Chapter 3
Chapter 3 in [Whi06] considers simple solutions of incompressible laminar shear flows. Incompressible flows are described by the physical laws for conservation of mass Newton’s second law of motion). Together these are usually termed the Navier-Stokes equations that mathematically can be represented as
(1)\[\begin{split}\begin{aligned}
\frac{\partial \boldsymbol{u}}{\partial t} + (\boldsymbol{u} \cdot \nabla)\boldsymbol{u} &= -\frac{1}{\rho} \nabla p + \nu \nabla ^2 \boldsymbol{u} + \boldsymbol{f} \\
\nabla \cdot \boldsymbol{u} &= 0,
\end{aligned}\end{split}\]
where \(\boldsymbol{u}\), \(p\), \(\rho\), \(\nu\) and \(\boldsymbol{f}\) are the velocity vector, pressure, density, kinematic viscosity
and body forces respectively. The equations as written are independent of coordinate system, but they look
exactly the same using Cartesian coordinates. Of equally great importance, at least in chapter 3, are the
Navier-Stokes equations in cylindrical coordinates. The cylindrical coordinates, \(r, \theta, z\), are given in
terms of the Cartesian coordinates \(x, y, z\) as
\[\begin{split}\begin{aligned}
x &= r \cos \theta, \\
y &= r \sin \theta, \\
z &= z .
\end{aligned}\end{split}\]
The Cartesian position vector is thus
\[ \boldsymbol{x} = r \cos \theta \boldsymbol{i} + r \sin \theta \boldsymbol{j} + z \boldsymbol{k}.\]
The unit vectors in cylindrical coordinates are
\[\begin{split}\begin{aligned}
\boldsymbol{i}_{r} &= \frac{\frac{\partial \boldsymbol{x}}{\partial r}}{|\frac{\partial \boldsymbol{x}}{\partial r}|} = \cos \theta \boldsymbol{i} + \sin \theta \boldsymbol{j}, \\
\boldsymbol{i}_{\theta} &= \frac{\frac{\partial \boldsymbol{x}}{\partial \theta}}{|\frac{\partial \boldsymbol{x}}{\partial \theta}|} = -\sin \theta \boldsymbol{i} + \cos \theta \boldsymbol{j}, \\
\boldsymbol{i}_{z} &= \frac{\frac{\partial \boldsymbol{x}}{\partial z}}{|\frac{\partial \boldsymbol{x}}{\partial z}|} = \boldsymbol{k}.
\end{aligned}\end{split}\]
The velocity vectors in Cartesian and cylindrical coordinates read respectively
\[\begin{split}\begin{aligned}
\boldsymbol{u}(x,y,z,t) &= u_x \boldsymbol{i} + u_y \boldsymbol{j} + u_z \boldsymbol{k} \\
\boldsymbol{u}(r,\theta,z,t) &= u_{r} \boldsymbol{i}_r + u_{\theta} \boldsymbol{i}_{\theta} + u_z \boldsymbol{i}_z
\end{aligned}\end{split}\]
The divergence of the velocity vector \(\nabla \cdot \boldsymbol{u}\) in Cartesian and cylindrical coordinates are given respectively by
\[\begin{split}\begin{aligned}
\nabla \cdot \boldsymbol{u} &= \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z} \\
\nabla \cdot \boldsymbol{u} &= \frac{1}{r}\frac{\partial r u_r}{\partial r} + \frac{1}{r}\frac{\partial u_{\theta}}{\partial \theta} + \frac{\partial u_z}{\partial z}
\end{aligned}\end{split}\]
The Laplacian in Cartesian and cylindrical coordinates are given, respectively, as
\[\begin{split}\begin{aligned}
\nabla^2 &= \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \\
\nabla^2 &= \frac{1}{r}\frac{\partial}{\partial r}\left( r\frac{\partial}{\partial r} \right) + \frac{1}{r^2}\frac{\partial^2}{\partial \theta ^2} + \frac{\partial^2}{\partial z^2}
\end{aligned}\end{split}\]
The advection \((\boldsymbol{u} \cdot \nabla)\) is given in Cartesian and cylindrical coordinates, respectively, as
\[\begin{split}\begin{aligned}
\boldsymbol{u} \cdot \nabla &= u_i \frac{\partial }{\partial x_i} \\
\boldsymbol{u} \cdot \nabla &= u_r\frac{\partial}{\partial r} + \frac{1}{r}u_{\theta}\frac{\partial}{\partial \theta} + u_z \frac{\partial}{\partial z}
\end{aligned}\end{split}\]
See mek2200 notes on notation for a more thorough discussion about notation.
Suggested assignment Navier-Stokes
Start from Navier-Stokes and derive the momentum equations for Cylindrical coordinates. The results are given in App B of [Whi06].
- Whi06(1,2)
Frank M. White. Viscous Fluid Flow. McGraw-Hill, third edition, 2006.
