Navier-Stokes equations for Newtonian fluid

Basis

The Navier-Stokes (N-S) equations are Newton's second law written for a fluid element, with the viscous stresses related to the velocity gradients by Stokes' relations for a Newtonian fluid:

$$\tau_{xy} = \mu\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right), \qquad \sigma_{xx} = -p + 2\mu\frac{\partial u}{\partial x}$$

Steps

1. Take a fluid element $dx\,dy\,dz$ with normal and shear stresses on its faces.
2. Net surface force in the $x$-direction $=$ (net normal stress gradient + shear stress gradients) $\times\ dx\,dy\,dz$.
3. Add the body force $\rho g_x\,dx\,dy\,dz$.
4. Equate the sum to mass $\times$ acceleration, $\rho\,\dfrac{Du}{Dt}\,dx\,dy\,dz$.
5. Substitute the Newtonian stress relations, and use incompressible continuity ($\nabla\cdot\vec V = 0$) with constant $\mu$ to simplify.

x-component (incompressible, constant viscosity)

$$\rho\left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}\right) = \rho g_x - \frac{\partial p}{\partial x} + \mu\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right)$$

In vector form: $\rho\dfrac{D\vec V}{Dt} = \rho\vec g - \nabla p + \mu\nabla^2\vec V$.

The y and z components are written in the same way.

Assumptions

Newtonian fluid, incompressible flow, constant viscosity and a continuum. The equations are exact under these assumptions: they apply to laminar flow and to the instantaneous velocities in turbulent flow.

Reduction

• Euler's equation: for an inviscid fluid, $\mu = 0$:

$$\rho\frac{D\vec V}{Dt} = \rho\vec g - \nabla p$$

• Hydrostatic equation: for a fluid at rest, $\vec V = 0$, and with $z$ upward, $\partial p/\partial z = -\rho g$ and $\partial p/\partial x = \partial p/\partial y = 0$.

The N-S equations have exact solutions only for simple cases, such as laminar flow between parallel plates and in circular pipes (Hagen-Poiseuille flow).