Euler's Equations of Motion for Inviscid Flows

In inviscid flows, viscous shearing stresses are neglected. This results in Euler’s Equation of Motion.

Thus, the shear terms in our previously derived differential fluid element equations of motion are cancelled

$$\begin{array}{rl}x:& \frac{\partial {\sigma }_{xx}}{\partial x}+\text{cancelto}0\frac{\partial {\tau }_{yx}}{\partial y}+\text{cancelto}0\frac{\partial {\tau }_{zx}}{\partial z}+\rho g_x=\rho \left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)\\ y:& \text{cancelto}0\frac{\partial {\tau }_{xy}}{\partial x}+\frac{\partial {\sigma }_{yy}}{\partial y}+\text{cancelto}0\frac{\partial {\tau }_{zy}}{\partial z}+\rho g_y=\rho \left(\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right)\\ z:& \text{cancelto}0\frac{\partial {\tau }_{xz}}{\partial x}+\text{cancelto}0\frac{\partial {\tau }_{yz}}{\partial y}+\frac{\partial {\sigma }_{zz}}{\partial z}+\rho g_z=\rho \left(\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\right)\end{array}$$

Next, we can note that the normal stresses are purely pressure forces ${\sigma }_{xx}={\sigma }_{yy}={\sigma }_{zz}=-p$, which further simplifies our momentum equations to:

$$\begin{array}{rl}x:& \rho g_x-\frac{\partial p}{\partial x}=\rho \left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)\\ y:& \rho g_y-\frac{\partial p}{\partial y}=\rho \left(\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right)\\ z:& \rho g_z-\frac{\partial p}{\partial z}=\rho \left(\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\right)\end{array}$$

which are the Euler Equations of Motion for inviscid flow.

In a compact vector notation, we can write

$$\rho \mathbf{g}-\nabla p=\rho \left[\frac{\partial \mathbf{V}}{\partial t}+(\mathbf{V}\cdot \nabla )\mathbf{V}\right]$$

where

• $\mathbf{g}$ is the gravitational acceleration.
• $\nabla p$ is the pressure gradient.
• $\mathbf{V}$ is the velocity vector.
• $(\mathbf{V}\cdot \nabla )\mathbf{V}$ is the convective acceleration.