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| The '''Cauchy momentum equation''' is a vector [[partial differential equation]] put forth by [[Cauchy]] that describes the non-relativistic momentum transport in any [[Continuum mechanics|continuum]]:<ref>{{cite book
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| | last = Acheson
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| | first = D. J.
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| | title = Elementary Fluid Dynamics
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| | publisher = [[Oxford University Press]]
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| | year = 1990
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| |page = 205
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| | isbn = 0-19-859679-0}}</ref>
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| :<math>\rho \frac{D \mathbf{v}}{D t} = \nabla \cdot \boldsymbol{\sigma} + \mathbf{f}</math> | |
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| or, with the [[convective derivative|material derivative]] expanded out,
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| :<math>\rho \left[\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v}\right] = \nabla \cdot \boldsymbol{\sigma} + \mathbf{f}</math>
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| where <math>\rho</math> is the [[density]] of the continuum, <math>\boldsymbol{\sigma}</math> is the [[Cauchy stress tensor|stress tensor]], and <math>\mathbf{f}</math> contains all of the [[body force]]s per unit volume (often simply [[density]] times [[gravity]]). <math>\mathbf{v}</math> is the velocity vector field, which depends on time and space.
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| The stress tensor is sometimes split into pressure and the deviatoric stress tensor:
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| :<math>\boldsymbol{\sigma} = -p\mathbb{I} + \mathbb{T}</math>
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| where <math>\scriptstyle \mathbb{I}</math> is the <math>\scriptstyle 3 \times 3</math> [[identity matrix]] and <math>\scriptstyle \mathbb{T}</math> the deviatoric stress tensor. The divergence of the stress tensor can be written as
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| :<math>\nabla \cdot \boldsymbol{\sigma} = -\nabla p + \nabla \cdot\mathbb{T}.</math> | |
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| All non-relativistic momentum conservation equations, such as the [[Navier–Stokes equation]], can be derived by beginning with the Cauchy momentum equation and specifying the stress tensor through a [[constitutive relation]].
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| ==Derivation==
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| Applying [[Newton's second law]] (<math>i^{th}</math> component) to a [[control volume]] in the continuum being modeled gives:
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| :<math>m a_i = F_i\,</math> | |
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| :<math>\rho \int_{\Omega} \frac{d u_i}{d t} \, dV = \int_{\Omega} \nabla_j\sigma_i^j \, dV + \int_{\Omega} f_i \, dV</math>
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| :<math> \int_{\Omega} (\rho \frac{d u_i}{d t} - \nabla_j\sigma_i^j - f_i )\, dV = 0</math>
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| :<math> \rho \dot{u_i} - \nabla_j\sigma_i^j - f_i = 0 </math>
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| where <math>\Omega</math> represents the control volume. Since this equation must hold for any control volume, it must be true that the integrand is zero, from this the Cauchy momentum equation follows. The main challenge in deriving this equation is establishing that the [[tensor derivative|derivative]] of the stress tensor is one of the forces that constitutes <math>F_i</math>.
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| ===Cartesian coordinates===
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| :<math>\begin{align}
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| x:\;\; \rho \left(\frac{\partial u_x}{\partial t} + u_x \frac{\partial u_x}{\partial x} + u_y \frac{\partial u_x}{\partial y} + u_z \frac{\partial u_x}{\partial z}\right)
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| &= -\frac{\partial P}{\partial x} + \frac{\partial \tau_{xx}}{\partial x} + \frac{\partial \tau_{xy}}{\partial y} + \frac{\partial \tau_{xz}}{\partial z} + \rho g_x
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| \\
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| y:\;\; \rho \left(\frac{\partial u_y}{\partial t} + u_x \frac{\partial u_y}{\partial x} + u_y \frac{\partial u_y}{\partial y}+ u_z \frac{\partial u_y}{\partial z}\right)
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| &= -\frac{\partial P}{\partial y} + \frac{\partial \tau_{yx}}{\partial x} + \frac{\partial \tau_{yy}}{\partial y} + \frac{\partial \tau_{yz}}{\partial z} + \rho g_y
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| \\
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| z:\;\; \rho \left(\frac{\partial u_z}{\partial t} + u_x \frac{\partial u_z}{\partial x} + u_y \frac{\partial u_z}{\partial y}+ u_z \frac{\partial u_z}{\partial z}\right)
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| &= -\frac{\partial P}{\partial z} + \frac{\partial \tau_{zx}}{\partial x} + \frac{\partial \tau_{zy}}{\partial y} + \frac{\partial \tau_{zz}}{\partial z} + \rho g_z.
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| \end{align}
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| </math>
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| ===Cylindrical coordinates===
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| :<math>
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| r:\;\;\rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_r}{\partial \phi} + u_z \frac{\partial u_r}{\partial z} - \frac{u_{\phi}^2}{r}\right) =
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| -\frac{\partial P}{\partial r} + \frac{1}{r}\frac{\partial {(r{\tau_{rr})}}}{\partial r} + \frac{1}{r}\frac{\partial {\tau_{\phi r}}}{\partial \phi} + \frac{\partial {\tau_{z r}}}{\partial z} - \frac {\tau_{\phi \phi}}{r} + \rho g_r</math> | |
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| :<math>
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| \phi:\;\;\rho \left(\frac{\partial u_{\phi}}{\partial t} + u_r \frac{\partial u_{\phi}}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_{\phi}}{\partial \phi} + u_z \frac{\partial u_{\phi}}{\partial z} + \frac{u_r u_{\phi}}{r}\right) =
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| -\frac{1}{r}\frac{\partial P}{\partial \phi} +\frac{1}{r}\frac{\partial {\tau_{\phi \phi}}}{\partial \phi} + | |
| \frac{1}{r^2}\frac{\partial {(r^2{\tau_{r \phi})}}}{\partial r} + \frac{\partial {\tau_{z r}}}{\partial z} + \rho g_{\phi}</math>
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| :<math>
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| z:\;\;\rho \left(\frac{\partial u_z}{\partial t} + u_r \frac{\partial u_z}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_z}{\partial \phi} + u_z \frac{\partial u_z}{\partial z}\right) =
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| -\frac{\partial P}{\partial z} + \frac{\partial {\tau_{z z}}}{\partial z} + \frac{1}{r}\frac{\partial {\tau_{\phi z}}}{\partial \phi} + \frac{1}{r}\frac{\partial {(r{\tau_{rz})}}}{\partial r} + \rho g_z.</math>
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| By expressing the shear stress in terms of [[viscosity]] and fluid [[Shear velocity|velocity]], and assuming constant density and viscosity, the Cauchy momentum equation will lead to the [[Navier–Stokes equations]]. By assuming [[inviscid flow]], the Navier–Stokes equations can further simpify to the Euler equations.
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| == See also ==
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| *[[Navier–Stokes equations]]
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| *[[Computational fluid dynamics]]
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| ==References==
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| {{Reflist}}
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| [[Category:Concepts in physics]]
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| [[Category:Continuum mechanics]]
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| [[Category:Partial differential equations]]
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