Momentum Conservation

Time derivative of momentum. How the momentum changes along a streamline in the flow. \( \rho\dfrac{\partial\vec{v}}{\partial t} \) \( 0 \, \) \( + \) \( \rho(\vec{v}\cdot\nabla)\vec{v} \) \( = \) \( - \) \( \nabla P \) \( + \) \( \eta\nabla^{2}\vec{v} \) \( + \) \( \rho\vec{f} \) Pressure gradient. Viscous term Body forces (e.g. gravity).

(Incompressible Newtonian Differential Vector Form)

\( \dfrac{\partial{v_x}}{\partial{t}} \) \( 0 \, \) \( + \) \( v_x \dfrac{\partial{v_x}}{\partial{x}} + v_y \dfrac{\partial{v_x}}{\partial{y}} + v_z \dfrac{\partial{v_x}}{\partial{z}} \) \( = \) \( -\dfrac{1}{\rho} \dfrac{\partial{P}}{\partial{x}} \) \( + \) \( \nu ( \dfrac{\partial^2{v_x}}{\partial{x}^2} + \dfrac{\partial^2{v_x}}{\partial{y}^2} + \dfrac{\partial^2{v_x}}{\partial{z}^2} ) \) \( + \) \( f_x \)

\( \dfrac{\partial{v_y}}{\partial{t}} \) \( 0 \, \) \( + \) \( v_x \dfrac{\partial{v_y}}{\partial{x}} + v_y \dfrac{\partial{v_y}}{\partial{y}} + v_z \dfrac{\partial{v_y}}{\partial{z}} \) \( = \) \( -\dfrac{1}{\rho} \dfrac{\partial{P}}{\partial{y}} \) \( + \) \( \nu ( \dfrac{\partial^2{v_y}}{\partial{x}^2} + \dfrac{\partial^2{v_y}}{\partial{y}^2} + \dfrac{\partial^2{v_y}}{\partial{z}^2} ) \) \( + \) \( f_y \)

\( \dfrac{\partial{v_z}}{\partial{t}} \) \( 0 \, \) \( + \) \( v_x \dfrac{\partial{v_z}}{\partial{x}} + v_y \dfrac{\partial{v_z}}{\partial{y}} + v_z \dfrac{\partial{v_z}}{\partial{z}} \) \( = \) \( -\dfrac{1}{\rho} \dfrac{\partial{P}}{\partial{z}} \) \( + \) \( \nu ( \dfrac{\partial^2{v_z}}{\partial{x}^2} + \dfrac{\partial^2{v_z}}{\partial{y}^2} + \dfrac{\partial^2{v_z}}{\partial{z}^2} ) \) \( + \) \( f_z \)

Compared to cylindrical or spherical form, the pattern between component equations is more clear in cartesian form. To make this pattern further apparent see the terms highlighted in red below. For the y-momentum equation simply replace every red x with a y. Likewise, for the z-momentum equation replace every red x with a z.

\( \dfrac{\partial{v_{\color{red}x}}}{\partial{t}} \) \( + \) \( v_x \dfrac{\partial{v_{\color{red}x}}}{\partial{x}} + v_y \dfrac{\partial{v_{\color{red}x}}}{\partial{y}} + v_z \dfrac{\partial{v_{\color{red}x}}}{\partial{z}} \) \( = \) \( -\dfrac{1}{\rho} \dfrac{\partial{P}}{\partial{{\color{red}x}}} \) \( + \) \( \nu ( \dfrac{\partial^2{v_{\color{red}x}}}{\partial{x}^2} + \dfrac{\partial^2{v_{\color{red}x}}}{\partial{y}^2} + \dfrac{\partial^2{v_{\color{red}x}}}{\partial{z}^2} ) \) \( + \) \( f_{\color{red}x} \)

(Incompressible Newtonian Differential Form in Cartesian Coordinates)

\( \dfrac{\partial{v_r}}{\partial{t}} \) \( 0 \, \) \( + \) \( v_r \dfrac{\partial{v_r}}{\partial{r}} + \dfrac{v_{\phi}}{r} \dfrac{\partial{v_r}}{\partial{\phi}} + v_z \dfrac{\partial{v_r}}{\partial{z}} - \dfrac{v_{\phi}^2}{r} \) \( = \) \( -\dfrac{1}{\rho} \dfrac{\partial{P}}{\partial{r}} \) \( + \) \( \nu ( \dfrac{\partial^2{v_r}}{\partial{r}^2} + \dfrac{1}{r^2} \dfrac{\partial^2{v_r}}{\partial{\phi}^2} + \dfrac{\partial^2{v_r}}{\partial{z}^2} + \dfrac{1}{r} \dfrac{\partial{v_r}}{\partial{r}} - \dfrac{2}{r^2} \dfrac{\partial{v_{\phi}}}{\partial{\phi}} - \dfrac{v_r}{r^2} ) \) \( + \) \( f_r \)

