x}(x)+y\,\theta _{x}(x)\,.} The governingequations for the plate then reduce to two coupled ordinary differential equations: b D d 4 w x d x 4 = q 1 ( x...
discretizes and simulates an orthogonal 3-D form of the governing groundwater flow equation. However, it has an option to run in a "quasi-3D" mode if...
an infinite cylinder is related to the Strouhal number by the following equation: S t = f D V {\displaystyle \mathrm {St} ={\frac {fD}{V}}} Where S t {\displaystyle...
governingequations as the time derivative of the properties are absent. For Studying Finite-volume method for unsteady flow there is some governing equations...
terms of vector calculus this is the curl of the flow velocity). The governingequation is: D ω D t = ∂ ω ∂ t + ( u ⋅ ∇ ) ω = ( ω ⋅ ∇ ) u − ω ( ∇ ⋅ u ) +...
systems of governingequations of fluid dynamics (for sound waves in liquids and gases) and elasticity (for sound waves in solids). These equations are generally...
and r {\displaystyle r} is the second phase particle radius. This governingequation shows that for dislocation bowing the strength is inversely proportional...
frictional loss is described using the Darcy–Weisbach equation. One obtains a governingequation of dividing flow as follows: where W {\displaystyle W\...
example of explicit time integration where the function that defines governingequation is evaluated at the current time. Later, the method was given a more...
homogeneous part of the governingequation exactly and for the particular part approximately. Recall that the governingequation for a sandwich beam is...
w}{\partial y}}} In the Kirchhoff–Love plate theory for plates the governingequations are N α β , α = 0 {\displaystyle N_{\alpha \beta ,\alpha }=0} and...
derivative for diffusion nonlocality. The time-space fractional diffusion governingequation can be written as ∂ α u ∂ t α = − K ( − Δ ) β u . {\displaystyle {\frac...
of the surrounding fluid (continuous phase) is solved through the governingequations, while the particulates (dispersed phase) are tracked independently...
In metallurgy, solid solution strengthening is a type of alloying that can be used to improve the strength of a pure metal. The technique works by adding...
coupled equations. The equations can be divided into three categories according to the nature and intended role: governingequation, auxiliary equations and...
10–103 Hz. Using the governingequation as the Navier-Stokes equation being subject to the no-slip boundary condition, the equation is: ∂ u ∂ t + ( u ⋅...
heat exchange in liquid–gas heat exchanger systems. To derive the governingequation of an annular fin, certain assumptions must be made. The fin must...
similar unsteady flows. The governingequations of a steady problem have one dimension fewer (time) than the governingequations of the same problem without...
{}T\right)+\Phi {}} One way to fix this problem is to change the governingequation; known as preconditioning; which can also increases the accuracy....
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