The balance of mass can be expressed as:
where
is the current mass density,
is
the material time derivative of
, and
is the
velocity of physical particles in the body
bounded by
the surface
.
We can show how this relation is derived by recalling that the general equation for the balance of a physical quantity
is given by
![{\displaystyle {\cfrac {d}{dt}}\left[\int _{\Omega }f(\mathbf {x} ,t)~{\text{dV}}\right]=\int _{\partial {\Omega }}f(\mathbf {x} ,t)[u_{n}(\mathbf {x} ,t)-\mathbf {v} (\mathbf {x} ,t)\cdot \mathbf {n} (\mathbf {x} ,t)]~{\text{dA}}+\int _{\partial {\Omega }}g(\mathbf {x} ,t)~{\text{dA}}+\int _{\Omega }h(\mathbf {x} ,t)~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcb120ab19e9001d85370ccb115c509cec72cf33)
To derive the equation for the balance of mass, we assume that the
physical quantity of interest is the mass density
.
Since mass is neither created or destroyed, the surface and interior
sources are zero, i.e.,
. Therefore, we have
![{\displaystyle {\cfrac {d}{dt}}\left[\int _{\Omega }\rho (\mathbf {x} ,t)~{\text{dV}}\right]=\int _{\partial {\Omega }}\rho (\mathbf {x} ,t)[u_{n}(\mathbf {x} ,t)-\mathbf {v} (\mathbf {x} ,t)\cdot \mathbf {n} (\mathbf {x} ,t)]~{\text{dA}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03f94d594114f237d5c144e08d373bc8cc993585)
Let us assume that the volume
is a control volume (i.e., it
does not change with time). Then the surface
has a zero
velocity (
) and we get
![{\displaystyle \int _{\Omega }{\frac {\partial \rho }{\partial t}}~{\text{dV}}=-\int _{\partial {\Omega }}\rho ~(\mathbf {v} \cdot \mathbf {n} )~{\text{dA}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f92d33880c6d67f5ad0f8c33b721aba2aec528d)
Using the divergence theorem
![{\displaystyle \int _{\Omega }{\boldsymbol {\nabla }}\bullet \mathbf {v} ~{\text{dV}}=\int _{\partial {\Omega }}\mathbf {v} \cdot \mathbf {n} ~{\text{dA}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c61c7b936f027d20c16b2df534c290fc80b3605)
we get
![{\displaystyle \int _{\Omega }{\frac {\partial \rho }{\partial t}}~{\text{dV}}=-\int _{\Omega }{\boldsymbol {\nabla }}\bullet (\rho ~\mathbf {v} )~{\text{dV}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9effb7f2264806f9a88ddb46dd4f15cbf4a3005)
or,
![{\displaystyle \int _{\Omega }\left[{\frac {\partial \rho }{\partial t}}+{\boldsymbol {\nabla }}\bullet (\rho ~\mathbf {v} )\right]~{\text{dV}}=0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/991491fdad14216d8c35799c2f7deb776e4a3089)
Since
is arbitrary, we must have
![{\displaystyle {\frac {\partial \rho }{\partial t}}+{\boldsymbol {\nabla }}\bullet (\rho ~\mathbf {v} )=0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27d0697a58b6ae8cdc4723fd885d77b7e8118715)
Using the identity
![{\displaystyle {\boldsymbol {\nabla }}\bullet (\varphi ~\mathbf {v} )=\varphi ~{\boldsymbol {\nabla }}\bullet \mathbf {v} +{\boldsymbol {\nabla }}\varphi \cdot \mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b0fda4d348a881cc28f0b0353c3ad9cf3110150)
we have
![{\displaystyle {\frac {\partial \rho }{\partial t}}+\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} +{\boldsymbol {\nabla }}\rho \cdot \mathbf {v} =0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8895dd34551a76962f75d47633b915244a486c)
Now, the material time derivative of
is defined as
![{\displaystyle {\dot {\rho }}={\frac {\partial \rho }{\partial t}}+{\boldsymbol {\nabla }}\rho \cdot \mathbf {v} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c755826463c2c7fc56f0fd6229cbfc7c16ec69a)
Therefore,
![{\displaystyle {{\dot {\rho }}+\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} =0~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04cde76f2ba8154cc79f1a3d6196b2610afc130b)