# Deriving the Navier-Stokes Equation in Fluid Dynamics

I am trying to understand the derivation of the following equation which describes the motion of Newtonian viscous fluids:

Equation 1: $$\rho \dfrac{D u}{Dt} = \rho g_x - \dfrac{\partial p}{\partial x} + \mu \nabla^2 u$$

Where $$\mu$$ is the viscosity of the fluid. I am following the proof written in Hibbeler's Fluid Dynamics which uses the following steps.

First the following formula is stated based on the free-body diagram below: (I am just writing the equations related to the $$x$$ component of the velocity, namely $$u$$)

Equation 2: $$\rho \dfrac{D u}{Dt} = \rho g_x + \dfrac{\partial \sigma_{xx}}{\partial x} + \dfrac{\partial \tau_{yx}}{\partial y} + \dfrac{\partial \tau_{zx}}{\partial z}$$ In the second step, the normal stress and shear stress variables in the previous equation are related to the velocity and viscosity of the fluid.

$$\sigma_{xx} = -p + 2\mu \dfrac{\partial u}{\partial x}$$

$$\tau_{yx} = \mu (\dfrac{\partial u}{\partial y} + \dfrac{\partial v}{\partial x})$$

$$\tau_{zx} = \mu (\dfrac{\partial u}{\partial z} + \dfrac{\partial w}{\partial x})$$

Here the book claims that by replacing the last three equations in equation(2), we yield equation(1).

I do not understand how the last three equations are derived. Moreover, after replacing these last three equations in equation(2), the result is not even similar to equation(1). This is what I got after doing so:

$$\dfrac{Du}{Dt} = \rho g_x - \dfrac{\partial p}{\partial x} + \mu (\nabla^2 u + \dfrac{\partial^2 u}{\partial^2 x} + \dfrac{\partial^2 v}{\partial x \partial y} + \dfrac{\partial^2 w}{\partial x \partial z})$$

What am I doing wrong? Is it possible the three equations presented in step 2 are wrong? Is there a source which explicitly explains this topic?

Thanks for your help!

## 1 Answer

There's nothing wrong. Simply notice that $$\dfrac{\partial^2 u}{\partial^2 x} + \dfrac{\partial^2 v}{\partial x \partial y} + \dfrac{\partial^2 w}{\partial x \partial z}=\frac{\partial}{\partial x} \nabla \cdot \mathbf u=0$$ where $$\mathbf u = (u,v,w)$$. The condition $$\nabla \cdot \mathbf u=0$$ Is due to incompressibility.

• I see. Thanks! Also, where do those equations come from in the first place? – Soroush khoubyarian Jul 30 '19 at 23:33
• I am referring only to those three equations, not the other ones. – Soroush khoubyarian Jul 30 '19 at 23:42
• For the derivation of the relationship between the stress tensor components and the velocity gradients, see Bird, Stewart, and Lightfoot, Transport Phenomena, Chapter 1. – Chet Miller Jul 31 '19 at 0:10
• Thanks! The Transport Phenomena textbook clarifies everything. – Soroush khoubyarian Jul 31 '19 at 0:29