13.1 Cartesian Coordinates

13.1.1 Continuity Equation: Compressible

We derived the continuity equation in section 10.4 for Cartesian coordinates. For compressible fluids, it is given by Eq. 10.16 as

$\begin{array}{ll}\frac{\partial \rho }{\partial t}+\overrightarrow{\nabla }\cdot \left(\rho \overrightarrow{\upsilon }\right)\hfill & =0\hfill \\ \frac{\partial \rho }{\partial t}+{\upsilon }_x\frac{\partial \rho }{\partial x}+{\upsilon }_y\frac{\partial \rho }{\partial y}+{\upsilon }_z\frac{\partial \rho }{\partial z}+\rho \left(\frac{\partial {\upsilon }_x}{\partial x}+\frac{\partial {\upsilon }_y}{\partial y}+\frac{\partial {\upsilon }_z}{\partial z}\right)\hfill & =0\hfill \end{array}$

13.1.2 Continuity Equation: Incompressible

For incompressible fluids, the equation simplifies to Eq. 10.17 given by

$\begin{array}{ll}\overrightarrow{\nabla }\cdot \overrightarrow{\upsilon }\hfill & =0\hfill \\ \frac{\partial {\upsilon }_x}{\partial x}+\frac{\partial {\upsilon }_y}{\partial y}+\frac{\partial {\upsilon }_z}{\partial z}\hfill & =0\hfill \end{array}$

13.1.3 Navier-Stokes Equation: Incompressible Newtonian Fluids

We derived the Navier-Stokes equation in Cartesian coordinates in section 11.7. It is given by Eq. 11.40 in its general form as

$\rho \left(\frac{\partial \overrightarrow{\upsilon }}{\partial t}+\left(\overrightarrow{\upsilon }\cdot \overrightarrow{\nabla }\right)\overrightarrow{\upsilon }\right)=\overrightarrow{k}-\overrightarrow{\nabla }p+\eta \text{Δ}\overrightarrow{\upsilon }$

which is written out in Cartesian coordinates

x-axis

$\rho \left(\frac{\partial {\upsilon }_x}{\partial t}+{\upsilon }_x\frac{\partial {\upsilon }_x}{\partial x}+{\upsilon }_y\frac{\partial {\upsilon }_x}{\partial y}+{\upsilon }_z\frac{\partial {\upsilon }_x}{\partial z}\right)=k_x-\frac{\partial p}{\partial x}+\eta \left(\frac{{\partial }^2{\upsilon }_x}{\partial x^2}+\frac{{\partial }^2{\upsilon }_x}{\partial y^2}+\frac{{\partial }^2{\upsilon }_x}{\partial z^2}\right)$

y-axis

$\rho \left(\frac{\partial {\upsilon }_y}{\partial t}+{\upsilon }_x\frac{\partial {\upsilon }_y}{\partial x}+{\upsilon }_y\frac{\partial {\upsilon }_y}{\partial y}+{\upsilon }_z\frac{\partial {\upsilon }_y}{\partial z}\right)=k_y-\frac{\partial p}{\partial y}+\eta \left(\frac{{\partial }^2{\upsilon }_y}{\partial x^2}+\frac{{\partial }^2{\upsilon }_y}{\partial y^2}+\frac{{\partial }^2{\upsilon }_y}{\partial z^2}\right)$

z-axis

$\rho \left(\frac{\partial {\upsilon }_z}{\partial t}+{\upsilon }_x\frac{\partial {\upsilon }_z}{\partial x}+{\upsilon }_y\frac{\partial {\upsilon }_z}{\partial y}+{\upsilon }_z\frac{\partial {\upsilon }_z}{\partial z}\right)=k_z-\frac{\partial p}{\partial z}+\eta \left(\frac{{\partial }^2{\upsilon }_z}{\partial x^2}+\frac{{\partial }^2{\upsilon }_z}{\partial y^2}+\frac{{\partial }^2{\upsilon }_z}{\partial z^2}\right)$

Before venturing to convert these equations into different coordinate systems, be aware of the meaning of the individual terms. $\left(\overrightarrow{\upsilon }\cdot \overrightarrow{\nabla }\right)\overrightarrow{\upsilon }$ is a material derivative (see section 7.1.3.10), whereas $\overrightarrow{\nabla }p$ is a gradient (see section 7.1.3.2), and $\eta \text{Δ}\overrightarrow{\upsilon }$ is a vector Laplacian (see section 7.1.3.7). We must be sure to pick the correctly converted versions of these operators when transferring the equations into different coordinate systems.

