The balance of linear momentum can be expressed as:
where
is the mass density,
is the velocity,
is the Cauchy stress, and
is the body force
density.
Recall the general equation for the balance of a physical quantity
![{\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)
In this case the physical quantity of interest is the momentum density,
i.e.,
. The source of momentum flux
at the surface is the surface traction, i.e.,
. The
source of momentum inside the body is the body force, i.e.,
. Therefore, we have
![{\displaystyle {\cfrac {d}{dt}}\left[\int _{\Omega }\rho ~\mathbf {v} ~{\text{dV}}\right]=\int _{\partial {\Omega }}\rho ~\mathbf {v} [u_{n}-\mathbf {v} \cdot \mathbf {n} ]~{\text{dA}}+\int _{\partial {\Omega }}\mathbf {t} ~{\text{dA}}+\int _{\Omega }\rho ~\mathbf {b} ~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a46f11b5cfa9b04cb07575c7f4e042233fcce4d6)
The surface tractions are related to the Cauchy stress by
![{\displaystyle \mathbf {t} ={\boldsymbol {\sigma }}\cdot \mathbf {n} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8be3270233f8edb37f4a2c5e1b713a1fededbbab)
Therefore,
![{\displaystyle {\cfrac {d}{dt}}\left[\int _{\Omega }\rho ~\mathbf {v} ~{\text{dV}}\right]=\int _{\partial {\Omega }}\rho ~\mathbf {v} [u_{n}-\mathbf {v} \cdot \mathbf {n} ]~{\text{dA}}+\int _{\partial {\Omega }}{\boldsymbol {\sigma }}\cdot \mathbf {n} ~{\text{dA}}+\int _{\Omega }\rho ~\mathbf {b} ~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adc3ef27ffa5e65209e677c035ea62b0e9cf1bb4)
Let us assume that
is an arbitrary fixed control volume. Then,
![{\displaystyle \int _{\Omega }{\frac {\partial }{\partial t}}(\rho ~\mathbf {v} )~{\text{dV}}=-\int _{\partial {\Omega }}\rho ~\mathbf {v} ~(\mathbf {v} \cdot \mathbf {n} )~{\text{dA}}+\int _{\partial {\Omega }}{\boldsymbol {\sigma }}\cdot \mathbf {n} ~{\text{dA}}+\int _{\Omega }\rho ~\mathbf {b} ~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa657ddc06c1ea6327416511b2560f6f5bb9739a)
Now, from the definition of the tensor product we have (for all vectors
)
![{\displaystyle (\mathbf {u} \otimes \mathbf {v} )\cdot \mathbf {a} =(\mathbf {a} \cdot \mathbf {v} )~\mathbf {u} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a3ebbf6a15197b98a14e8f5ac169c0ec474410)
Therefore,
![{\displaystyle \int _{\Omega }{\frac {\partial }{\partial t}}(\rho ~\mathbf {v} )~{\text{dV}}=-\int _{\partial {\Omega }}\rho ~(\mathbf {v} \otimes \mathbf {v} )\cdot \mathbf {n} ~{\text{dA}}+\int _{\partial {\Omega }}{\boldsymbol {\sigma }}\cdot \mathbf {n} ~{\text{dA}}+\int _{\Omega }\rho ~\mathbf {b} ~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/332f92b6f2951295ad65c91628fc89c2cb80cdd2)
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 have
![{\displaystyle \int _{\Omega }{\frac {\partial }{\partial t}}(\rho ~\mathbf {v} )~{\text{dV}}=-\int _{\Omega }{\boldsymbol {\nabla }}\bullet [\rho ~(\mathbf {v} \otimes \mathbf {v} )]~{\text{dV}}+\int _{\Omega }{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}~{\text{dV}}+\int _{\Omega }\rho ~\mathbf {b} ~{\text{dV}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34f59a0faa78bd490f4b097fe1846bee54252d62)
or,
![{\displaystyle \int _{\Omega }\left[{\frac {\partial }{\partial t}}(\rho ~\mathbf {v} )+{\boldsymbol {\nabla }}\bullet [(\rho ~\mathbf {v} )\otimes \mathbf {v} )]-{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} \right]~{\text{dV}}=0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03c321bc67c81041e73677bfadbe98c206a18988)
Since
is arbitrary, we have
![{\displaystyle {\frac {\partial }{\partial t}}(\rho ~\mathbf {v} )+{\boldsymbol {\nabla }}\bullet [(\rho ~\mathbf {v} )\otimes \mathbf {v} )]-{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} =0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/615dfb89380d7157ae3c429c3eff733eacf30c58)
Using the identity
![{\displaystyle {\boldsymbol {\nabla }}\bullet (\mathbf {u} \otimes \mathbf {v} )=({\boldsymbol {\nabla }}\bullet \mathbf {v} )\mathbf {u} +({\boldsymbol {\nabla }}\mathbf {u} )\cdot \mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/404939824430b3b2506784c13b54b0037b3068a0)
we get