\( \dfrac{\partial{v_{\phi}}}{\partial{t}} \) \( 0 \, \) \( + \) \( v_r \dfrac{\partial{v_{\phi}}}{\partial{r}} + \dfrac{v_{\phi}}{r} \dfrac{\partial{v_{\phi}}}{\partial{\phi}} + v_z \dfrac{\partial{v_{\phi}}}{\partial{z}} + \dfrac{v_r v_{\phi}}{r} \) \( = \) \( -\dfrac{1}{\rho} \dfrac{1}{r} \dfrac{\partial{P}}{\partial{\phi}} \) \( + \) \( \nu ( \dfrac{\partial^2{v_{\phi}}}{\partial{r}^2} + \dfrac{1}{r^2} \dfrac{\partial^2{v_{\phi}}}{\partial{\phi}^2} + \dfrac{\partial^2{v_{\phi}}}{\partial{z}^2} + \dfrac{1}{r} \dfrac{\partial{v_{\phi}}}{\partial{r}} + \dfrac{2}{r^2} \dfrac{\partial{v_r}}{\partial{\phi}} - \dfrac{v_{\phi}}{r^2} ) \) \( + \) \( f_{\phi} \)

\( \dfrac{\partial{v_z}}{\partial{t}} \) \( 0 \, \) \( + \) \( v_r \dfrac{\partial{v_z}}{\partial{r}} + \dfrac{v_{\phi}}{r} \dfrac{\partial{v_z}}{\partial{\phi}} + v_z \dfrac{\partial{v_z}}{\partial{z}} \) \( = \) \( -\dfrac{1}{\rho} \dfrac{\partial{P}}{\partial{z}} \) \( + \) \( \nu ( \dfrac{\partial^2{v_z}}{\partial{r}^2} + \dfrac{1}{r^2} \dfrac{\partial^2{v_z}}{\partial{\phi}^2} + \dfrac{\partial^2{v_z}}{\partial{z}^2} + \dfrac{1}{r} \dfrac{\partial{v_z}}{\partial{r}} ) \) \( + \) \( f_z \)

(Incompressible Newtonian Differential Form in Cylindrical Coordinates)

\( \dfrac{\partial{v_r}}{\partial{t}} \) \( 0 \, \) \( + \) \( v_r \dfrac{\partial{v_r}}{\partial{r}} + \dfrac{v_{\theta}}{r} \dfrac{\partial{v_r}}{\partial{\theta}} + \dfrac{v_{\phi}}{r \sin\theta} \dfrac{\partial{v_r}}{\partial{\phi}} - \dfrac{v_{\theta}^2 + v_{\phi}^2}{r} \) \( = \) \( -\dfrac{1}{\rho} \dfrac{\partial{P}}{\partial{r}} \) \( + \) \( \nu ( \dfrac{1}{r} \dfrac{\partial^2{(r v_r)}}{\partial{r}^2} + \dfrac{1}{r^2} \dfrac{\partial^2{v_r}}{\partial{\theta}^2} + \dfrac{1}{r^2 \sin^2\theta} \dfrac{\partial^2{v_r}}{\partial{\phi}^2} + \dfrac{\cot\theta}{r^2} \dfrac{\partial{v_r}}{\partial{\theta}} - \dfrac{2}{r^2} \dfrac{\partial{v_{\theta}}}{\partial{\theta}} - \dfrac{2}{r^2 \sin\theta} \dfrac{\partial{v_{\phi}}}{\partial{\phi}} - \dfrac{2 v_r}{r^2} - \dfrac{2 \cot\theta}{r^2} v_{\theta} ) \) \( + \) \( f_r \)

