Cauchy momentum equation

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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:[1]

\rho \frac{D \mathbf{v}}{D t} = \nabla \cdot \boldsymbol{\sigma} +  \mathbf{f}

or, with the material derivative expanded out,

\rho \left[\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v}\right] = \nabla \cdot \boldsymbol{\sigma} +  \mathbf{f}

where \rho is the density of the continuum, \boldsymbol{\sigma} is the stress tensor, and \mathbf{f} contains all of the body forces per unit volume (often simply density times gravity). \mathbf{v} is the velocity vector field, which depends on time and space.The expression \mathbf{v} \cdot \nabla \mathbf{v} denotes the vector field with components  v^j \partial_j v^i and \nabla \cdot \sigma stands for the vector field with components  \partial_j \sigma_j^i .

The stress tensor is usually split into pressure and its deviatoric tensor:

\boldsymbol{\sigma} = -p\mathbb{I} + \boldsymbol{\tau}

where \scriptstyle \mathbb{I} is the \scriptstyle 3 \times 3 identity matrix and \boldsymbol{\tau} the shear tensor. The divergence of the stress tensor can be written as

\nabla \cdot \boldsymbol{\sigma} = -\nabla p + \nabla \boldsymbol{\tau}.

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. By expressing the shear tensor in terms of viscosity and fluid 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 simplify to the Euler equations.

Derivation[edit]

Applying Newton's second law (i^{th} component) to a control volume in the continuum being modeled gives:

m a_i = F_i\,
\rho \int_{\Omega} \frac{d u_i}{d t} \, dV = \int_{\Omega} \nabla_j\sigma_i^j \, dV + \int_{\Omega} f_i \, dV
 \int_{\Omega} (\rho \frac{d u_i}{d t} - \nabla_j\sigma_i^j - f_i )\, dV = 0
 \rho \dot{u_i} - \nabla_j\sigma_i^j - f_i = 0

where \Omega 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 step (not done above) in deriving this equation is establishing that the derivative of the stress tensor is one of the forces that constitutes F_i.[1]

3D explicit forms[edit]

Cartesian coordinates[edit]

\begin{align}
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)
    &= -\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 
\\
 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)
    &= -\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
 \\
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)
    &= -\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.
\end{align}

Cylindrical coordinates[edit]


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) =
-\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

\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) =
-\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 \phi}}}{\partial z} + \rho g_{\phi}

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) =
-\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.

See also[edit]

References[edit]

  1. ^ a b Acheson, D. J. (1990). Elementary Fluid Dynamics. Oxford University Press. p. 205. ISBN 0-19-859679-0.