13.2 Cylindrical Coordinates

13.2.1 Continuity Equation: Compressible

In order to express the continuity equation in cylindrical coordinates, we need the divergence in cylindrical coordinates that we derived in section 7.6.1 and noted in Eq. 7.94. Using this equation, the continuity equation for compressible fluids becomes

$\begin{array}{ll}\frac{\partial \rho }{\partial t}+\overrightarrow{\nabla }\cdot \left(\rho \overrightarrow{\upsilon }\right)\hfill & =0\hfill \\ \frac{\partial \rho }{\partial t}+\frac{1}{r}\frac{\partial }{\partial r}\left(\rho r{\upsilon }_r\right)+\frac{1}{r}\frac{\partial \left(\rho {\upsilon }_{\phi }\right)}{\partial \phi }+\frac{\partial \left(\rho {\upsilon }_z\right)}{\partial z}\hfill & =0\hfill \end{array}$

13.2.2 Continuity Equation: Incompressible

Likewise, the continuity equation for incompressible fluids is given by

$\begin{array}{ll}\overrightarrow{\nabla }\cdot \overrightarrow{\upsilon }\hfill & =0\hfill \\ \frac{1}{r}\frac{\partial }{\partial r}\left(r{\upsilon }_r\right)+\frac{1}{r}\frac{\partial {\upsilon }_{\phi }}{\partial \phi }+\frac{\partial {\upsilon }_z}{\partial z}\hfill & =0\hfill \\ \frac{\partial }{\partial r}+\frac{1}{r}{\upsilon }_r+\frac{1}{r}\frac{\partial {\upsilon }_{\phi }}{\partial \phi }+\frac{\partial {\upsilon }_z}{\partial z}\hfill & =0\hfill \end{array}$

13.2.3 Navier-Stokes Equation: Incompressible Newtonian Fluids

Pressure Gradient. In order to express Eq. 11.40 in cylindrical coordinates, we require the gradient of the pressure field given by Eq. 7.90 as

$\overrightarrow{\text{∇}}P=\frac{\partial p}{\partial r}{\overrightarrow{e}}_r+\frac{1}{r}\frac{\partial p}{\partial \phi }{\overrightarrow{e}}_{\phi }+\frac{\partial p}{\partial z}{\overrightarrow{e}}_z$

Material Derivative. For the material derivative, we require Eq. 7.106, using $\overrightarrow{F}=\overrightarrow{G}=\overrightarrow{\upsilon }$ in which case we find

$\left(\overrightarrow{\upsilon }\cdot \overrightarrow{\nabla }\right)\overrightarrow{\upsilon }=\left({\upsilon }_r\frac{\partial {\upsilon }_r}{\partial r}+\frac{{\upsilon }_{\phi }}{r}\frac{\partial {\upsilon }_r}{\partial \phi }-\frac{{\upsilon }_{\phi }^2}{r}+{\upsilon }_z\frac{\partial {\upsilon }_r}{\partial z}\right){\overrightarrow{e}}_r+\left({\upsilon }_r\frac{\partial {\upsilon }_{\phi }}{\partial r}+\frac{{\upsilon }_{\phi }}{r}\frac{\partial {\upsilon }_{\phi }}{\partial r}+\frac{{\upsilon }_r{\upsilon }_{\phi }}{r}+{\upsilon }_z\frac{\partial {\upsilon }_{\phi }}{\partial z}\right){\overrightarrow{e}}_{\phi }+\left({\upsilon }_r\frac{\partial {\upsilon }_z}{\partial r}+\frac{{\upsilon }_{\phi }}{r}\frac{\partial {\upsilon }_z}{\partial \phi }+{\upsilon }_z\frac{\partial {\upsilon }_r}{\partial z}\right){\overrightarrow{e}}_z$

Vector Laplacian. For the vector Laplacian, we use Eq. 7.103, which yields

$\text{Δ}\overrightarrow{\upsilon }=\left(\frac{{\partial }^2{\upsilon }_r}{\partial r^2}+\frac{1}{r}\frac{\partial {\upsilon }_r}{\partial r}+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_r}{\partial {\phi }^2}+\frac{{\partial }^2{\upsilon }_r}{\partial z^2}-\frac{{\upsilon }_r}{r^2}-\frac{2}{r^2}\frac{\partial {\upsilon }_{\phi }}{\partial \phi }\right){\overrightarrow{e}}_r+\left(\frac{{\partial }^2{\upsilon }_{\phi }}{\partial r^2}+\frac{1}{r}\frac{\partial {\upsilon }_{\phi }}{\partial r}+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_{\phi }}{\partial {\phi }^2}+\frac{{\partial }^2{\upsilon }_{\phi }}{\partial z^2}-\frac{{\upsilon }_{\phi }}{r^2}+\frac{2}{r^2}\frac{\partial {\upsilon }_r}{\partial \phi }\right){\overrightarrow{e}}_{\phi }+\left(\frac{{\partial }^2{\upsilon }_z}{\partial z^2}+\frac{1}{r}\frac{\partial {\upsilon }_z}{\partial r}+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_z}{\partial {\phi }^2}+\frac{{\partial }^2{F}_z}{\partial z^2}\right){\overrightarrow{e}}_z$

