Incompressible Navier-Stokes Eqns.
The incompressible Navier-Stokes equations are a simpler form of the Navier-Stokes equations that describe the motion of incompressible (\( \small \rho \) = constant) fluid substances. This page focuses on the 3-dimensional equations.
Cartesian Coordinates #
For cartesian coordinates \( \small x, y, z \) and time \( \small t \), the incompressible Navier-Stokes equations are:
Continuity Equation #
$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 $$
X-momentum Equation #
$$ \rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}\right) = - \frac{\partial P}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \rho g_x$$
Y-momentum Equation #
$$ \rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z}\right) = - \frac{\partial P}{\partial y} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right) + \rho g_y$$
Z-momentum Equation #
$$ \rho \left(\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z}\right) = - \frac{\partial P}{\partial z} + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right) + \rho g_z$$
Where
- \(\small \rho\) is the fluid density,
- \(\small P\) is the pressure,
- \(\small u\), \(\small v\), and \(\small w\) are the velocity components in the \(x\), \(y\), and \(z\) directions, respectively.
- \(\small g_x\), \(\small g_y\), and \(\small g_z\) are the components of the gravitational acceleration vector, and
- \(\small \mu\) is the dynamic viscosity of the fluid.
Cylindrical Coordinates #
For cylindrical coordinates \( \small r, \theta, z \) and time \( \small t \), the incompressible Navier-Stokes equations are:
Continuity Equation #
$$ \frac{1}{r} \frac{\partial (r u_r)}{\partial r} + \frac{1}{r} \frac{\partial (u_{\theta})}{\partial \theta} + \frac{\partial (u_{z})}{\partial z} = 0 $$
R-component #
$$ \rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\theta}}{r} \frac{\partial u_r}{\partial \theta} - \frac{u_{\theta}^2}{r} + u_z \frac{\partial u_r}{\partial z} \right) = - \frac{\partial P}{\partial r} + \rho g_r+ \mu \left[ \frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial u_r}{\partial r}\right) - \frac{u_r}{r^2} + \frac{1}{r^2} \frac{\partial^2 u_r}{\partial \theta^2} -\frac{2}{r^2} \frac{\partial u_{\theta}}{\partial \theta} + \frac{\partial^2 u_r}{\partial z^2} \right] $$
\(\small \theta\)-component #
Here is the transcribed equation in the requested single-line LaTeX format:
$$ \rho \left(\frac{\partial u_{\theta}}{\partial t} + u_{r} \frac{\partial u_{\theta}}{\partial r} + \frac{u_{\theta}}{r} \frac{\partial u_{\theta}}{\partial \theta} + \frac{u_{r} u_{\theta}}{r} + u_{z} \frac{\partial u_{\theta}}{\partial z} \right) = - \frac{1}{r} \frac{\partial P}{\partial \theta} + \rho g_{\theta} + \mu \left[ \frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial u_{\theta}}{\partial r}\right) - \frac{u_{\theta}}{r^{2}} + \frac{1}{r^{2}} \frac{\partial^{2} u_{\theta}}{\partial \theta^{2}} + \frac{2}{r^{2}} \frac{\partial u_{r}}{\partial \theta} + \frac{\partial^{2} u_{\theta}}{\partial z^{2}} \right] $$
Z-component #
$$ \rho \left(\frac{\partial u_z}{\partial t} + u_r \frac{\partial u_z}{\partial r} + \frac{u_{\theta}}{r} \frac{\partial u_z}{\partial \theta} + u_z \frac{\partial u_z}{\partial z} \right) = - \frac{\partial P}{\partial z} + \rho g_z + \mu \left[ \frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial u_z}{\partial r}\right) + \frac{1}{r^2} \frac{\partial^2 u_z}{\partial \theta^2} + \frac{\partial^2 u_z}{\partial z^2} \right] $$
Where:
- \(\small u_r\), \(\small u_{\theta}\), and \(\small u_z\) are the velocity components in the \(r\), \(\theta\), and \(z\) directions, and
- \(\small g_r\), \(\small g_{\theta}\), and \(\small g_z\) are the components of the gravitational acceleration vector in the \(r\), \(\theta\), and \(z\) directions, respectively.