\( \dfrac{\partial{v_{\theta}}}{\partial{t}} \) \( 0 \, \) \( + \) \( v_r \dfrac{\partial{v_{\theta}}}{\partial{r}} + \dfrac{v_{\theta}}{r} \dfrac{\partial{v_{\theta}}}{\partial{\theta}} + \dfrac{v_{\phi}}{r \sin\theta} \dfrac{\partial{v_{\theta}}}{\partial{\phi}} + \dfrac{v_r v_{\theta}}{r} - \dfrac{v_{\phi}^2 \cot\theta}{r} \) \( = \) \( -\dfrac{1}{\rho} \dfrac{1}{r} \dfrac{\partial{P}}{\partial{\theta}} \) \( + \) \( \nu ( \dfrac{1}{r} \dfrac{\partial^2{(r v_{\theta})}}{\partial{r}^2} + \dfrac{1}{r^2} \dfrac{\partial^2{v_{\theta}}}{\partial{\theta}^2} + \dfrac{1}{r^2 \sin^2\theta} \dfrac{\partial^2{v_{\theta}}}{\partial{\phi}^2} + \dfrac{\cot\theta}{r^2} \dfrac{\partial{v_{\theta}}}{\partial{\theta}} - \dfrac{2 \cos\theta}{r^2 \sin^2\theta} \dfrac{\partial{v_{\phi}}}{\partial{\phi}} + \dfrac{2}{r^2} \dfrac{\partial{v_r}}{\partial{\theta}} - \dfrac{v_{\theta}}{r^2 \sin^2\theta} ) \) \( + \) \( f_{\theta} \)

\( \dfrac{\partial{v_{\phi}}}{\partial{t}} \) \( 0 \, \) \( + \) \( v_r \dfrac{\partial{v_{\phi}}}{\partial{r}} + \dfrac{v_{\theta}}{r} \dfrac{\partial{v_{\phi}}}{\partial{\theta}} + \dfrac{v_{\phi}}{r \sin\theta} \dfrac{\partial{v_{\phi}}}{\partial{\phi}} + \dfrac{v_r v_{\phi}}{r} + \dfrac{v_{\theta} v_{\phi} \cot\theta}{r} \) \( = \) \( -\dfrac{1}{\rho} \dfrac{1}{r \sin\theta} \dfrac{\partial{P}}{\partial{\phi}} \) \( + \) \( \nu ( \dfrac{1}{r} \dfrac{\partial^2{(r v_{\phi})}}{\partial{r}^2} + \dfrac{1}{r^2} \dfrac{\partial^2{v_{\phi}}}{\partial{\theta}^2} + \dfrac{1}{r^2 \sin^2\theta} \dfrac{\partial^2{v_{\phi}}}{\partial{\phi}^2} + \dfrac{\cot\theta}{r^2} \dfrac{\partial{v_{\phi}}}{\partial{\theta}} + \dfrac{2}{r^2 \sin\theta} \dfrac{\partial{v_r}}{\partial{\phi}} + \dfrac{2 \cos\theta}{r^2 \sin^2\theta} \dfrac{\partial{v_{\theta}}}{\partial{\phi}} - \dfrac{v_{\phi}}{r^2 \sin^2\theta} ) \) \( + \) \( f_{\phi} \)

(Incompressible Newtonian Differential Form in Spherical Coordinates)

\( \dfrac{d}{dt} {\displaystyle \iiint_V \rho\vec{v} \;dV} \) \( = \) \( {\displaystyle \iint_S \rho\vec{v} (\vec{v}\cdot\hat{n}) + P\hat{n} - \overline{\overline{\tau}}\cdot\hat{n} \;dS} \) \( + \) \( {\displaystyle \iiint_V \rho\vec{f} \;dV} \)

(General Integral Form in Vector Notation)

Material derivative of velocity (i.e. how does velocity change in space and time). \( \rho\dfrac{D\vec{v}}{Dt} \) \( = \) \( \rho\vec{f} \) \( + \) \( \nabla\cdot\overline{\overline{\sigma}} \) Fluid stresses (i.e. how stresses contract or expand the fluid locally).

(General Differential Form in Vector Notation)

Material derivative of density (i.e. how does density change in space and time). \( \rho\dfrac{d\vec{v}}{dt} \) \( + \) \( \rho(\vec{v}\cdot\nabla)\vec{v} \) \( = \) \( \rho\vec{f} \) \( + \) \( \nabla\cdot\overline{\overline{\sigma}} \) Material derivative of density (i.e. how does density change in space and time).

(General Differential Form in Vector Notation)

Material derivative of density (i.e. how does density change in space and time). \( \rho\dfrac{d\vec{v}}{dt} \) \( + \) \( \rho(\vec{v}\cdot\nabla)\vec{v} \) \( = \) \( \rho\vec{f} \) \( - \) \( \nabla P \) \( + \) \( \nabla\cdot\overline{\overline{\tau}} \) Material derivative of density (i.e. how does density change in space and time).

(Incompressible Differential Form in Vector Notation)

Fluid Fluency

An Online Repository for Fluid Physics Notes

by Rónán Gissler

Momentum Conservation