We can derive a slightly more compact notation using the fact that

$\begin{array}{ll}\frac{\partial }{\partial r}\left(\frac{1}{r}\frac{\partial r{\upsilon }_r}{\partial r}\right)\hfill & =\frac{\partial }{\partial r}\left(\frac{1}{r}\left({\upsilon }_r+r\frac{\partial {\upsilon }_r}{\partial r}\right)\right)=\frac{\partial }{\partial r}\left(\frac{{\upsilon }_r}{r}+\frac{\partial {\upsilon }_r}{\partial r}\right)=\frac{1}{r}\frac{{\upsilon }_r}{r}-\frac{{\upsilon }_r}{r^2}+\frac{{\partial }^2{\upsilon }_r}{\partial r^2}\hfill \\ \frac{\partial }{\partial r}\left(\frac{1}{r}\frac{\partial r{\upsilon }_{\phi }}{\partial r}\right)\hfill & =\frac{\partial }{\partial r}\left(\frac{1}{r}\left({\upsilon }_{\phi }+r\frac{\partial {\upsilon }_{{}_{\phi }}}{\partial r}\right)\right)=\frac{\partial }{\partial r}\left(\frac{{\upsilon }_{\phi }}{r}+\frac{\partial {\upsilon }_{\phi }}{\partial r}\right)=\frac{1}{r}\frac{{\upsilon }_{{}_{\phi }}}{r}-\frac{{\upsilon }_{{}_{\phi }}}{r^2}+\frac{{\partial }^2{\upsilon }_{\phi }}{\partial r^2}\hfill \\ \frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial r{\upsilon }_z}{\partial r}\right)\hfill & =\frac{1}{r}\left(\frac{\partial {\upsilon }_z}{\partial r}+r\frac{{\partial }^{2}r{\upsilon }_z}{\partial r^2}\right)\hfill \\ \hfill & =\frac{1}{r}\frac{\partial {\upsilon }_z}{\partial r}+r\frac{{\partial }^{2}r{\upsilon }_z}{\partial r^2}\hfill \end{array}$

which allows us to write

$\begin{array}{ll}\text{Δ}\overrightarrow{\upsilon }=\hfill & \left(\frac{\partial }{\partial r}\left(\frac{1}{r}\frac{{\partial }^2{\upsilon }_r}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_r}{\partial {\phi }^2}-\frac{2}{r^2}\frac{\partial {\upsilon }_{\phi }}{\partial \phi }+\frac{{\partial }^2{\upsilon }_r}{\partial z^2}\right){\overrightarrow{e}}_r+\hfill \\ \hfill & \left(\frac{\partial }{\partial r}\left(\frac{1}{r}\frac{\partial \left(r{\upsilon }_{\phi }\right)}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_{\phi }}{\partial {\phi }^2}+\frac{2}{r^2}\frac{\partial {\upsilon }_r}{\partial \phi }+\frac{{\partial }^2{\upsilon }_{\phi }}{\partial z^2}\right){\overrightarrow{e}}_{\phi }+\hfill \\ \hfill & \left(\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial \left(r{\upsilon }_z\right)}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_z}{\partial {\phi }^2}+\frac{{\partial }^2{F}_z}{\partial r^2}\right){\overrightarrow{e}}_z\hfill \end{array}$

Navier-Stokes Equation. We can now write the Navier-Stokes equation for incompressible Newtonian fluids in cylindrical coordinates as

r-axis

$\begin{array}{l}\rho \left(\frac{\partial {\upsilon }_r}{\partial t}+{\upsilon }_r\frac{\partial {\upsilon }_r}{\partial r}+\frac{{\upsilon }_{\phi }}{r}\frac{\partial {\upsilon }_r}{\partial \phi }-\frac{{\upsilon }_{\phi }^2}{r}+{\upsilon }_z\frac{\partial {\upsilon }_r}{\partial z}\right)\hfill \\ =k_r-\frac{\partial p}{\partial r}+\eta \left(\frac{\partial }{\partial r}\left(\frac{1}{r}\frac{\partial \left(r{\upsilon }_r\right)}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_r}{\partial {\phi }^2}-\frac{2}{r^2}\frac{\partial {\upsilon }_{\phi }}{\partial \phi }+\frac{{\partial }^2{\upsilon }_r}{\partial z^2}\right)\hfill \end{array}$

