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Concept
Cauchy momentum equation
Summary
The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum.
In convective (or Lagrangian) form the Cauchy momentum equation is written as:
where
is the flow velocity vector field, which depends on time and space, (unit: )
is time, (unit: )
is the material derivative of , equal to , (unit: )
is the density at a given point of the continuum (for which the continuity equation holds), (unit: )
is the stress tensor, (unit: )
is a vector containing all of the accelerations caused by body forces (sometimes simply gravitational acceleration), (unit: )
is the divergence of stress tensor. (unit: )
Commonly used SI units are given in parentheses although the equations are general in nature and other units can be entered into them or units can be removed at all by nondimensionalization.
Note that only we use column vectors (in the Cartesian coordinate system) above for clarity, but the equation is written using physical components (which are neither covariants ("column") nor contravariants ("row") ). However, if we chose a non-orthogonal curvilinear coordinate system, then we should calculate and write equations in covariant ("row vectors") or contravariant ("column vectors") form.
After an appropriate change of variables, it can also be written in conservation form:
where j is the momentum density at a given space-time point, F is the flux associated to the momentum density, and s contains all of the body forces per unit volume.
Let us start with the generalized momentum conservation principle which can be written as follows: "The change in system momentum is proportional to the resulting force acting on this system". It is expressed by the formula:
where is momentum in time t, is force averaged over . After dividing by and passing to the limit we get (derivative):
Let us analyse each side of the equation above.
We split the forces into body forces and surface forces
Surface forces act on walls of the cubic fluid element.
Official source
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