Keywords

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Equation 3.2 should appear as follows:

$$\frac{\partial (\rho {u}_{i})} {\partial t} +{ \sum \limits_{j}}\frac{\partial (\rho {u}_{i}{u}_{j})} {\partial {x}_{j}} ={ \sum \limits_{j}}\frac{\partial {\tau }_{ij}} {\partial {x}_{j}} + {F}_{i}{\mbox{, Nm}}^{-3},$$
(3.2)

Equation 3.3 should appear as follows:

$$\begin{array}{rcl} {\tau }_{ij}& =& \left (-p -\frac{2} {3}\mu \sum \frac{\partial {u}_{j}} {\partial {x}_{j}}\right ){\delta }_{ij} + \mu \left (\frac{\partial {u}_{i}} {\partial {x}_{j}} + \frac{\partial {u}_{j}} {\partial {x}_{i}} \right ) =- p{\delta }_{ij} - {\tau }_{ij}^{{_\ast}}, \\ {\delta }_{ij}& =& \mbox{ Kronecker delta: }{\delta }_{ii} = 1,{\delta }_{ij} = 0, \\ {\tau }_{ij}^{{_\ast}}& =& \left (\frac{2} {3}\mu \sum \frac{\partial {u}_{j}} {\partial {x}_{j}}\right ){\delta }_{ij} - \mu \left (\frac{\partial {u}_{i}} {\partial {x}_{j}} + \frac{\partial {u}_{j}} {\partial {x}_{i}} \right ). \end{array}$$
(3.3)

Equation 3.4 should appear as follows:

$$\frac{\partial (\rho {u}_{i})} {\partial t} +{ \sum \limits_{j}}\frac{\partial (\rho {u}_{i}{u}_{j})} {\partial {x}_{j}} = -\frac{\partial p} {\partial {x}_{j}} -{\sum \limits_{j}}\frac{\partial {\tau }_{ij}^{{_\ast}}} {\partial {x}_{j}} + {F}_{i}{\mbox{, Nm}}^{-3}.$$
(3.4)

Equation 3.6 should appear as follows:

$$\begin{array}{rcl} {\sum \limits_{i}} \frac{{\partial}^{2}p} {\partial {x}_{i}^{2}}& =&- {\sum\limits_{i}}\frac{{\partial }^{2}(\rho {u}_{i})} {\partial t\partial {x}_{i}} -{\sum \limits_{i}}{ \sum \limits_{j}}\frac{\partial (\rho{u}_{i}{u}_{j})} {\partial {x}_{i}\partial {x}_{j}} -{\sum\limits_{i}}{ \sum \limits_{j}} \frac{{\partial }^{2}{\tau}_{ij}^{{_\ast}}} {\partial {x}_{i}\partial {x}_{j}} +{ \sum\limits_{i}}\frac{\partial {F}_{i}} {\partial{x}_{i}}.\end{array}$$
(3.6)

Equation 3.10 should be replaced by the following text: Strictly speaking, the second term in the above equation should have \(T^*_{ij}\), but here it has been replaced by \(T_{ij}\), because Lighthill considers quadruple effect only through the convective term and through the fluctuation of stress terms.