ϕ-axis

$\begin{array}{l}\rho \left(\frac{\partial {\upsilon }_{\phi }}{\partial t}+{\upsilon }_r\frac{\partial {\upsilon }_{\phi }}{\partial r}+\frac{{\upsilon }_{\phi }}{r}\frac{\partial {\upsilon }_{\phi }}{\partial \phi }+\frac{{\upsilon }_r{\upsilon }_{\phi }}{r}+{\upsilon }_z\frac{\partial {\upsilon }_{\phi }}{\partial z}\right)\hfill \\ =k_{\phi }-\frac{1}{r}\frac{\partial p}{\partial \phi }+\eta \left(\frac{\partial }{\partial r}\left(\frac{1}{r}\frac{\partial \left(r{\upsilon }_{\phi }\right)}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_{\phi }}{\partial {\phi }^2}-\frac{2}{r^2}\frac{\partial {\upsilon }_r}{\partial \phi }+\frac{{\partial }^2{\upsilon }_{\phi }}{\partial z^2}\right)\hfill \end{array}$

z-axis

$\begin{array}{l}\rho \left(\frac{\partial {\upsilon }_z}{\partial t}+{\upsilon }_r\frac{\partial {\upsilon }_z}{\partial r}+\frac{{\upsilon }_{\phi }}{r}\frac{\partial {\upsilon }_z}{\partial \phi }+{\upsilon }_z\frac{\partial {\upsilon }_z}{\partial z}\right)\hfill \\ =k_z-\frac{\partial p}{\partial z}+\eta \left(\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial \left(r{\upsilon }_z\right)}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_z}{\partial {\phi }^2}+\frac{{\partial }^2{F}_z}{\partial z^2}\right)\hfill \end{array}$

13.3.1 Continuity Equation: Compressible

For the continuity equation for compressible fluids, we therefore find

$\begin{array}{ll}\frac{\partial \rho }{\partial t}+\overrightarrow{\nabla }\cdot \left(\rho \overrightarrow{\upsilon }\right)\hfill & =0\hfill \\ \frac{\partial \rho }{\partial t}+\frac{1}{r}\frac{\partial }{\partial r}\left(\rho r{\upsilon }_r\right)+\frac{1}{r}\frac{\partial \left(\rho {\upsilon }_{\phi }\right)}{\partial \phi }\hfill & =0\hfill \end{array}$

13.3.2 Continuity Equation: Incompressible

In comparison, the continuity equation for incompressible fluids is given by

$\begin{array}{ll}\overrightarrow{\nabla }\cdot \overrightarrow{\upsilon }\hfill & =0\hfill \\ \frac{1}{r}\frac{\partial }{\partial r}\left(r{\upsilon }_r\right)+\frac{1}{r}\frac{\partial {\upsilon }_{\phi }}{\partial \phi }\hfill & =0\hfill \\ \frac{\partial {\upsilon }_r}{\partial r}+\frac{1}{r}{\upsilon }_r+\frac{1}{r}\frac{\partial {\upsilon }_{\phi }}{\partial \phi }\hfill & =0\hfill \end{array}$

13.3.3 Navier-Stokes Equation: Incompressible Newtonian Fluids

The Navier-Stokes equation for incompressible Newtonian fluids in polar coordinates can simply be copied from the equation for cylindrical coordinates by ignoring all contributions in z-direction. We thus find

r-axis

$\begin{array}{l}\rho \left(\frac{\partial {\upsilon }_r}{\partial t}+{\upsilon }_r\frac{\partial {\upsilon }_r}{\partial r}+\frac{{\upsilon }_{\phi }}{r}\frac{\partial {\upsilon }_r}{\partial \phi }-\frac{{\upsilon }_{\phi }^2}{r}\right)\hfill \\ =k_r-\frac{\partial p}{\partial r}+\eta \left(\frac{\partial }{\partial r}\left(\frac{1}{r}\frac{\partial \left(r{\upsilon }_r\right)}{\partial r}\right)\right)+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_r}{\partial {\phi }^2}-\frac{2}{r^2}\frac{\partial {\upsilon }_{\phi }}{\partial \phi }\hfill \end{array}$

ϕ-axis

$\begin{array}{l}\rho \left(\frac{\partial {\upsilon }_{\phi }}{\partial t}+{\upsilon }_r\frac{\partial {\upsilon }_{\phi }}{\partial r}+\frac{{\upsilon }_{\phi }}{r}\frac{\partial {\upsilon }_{\phi }}{\partial \phi }-\frac{{\upsilon }_r{\upsilon }_{\phi }}{r}\right)\hfill \\ =k_{\phi }-\frac{1}{r}\frac{\partial p}{\partial \phi }+\eta \left(\frac{\partial }{\partial r}\left(\frac{1}{r}\frac{\partial \left(r{\upsilon }_{\phi }\right)}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_{\phi }}{\partial {\phi }^2}+\frac{2}{r^2}\frac{\partial {\upsilon }_r}{\partial \phi }\right)\hfill \end{array}$

13.4 Spherical Coordinates

13.4.1 Continuity Equation: Compressible

For spherical coordinates,¹ we derived the divergence in Eq. 7.122, which allows us to express the continuity equation for compressible fluids as

