Continuity
![{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {v} )=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1192e2016cbf0f9049460d0cdd78084db23542d)
or, using the substantive derivative:
![{\displaystyle {\frac {D\rho }{Dt}}+\rho (\nabla \cdot \mathbf {v} )=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa76dc6ff3cf038eb0b5a4c747e509f27b586a93)
For an incompressible fluid,
is constant, hence the velocity field must be divergence-free:
![{\displaystyle \nabla \cdot \mathbf {v} =0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73d339fc17d5e07c79bdf7f25f82567746462400)
Momentum
(Also known as the Navier-Stokes equation.)
![{\displaystyle \rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=-\nabla p+\nabla \cdot \mathbb {T} +\mathbf {f} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/416150a0e149a30ff7f2cb2177202690243129d7)
or, using the substantive derivative:
![{\displaystyle \rho \left({\frac {D\mathbf {v} }{Dt}}\right)=-\nabla p+\nabla \cdot \mathbb {T} +\mathbf {f} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/707098f874410ba0340ce388150ef1777a68333a)
where
is a volumetric force (e.g.
for gravity), and
is the stress tensor.
The vector quantity
is the shear stress. For a Newtonian incompressible fluid,
![{\displaystyle \nabla \mathbb {T} =\mu \nabla ^{2}\mathbf {v} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2de375c0677f4b564933f19c1e8552a67d236950)
with
being the (dynamic) viscosity.
For an inviscid fluid, the momentum equation becomes Euler's equation for ideal fluids:
![{\displaystyle \rho \left({\frac {D\mathbf {v} }{Dt}}\right)=-\nabla p+\mathbf {f} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3b11b718cd15dddea5ab4fd71973d44248b149)
References