Equations of Motion of an Elastic Body with Cavities Partially Filled with an Ideal Fluid

Abstract

Chapter 1 considered the problem of finding the pressure p (x, y, z, t) in regions occupied by the fluid masses following the specified elastic displacements $\overrightarrow{u}$ (x, y, z, t). Here, let us consider the problem of finding these displacements for the given pressure distribution p (x, y, z, t)in regions occupied by the fluid masses. According to D’Alembert’s principle, equations defining the vectorial function $\overrightarrow{u}$(x, y, z, t) can be set up as equilibrium equations for an elastic body by supplementing the volume forces acting on the body by the volume inertia forces. The volume force vector $\overrightarrow{w}$(x,y, z, t) in this case will have the form

$$\mathop \omega \limits^ \to= \mathop \omega \limits^ \to+ \frac{{\mathop {d\omega }\limits^ \to}}{{dt}}x\mathop r\limits^ \to+ \frac{{{\partial ^2}u}}{{\partial {t^2}}}$$

(2.1.1)

where $\overrightarrow{w}$ (x,y, z, t) is the acceleration of a particle of the elastic body with coordinates x, y, z at time t. It is assumed that the moving coordinate system x,y, z rotates slowly. Disregarding the Coriolis acceleration and the acceleration of transport $\overrightarrow{w}\times \left( \overrightarrow{w}\times \overrightarrow{r} \right)$, we will set in Eq. (2.1.1)

$$\overrightarrow{w}={{\overrightarrow{w}}_{0}}+\frac{d\overrightarrow{w}}{dt}\times \overrightarrow{r}+\frac{{{\partial }^{2}}\overrightarrow{u}}{\partial {{t}^{2}}}$$

(2.1.2)

in the same manner as in deriving the force and moment equations in Chap. 1. Equation (2.1.1) in this case will take the form

$$\overrightarrow{Q}=-\rho \left( {{\overrightarrow{w}}_{0}}-\overrightarrow{g}+\frac{d\overrightarrow{w}}{dt}\times \overrightarrow{r}+\frac{{{\partial }^{2}}\overrightarrow{u}}{\partial {{t}^{2}}} \right)$$

(2.1.3)