$\begin{array}{ll}\frac{\partial \rho }{\partial t}+\overrightarrow{\nabla }\cdot \left(\rho \overrightarrow{\upsilon }\right)\hfill & =0\hfill \\ \frac{\partial \rho }{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\rho F_r\right)+\frac{1}{r\text{sin}\theta }\left(\frac{\partial }{\partial \theta }\left(p{F}_0\text{sin}\theta +\frac{\partial \rho F_{\phi }}{\partial \phi }\right)\right)\hfill & =0\hfill \end{array}$

13.4.2 Continuity Equation: Incompressible

For incompressible fluids, the continuity equation simplifies to

$\begin{array}{ll}\overrightarrow{\nabla }\cdot \left(\rho \overrightarrow{\upsilon }\right)\hfill & =0\hfill \\ \frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2{F}_r\right)+\frac{1}{r\text{sin}\theta }\left(\frac{\partial }{\partial \theta }\left(F_0\text{sin}\theta \right)+\frac{\partial F_{\phi }}{\partial \phi }\right)\hfill & =0\hfill \end{array}$

13.4.3 Navier-Stokes Equation: Incompressible Newtonian Fluids

Pressure Gradient. The pressure gradient in spherical coordinates is given by Eq. 7.118 as

$\overrightarrow{\nabla }p=\frac{\partial p}{\partial r}{\overrightarrow{e}}_r+\frac{1}{r}\frac{\partial p}{\partial \theta }{\overrightarrow{e}}_{\theta }+\frac{1}{r\text{sin}\theta }\frac{\partial p}{\partial \phi }{\overrightarrow{e}}_{\phi }$

Material Derivative. For the material derivative, we use Eq. 7.143 and set $\overrightarrow{F}=\overrightarrow{G}=\overrightarrow{\upsilon }$, in which case we find

$\begin{array}{ll}\left(\overrightarrow{\upsilon }\cdot \overrightarrow{\nabla }\right)\overrightarrow{\upsilon }=\hfill & \left({\upsilon }_r\frac{\partial {\upsilon }_r}{\partial r}+\frac{{\upsilon }_{\theta }}{r}\frac{\partial {\upsilon }_r}{\partial \theta }+\frac{{\upsilon }_{\phi }}{r\text{sin}\theta }\frac{\partial {\upsilon }_r}{\partial \phi }-\frac{{\upsilon }_{\theta }{\upsilon }_{\theta }+{\upsilon }_{\phi }{\upsilon }_{\phi }}{r}\right){\overrightarrow{e}}_r+\hfill \\ \hfill & \left({\upsilon }_r\frac{\partial {\upsilon }_{\theta }}{\partial r}+\frac{{\upsilon }_{\theta }}{r}\frac{\partial {\upsilon }_{\theta }}{\partial \theta }+\frac{{\upsilon }_{\phi }}{r\text{sin}\theta }\frac{\partial {\upsilon }_{\theta }}{\partial \phi }+\frac{{\upsilon }_{\theta }{\upsilon }_r}{r}-\frac{{\upsilon }_{\phi }{\upsilon }_{\phi }}{r}\frac{\cos \theta }{\text{sin}\theta }\right){\overrightarrow{e}}_{\theta }+\hfill \\ \hfill & \left({\upsilon }_r\frac{\partial {\upsilon }_{\phi }}{\partial r}+\frac{{\upsilon }_{\theta }}{r}\frac{\partial {\upsilon }_{\phi }}{\partial \theta }+\frac{{\upsilon }_{\phi }}{r\text{sin}\theta }\frac{\partial {\upsilon }_{\theta }}{\partial \phi }+\frac{{\upsilon }_{\phi }{\upsilon }_r}{r}+\frac{{\upsilon }_{\phi }{\upsilon }_{\theta }}{r}\frac{\cos \theta }{\text{sin}\theta }\right){\overrightarrow{e}}_{\phi }\hfill \end{array}$

