We may describe the deformation of a fluid element by two primary metrics, the divergence and curl of the velocity field. We use divergence to obtain the volumetric dilation rate, $d_V$, which is zero for incompressible flow. We use curl to obtain vorticity, $\boldsymbol\zeta$, and the rotation vector, $\boldsymbol\omega$, both of which are zero for irrotational flow.
$$ d_V=\nabla\cdot\bold u $$
$$ 2\boldsymbol\omega=\boldsymbol\zeta=\nabla\times\bold u $$
Now that we have introduced the concept of an irrotational flow, we may expand our application of Bernoulli’s equation. For steady, incompressible, inviscid flow along a streamline, the total head is constant, and so Bernoulli’s equation may be used to compare points along a streamline. Additionally, for irrotational flow, the total head is constant for the whole flow, and so Bernoulli’s equation may be used to compare points along different streamlines in irrotational flow.
$$ p+\frac 12\rho u^2+gz=c $$
The differential expression for conservation of mass is given by the continuity equation $(\bold 1)$. This is expressed in cylindrical coordinates by $(\bold 4)$. For steady flow, the first term goes to zero because no flow property varies with time, giving $(\bold 2)$. For incompressible flow, the first term goes to zero because density does not vary, which also allows density to divide out of the second term, giving $(\bold 3)$.
$$ (\bold 1)\>\>\>0=\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\bold u) $$
$$ (\bold 2)\>\>\>0=\nabla\cdot(\rho\bold u) $$
$$ (\bold 3)\>\>\>0=\nabla\cdot\bold u $$
$$ (\bold 4)\>\>\>0=\frac{\partial\rho}{\partial t}+\frac 1r\frac{\partial(r\rho v_r)}{\partial r}+\frac 1r\frac{\partial(\rho v_\theta)}{\partial\theta}+\frac{\partial(\rho w)}{\partial z} $$
The Navier-Stokes equation governs all flows. We will use a form for incompressible flows with constant viscosity. Because the material derivative includes $\bold u\cdot\nabla\bold u$, it is a second order PDE, for which there is no general analytical method to solve. Below is the cartesian expression with del operators and the expanded cartesian form.
$$ \rho\frac{D\bold u}{Dt}= -\nabla p+ \mu\nabla^2\bold u+ \rho\bold g $$
$$ \rho(\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}u+\frac{\partial u}{\partial y}v+\frac{\partial u}{\partial z}w)=\rho g_x-\frac{\partial p}{\partial x}+\mu(\frac{\partial^2 u}{\partial^2x}+\frac{\partial^2u}{\partial^2y}+\frac{\partial^2u}{\partial^2z}) $$
$$ \rho(\frac{\partial v}{\partial t}+\frac{\partial v}{\partial x}u+\frac{\partial v}{\partial y}v+\frac{\partial v}{\partial z}w)=\rho g_y-\frac{\partial p}{\partial y}+\mu(\frac{\partial^2 v}{\partial^2x}+\frac{\partial^2v}{\partial^2y}+\frac{\partial^2v}{\partial^2z}) $$
$$ \rho(\frac{\partial w}{\partial t}+\frac{\partial w}{\partial x}u+\frac{\partial w}{\partial y}v+\frac{\partial w}{\partial z}w)=\rho g_z-\frac{\partial p}{\partial z}+\mu(\frac{\partial^2 w}{\partial^2x}+\frac{\partial^2w}{\partial^2y}+\frac{\partial^2w}{\partial^2z}) $$