![{\displaystyle {\frac {\partial \rho }{\partial t}}~\mathbf {v} +\rho ~{\frac {\partial \mathbf {v} }{\partial t}}+({\boldsymbol {\nabla }}\bullet \mathbf {v} )(\rho \mathbf {v} )+{\boldsymbol {\nabla }}(\rho ~\mathbf {v} )\cdot \mathbf {v} -{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e127134be51fe87f4f8bef69628f9dcb37102fe)
or,
![{\displaystyle \left[{\frac {\partial \rho }{\partial t}}+\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} \right]\mathbf {v} +\rho ~{\frac {\partial \mathbf {v} }{\partial t}}+{\boldsymbol {\nabla }}(\rho ~\mathbf {v} )\cdot \mathbf {v} -{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c6a6d133032adf207945ec955ca922476fa4de2)
Using the identity
![{\displaystyle {\boldsymbol {\nabla }}(\varphi ~\mathbf {v} )=\varphi ~{\boldsymbol {\nabla }}\mathbf {v} +\mathbf {v} \otimes ({\boldsymbol {\nabla }}\varphi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5efac15fa759bef1c3a17a4062e524d9ae599428)
we get
![{\displaystyle \left[{\frac {\partial \rho }{\partial t}}+\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} \right]\mathbf {v} +\rho ~{\frac {\partial \mathbf {v} }{\partial t}}+\left[\rho ~{\boldsymbol {\nabla }}\mathbf {v} +\mathbf {v} \otimes ({\boldsymbol {\nabla }}\rho )\right]\cdot \mathbf {v} -{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8f4634efd67549ed2d10b81b08fc879960a230b)
From the definition
![{\displaystyle (\mathbf {u} \otimes \mathbf {v} )\cdot \mathbf {a} =(\mathbf {a} \cdot \mathbf {v} )~\mathbf {u} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1d709d150664ab5ffe0875f17806b424a2511c)
we have
![{\displaystyle [\mathbf {v} \otimes ({\boldsymbol {\nabla }}\rho )]\cdot \mathbf {v} =[\mathbf {v} \cdot ({\boldsymbol {\nabla }}\rho )]~\mathbf {v} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d51ffdfa1c0eef65a6d11cc792f7e0c9195e9412)
Hence,
![{\displaystyle \left[{\frac {\partial \rho }{\partial t}}+\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} \right]\mathbf {v} +\rho ~{\frac {\partial \mathbf {v} }{\partial t}}+\rho ~{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {v} +[\mathbf {v} \cdot ({\boldsymbol {\nabla }}\rho )]~\mathbf {v} -{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9020d02391b4cbf5edfcfc6c8ab79644a13835de)
or,
![{\displaystyle \left[{\frac {\partial \rho }{\partial t}}+{\boldsymbol {\nabla }}\rho \cdot \mathbf {v} +\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} \right]\mathbf {v} +\rho ~{\frac {\partial \mathbf {v} }{\partial t}}+\rho ~{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {v} -{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} =0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13599a24b1ac883c0ed676882f088c9209f86b41)
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 \left[{\dot {\rho }}+\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} \right]\mathbf {v} +\rho ~{\frac {\partial \mathbf {v} }{\partial t}}+\rho ~{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {v} -{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} =0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4484bb80e35493fc3a7c3978fc27e937c0c2aeb)
From the balance of mass, we have
![{\displaystyle {\dot {\rho }}+\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} =0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29f84fc3709ebd9ec30d1227f386a051192aedbf)
Therefore,
![{\displaystyle \rho ~{\frac {\partial \mathbf {v} }{\partial t}}+\rho ~{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {v} -{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} =0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3ab0ad14d543193bd9b30c11dee4089bb782f3)
The material time derivative of
is defined as
![{\displaystyle {\dot {\mathbf {v} }}={\frac {\partial \mathbf {v} }{\partial t}}+{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {v} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a99ae843473e9fa146f060b47d113def2439c05)
Hence,
![{\displaystyle {\rho ~{\dot {\mathbf {v} }}-{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} =0~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e288f5fbc61ad2427f67250fc4daa71da12bb647)