Cauchy momentum equation

From Wikipedia, the free encyclopedia

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 \sigma + \mathbf{f}

or, with the derivative expanded out,

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

where ρ is the density of the continuum, σ is the stress tensor, and \mathbf{f} contains all of the body forces (normally just gravity). \mathbf{v} is the velocity vector field, which depends on time and space.

The stress tensor is sometimes split into pressure and the deviatoric stress tensor:

\nabla \cdot \sigma = -\nabla p + \nabla \cdot\mathbb{T}

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.

[edit] Derivation

Applying Newton's second law (ith 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} \frac{\partial \sigma_{ij}}{\partial x_j} \, dV + \int_{\Omega} f_i \, dV
\rho \int_{\Omega} \frac{d u_i}{d t} - \frac{\partial \sigma_{ij}}{\partial x_j} - f_i \, dV = 0

where Ω 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 derivative of the the stress tensor is one of the forces that constitutes Fi.

[edit] References

  1. ^ Acheson, D. J. (1990). Elementary Fluid Mechanics. Clarendon Press. ISBN 0198596790.