Vector Laplacian. For the vector Laplacian, we use Eq. 7.135, which yields

$\begin{array}{ll}\text{Δ}\overrightarrow{\upsilon }=\hfill & \left(\frac{2}{r}\frac{\partial {\upsilon }_r}{\partial r}+\frac{{\partial }^2{\upsilon }_r}{\partial r^2}+\frac{1}{r^2}\frac{\cos \theta }{\text{sin}\theta }\frac{\partial {\upsilon }_r}{\partial \theta }+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_r}{\partial {\theta }^2}+\frac{1}{r^2{\text{sin}}^2\theta }\frac{{\partial }^2{\upsilon }_r}{\partial {\phi }^2}-\frac{2}{r^2}{\upsilon }_r-\frac{2}{r^2{\text{sin}}^2\theta }\frac{{\partial }^2{\upsilon }_{\phi }}{\partial \phi }-\frac{2}{r^2}\frac{\partial {\upsilon }_{\theta }}{\partial \theta }-\frac{2}{r^2}\frac{\cos \theta }{\text{sin}\theta }{\upsilon }_{\theta }\right){\overrightarrow{e}}_r+\hfill \\ \hfill & \left(\frac{2}{r}\frac{\partial {\upsilon }_{\theta }}{\partial r}+\frac{{\partial }^2{\upsilon }_{\theta }}{\partial r^2}+\frac{1}{r^2}\frac{\cos \theta }{\text{sin}\theta }\frac{\partial {\upsilon }_{\theta }}{\partial \theta }+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_{\theta }}{\partial {\theta }^2}+\frac{1}{r^2{\text{sin}}^2\theta }\frac{{\partial }^2{\upsilon }_{\theta }}{\partial {\phi }^2}-\frac{{\upsilon }_{\theta }}{r^2{\text{sin}}^2\theta }+\frac{2}{r^2}\frac{\partial {\upsilon }_r}{\partial \theta }-\frac{2}{r^2}\frac{\cos \theta }{{\text{sin}}^2\theta }\frac{\partial {\upsilon }_{\phi }}{\partial \phi }\right){\overrightarrow{e}}_{\theta }+\hfill \\ \hfill & \left(\frac{2}{r}\frac{\partial {\upsilon }_{\phi }}{\partial r}+\frac{{\partial }^2{\upsilon }_{\phi }}{\partial r^2}+\frac{1}{r^2}\frac{\cos \theta }{\text{sin}\theta }\frac{\partial {\upsilon }_{\phi }}{\partial \theta }+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_{\phi }}{\partial {\theta }^2}+\frac{1}{r^2{\text{sin}}^2\theta }\frac{{\partial }^2{\upsilon }_{\phi }}{\partial {\phi }^2}-\frac{{\upsilon }_{\phi }}{r^2{\text{sin}}^2\theta }+\frac{2}{r^2{\text{sin}}^2\theta }\frac{\partial {\upsilon }_r}{\partial \phi }+\frac{2}{r^2}\frac{\cos \theta }{\text{sin}\theta }\frac{\partial {\upsilon }_{\theta }}{\partial \phi }\right){\overrightarrow{e}}_{\phi }\hfill \end{array}$

You may come across a slightly more compact notation that uses the fact that

$\begin{array}{ll}\frac{\partial }{\partial r}\left(\frac{1}{r^2}\frac{\partial r^2{\upsilon }_r}{\partial r}\right)\hfill & =\frac{\partial }{\partial r}\left(\frac{2}{r}{\upsilon }_r+\frac{\partial {\upsilon }_r}{\partial r}\right)\hfill \\ \hfill & =-\frac{2}{r^2}{\upsilon }_r+\frac{2}{r}\frac{\partial {\upsilon }_r}{\partial r}+\frac{{\partial }^2{\upsilon }_r}{\partial r^2}\hfill \\ \frac{1}{r^2\text{sin}\theta }\frac{\partial }{\partial \theta }\left(\frac{\partial \left({\upsilon }_r\text{sin}\theta \right)}{\partial \theta }\right)\hfill & =\frac{1}{r^2\text{sin}\theta }\left(\cos \theta \frac{\partial {\upsilon }_r}{\partial \theta }+\text{sin}\theta \frac{{\partial }^2{\upsilon }_r}{\partial {\theta }^2}\right)\hfill \\ =\frac{1}{r^2}\frac{\cos \theta }{\text{sin}\theta }\frac{\partial {\upsilon }_r}{\partial \theta }+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_r}{\partial {\theta }^2}\hfill \\ -\frac{2}{r^2\text{sin}\theta }\frac{\partial }{\partial \theta }\left({\upsilon }_r\text{sin}\theta \right)\hfill & -=\frac{2}{r^2\text{sin}\theta }\left(\cos \theta {\upsilon }_{\theta }+\text{sin}\theta \frac{{\partial }^2{\upsilon }_{{}^{\theta }}}{\partial \theta }\right)\hfill \\ \hfill & =\frac{2}{r^2}\frac{\cos \theta }{\text{sin}\theta }{\upsilon }_{\theta }-\frac{2}{r^2}\frac{\partial {\upsilon }_{\theta }}{\partial \theta }\hfill \\ \frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial {\upsilon }_{\theta }}{\partial r}\right)\hfill & =\frac{1}{r^2}\left(2r\frac{\partial {\upsilon }_{\theta }}{\partial r}+r^2\frac{{\partial }^2{\upsilon }_{\theta }}{\partial r^2}\right)\hfill \\ \hfill & =\frac{2}{r}\frac{\partial {\upsilon }_{\theta }}{\partial r}+\frac{{\partial }^2{\upsilon }_{\theta }}{\partial r^2}\hfill \\ \frac{1}{r^2}\frac{\partial }{\partial \theta }\left(\frac{1}{\text{sin}\theta }\frac{\partial }{\partial \theta }\left({\upsilon }_{\theta }\text{sin}\theta \right)\right)\hfill & =\frac{1}{r^2}\frac{\partial }{\partial \theta }\left(\frac{\cos \theta }{\text{sin}\theta }{\upsilon }_{\theta }+\frac{\partial {\upsilon }_{\theta }}{\partial \theta }\right)\hfill \\ \hfill & =-\frac{1}{r^2{\text{sin}}^2\theta }{\upsilon }_{\theta }+\frac{1}{r^2}\frac{\cos \theta }{\text{sin}\theta }\frac{\partial {\upsilon }_{\theta }}{\partial \theta }+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_{\theta }}{\partial {\theta }^2}\hfill \\ \frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial {\upsilon }_{\phi }}{\partial r}\right)\hfill & =\frac{1}{r^2}\left(2r\frac{\partial {\upsilon }_{\phi }}{\partial r}+r^2\frac{{\partial }^2{\upsilon }_{\phi }}{\partial r^2}\right)\hfill \\ \hfill & =\frac{2}{r}\frac{\partial {\upsilon }_{\phi }}{\partial r}+\frac{{\partial }^2{\upsilon }_{\phi }}{\partial r^2}\hfill \\ \frac{1}{r^2}\frac{\partial }{\partial r}\left(\frac{1}{\text{sin}\theta }\frac{\partial }{\partial \theta }\left({\upsilon }_{\phi }\text{sin}\theta \right)\right)\hfill & =\frac{1}{r^2}\frac{\partial }{\partial r}\left(\frac{\cos \theta }{\text{sin}\theta }{\upsilon }_{\phi }+\frac{\partial {\upsilon }_{\phi }}{\partial \theta }\right)\hfill \\ \hfill & =-\frac{1}{r^2{\text{sin}}^2\theta }{\upsilon }_{\phi }+\frac{1}{r^2}\frac{\cos \theta }{\text{sin}\theta }\frac{\partial {\upsilon }_{\phi }}{\partial \theta }+\frac{1}{r^2}\frac{{\partial }^2{\upsilon }_{\theta }}{\partial {\theta }^2}\hfill \end{array}$

