Appendix E The divergence equation

We start with the momentum equation (2.2)

${\displaystyle \frac{\partial }{\partial t}}(\rho v_i)+{\displaystyle \frac{\partial }{\partial r_j}}(v_j\rho v_i)$

$=-{\displaystyle \frac{\partial }{\partial r_i}}p+{\displaystyle \frac{\partial }{\partial r_j}}{\sigma }_{ij}^{\prime }-\rho {\displaystyle \frac{\partial }{\partial r_i}}\varphi ,$

where we assumed $g_i=-\frac{\partial }{\partial r_i}\varphi$. If we make the substitutions $\frac{\partial }{\partial r_i}p\to \frac{\partial }{\partial r_j}p{\delta }_{ij}$ and $\frac{\partial }{\partial r_i}\varphi \to \frac{\partial }{\partial r_j}\varphi {\delta }_{ij}$ we can write it in the form

${\displaystyle \frac{\partial }{\partial t}}(\rho v_i)+{\displaystyle \frac{\partial }{\partial r_j}}(v_j\rho v_i+p{\delta }_{ij}-{\sigma }_{ij}^{\prime })=-\rho {\displaystyle \frac{\partial }{\partial r_j}}\varphi {\delta }_{ij}.$

Taking the divergence of this equation we get

${\displaystyle \frac{\partial }{\partial t}}\left[{\displaystyle \frac{\partial }{\partial r_i}}(\rho v_i)\right]+{\displaystyle \frac{{\partial }^2}{\partial r_i\partial r_j}}(v_j\rho v_i+p{\delta }_{ij}-{\sigma }_{ij}^{\prime })=-{\displaystyle \frac{\partial }{\partial r_i}}\left(\rho {\displaystyle \frac{\partial }{\partial r_j}}\varphi {\delta }_{ij}\right),$

where we assumed that $\frac{\partial }{\partial t}$ and $\frac{\partial }{\partial r_i}$ commute. Using the continuity equation (2.1) we get a interesting form of the fluiddynamic equations

${\displaystyle \frac{{\partial }^2}{\partial t^2}}\rho -{\displaystyle \frac{{\partial }^2}{\partial r_i\partial r_j}}(v_j\rho v_i+p{\delta }_{ij}-{\sigma }_{ij}^{\prime })=+{\displaystyle \frac{\partial }{\partial r_i}}\left(\rho {\displaystyle \frac{\partial }{\partial r_j}}\varphi {\delta }_{ij}\right).$

(E.1)

In case of no gravitation, the fluiddynamic equation can be written in a form showing some similarity to a wave equation

${\displaystyle \frac{{\partial }^2}{\partial t^2}}\rho -{\displaystyle \frac{{\partial }^2}{\partial r_i\partial r_j}}(v_j\rho v_i+p{\delta }_{ij}-{\sigma }_{ij}^{\prime })=0.$

But despite its simple form, this equation hides an extreme complexity.

Solving for pressure this equation is written like

${\displaystyle \frac{{\partial }^2}{\partial r_i^2}}p={\displaystyle \frac{{\partial }^2}{\partial t^2}}\rho -{\displaystyle \frac{{\partial }^2}{\partial r_i\partial r_j}}(\rho v_i{v}_j-{\sigma }_{ij}^{\prime })$

(E.2)

and sometimes called the equation for the instantaneous pressure.