which allows us to write

$\text{Δ}\overrightarrow{\upsilon }=\left(\frac{\partial }{\partial r}\left(\frac{1}{r^2}\frac{\partial \left(r^2{\upsilon }_r\right)}{\partial r}\right)+\frac{1}{r^2\text{sin}\theta }\frac{\partial }{\partial \theta }\left(\frac{\partial {\upsilon }_r}{\partial \theta }\text{sin}\theta \right)+\frac{1}{r^2{\text{sin}}^2\theta }\frac{{\partial }^2{\upsilon }_r}{\partial {\phi }^2}-\frac{1}{r^2\text{sin}\theta }\frac{\partial }{\partial \theta }\left({\upsilon }_{\theta }\text{sin}\theta \right)-\frac{2}{r^2\text{sin}\theta }\frac{\partial {\upsilon }_{\phi }}{\partial \phi }\right){\overrightarrow{e}}_r+$

$\begin{array}{l}\left(\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\left(\partial {\upsilon }_{\theta }\right)}{\partial r}\right)+\frac{1}{r^2}\frac{\partial }{\partial \theta }\left(r^2\frac{\partial {\upsilon }_{\theta }}{\partial \theta }\right)+\frac{2}{r^2}\frac{\partial \upsilon }{\partial \theta }+\frac{1}{r^2{\text{sin}}^2\theta }\frac{{\partial }^2{\upsilon }_{\theta }}{\partial {\phi }^2}-\frac{2}{r^2}\frac{\cos \theta }{{\text{sin}}^2\theta }\frac{\partial {\upsilon }_{\phi }}{\partial \phi }\right){\overrightarrow{e}}_{\theta }+\hfill \\ \left(\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\left(\partial {\upsilon }_{\phi }\right)}{\partial r}\right)+\frac{1}{r^2}\frac{\partial }{\partial \theta }\left(\frac{1}{\text{sin}\theta }\frac{\partial {\upsilon }_{\theta }}{\partial \theta }\left({\upsilon }_{\phi }\text{sin}\theta \right)\right)+\frac{1}{r^2{\text{sin}}^2\theta }\frac{{\partial }^2{\upsilon }_{\phi }}{\partial {\phi }^2}+\frac{2}{r^2\text{sin}\theta }\frac{\partial {\upsilon }_r}{\partial \phi }+\frac{2}{r^2}\frac{\cos \theta }{{\text{sin}}^2\theta }\frac{\partial {\upsilon }_{\theta }}{\partial \phi }\right){\overrightarrow{e}}_{\phi }\hfill \end{array}$

Navier-Stokes Equation. We can now write the Navier-Stokes equation for incompressible Newtonian fluids in spherical coordinates as

r-axis

$\begin{array}{l}\rho \left(\frac{\partial {\upsilon }_r}{\partial t}+{\upsilon }_r\frac{\partial {\upsilon }_r}{\partial r}+\frac{{\upsilon }_{\theta }}{r}\frac{\partial {\upsilon }_r}{\partial \theta }+\frac{{\upsilon }_{\phi }}{r\text{sin}\theta }\frac{\partial {\upsilon }_r}{\partial \phi }-\frac{{\upsilon }_{\theta }^2+{\upsilon }_{\phi }^2}{r}\right)\hfill \\ =k_r-\frac{\partial p}{\partial r}+\eta \left(\frac{\partial }{\partial r}\left(\frac{1}{r^2}\frac{\partial \left(r^2{\upsilon }_r\right)}{\partial r}\right)+\frac{1}{r^2\text{sin}\theta }\frac{\partial }{\partial \theta }\left(\text{sin}\theta \frac{\partial {\upsilon }_r}{\partial \theta }\right)+\frac{1}{r^2{\text{sin}}^2\theta }\frac{{\partial }^2{\upsilon }_r}{\partial {\phi }^2}-\frac{2}{r^2\text{sin}\theta }\frac{\partial \left({\upsilon }_{\theta }\text{sin}\theta \right)}{\partial \theta }-\frac{2}{r^2\text{sin}\theta }\frac{\partial {\upsilon }_{\phi }}{\partial \phi }\right)\hfill \end{array}$

θ-axis

$\begin{array}{l}\rho \left(\frac{\partial {\upsilon }_{\phi }}{\partial t}+{\upsilon }_r\frac{\partial {\upsilon }_{\theta }}{\partial r}+\frac{{\upsilon }_{\theta }}{r}\frac{\partial {\upsilon }_{\theta }}{\partial \theta }+\frac{{\upsilon }_{\phi }}{r\text{sin}\theta }\frac{\partial {\upsilon }_{\theta }}{\partial \phi }+\frac{{\upsilon }_r{\upsilon }_{\theta }}{r}-\frac{{\upsilon }_{\phi }^2}{r}\frac{\cos \theta }{\text{sin}\theta }\right)\hfill \\ =k_{\theta }-\frac{1}{r}\frac{\partial p}{\partial \theta }+\eta \left(\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial {\upsilon }_{\theta }}{\partial r}\right)+\frac{1}{r^2}\frac{\partial }{\partial \theta }\left(r^2\frac{\partial {\upsilon }_{\theta }}{\partial \theta }\right)+\frac{2}{r^2}\frac{\partial {\upsilon }_r}{\partial \theta }+\frac{1}{r^2{\text{sin}}^2\theta }\frac{{\partial }^2{\upsilon }_{\theta }}{\partial {\phi }^2}-\frac{2}{r^2}\frac{\cos \theta }{\text{sin}\theta }\frac{\partial {\upsilon }_{\phi }}{\partial \phi }\right)\hfill \end{array}$

ϕ-axis

$\begin{array}{l}\rho \left(\frac{\partial {\upsilon }_{\phi }}{\partial t}+{\upsilon }_r\frac{\partial {\upsilon }_{\phi }}{\partial r}+\frac{{\upsilon }_{\theta }}{r}\frac{\partial {\upsilon }_{\phi }}{\partial \theta }+\frac{{\upsilon }_{\phi }}{r\text{sin}\theta }\frac{\partial {\upsilon }_{\phi }}{\partial \phi }+\frac{{\upsilon }_r{\upsilon }_{\phi }}{r}+\frac{{\upsilon }_{\theta }{\upsilon }_{\phi }}{r}\frac{\cos \theta }{\text{sin}\theta }\right)\hfill \\ =k_{\phi }-\frac{1}{r\text{sin}\theta }\frac{\partial p}{\partial \phi }+\eta \left(\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial {\upsilon }_{\phi }}{\partial r}\right)+\frac{1}{r^2}\frac{\partial }{\partial \theta }\left(\frac{1}{r\text{sin}\theta }\frac{\partial }{\partial \theta }\left({\upsilon }_{\phi }\text{sin}\theta \right)\right)+\frac{1}{r^2\text{sin}\theta }\frac{{\partial }^2{\upsilon }_{\phi }}{\partial {\phi }^2}+\frac{2}{r^2\text{sin}\theta }\frac{\partial {\upsilon }_r}{\partial \phi }+\frac{2}{r^2}\frac{\cos \theta }{\text{sin}\theta }\frac{\partial {\upsilon }_{\theta }}{\partial \phi }\right)\hfill \end{array}$