# Sonoluninscence

Discussion in 'Physics & Math' started by curvature, Aug 12, 2018.

1. ### curvatureRegistered Member

Messages:
41
1). It's been suggested the Reyleigh-Plesset equation does not describe appropriate heat

2). That heat I suggest may come from cyclotron radiation from the spin of the cavity which has been shown to exceed the speed of light inside of the cavity.

3). I attempt to put corrective factors in including any forces from Van der Waals.

$\frac{1}{R^2} (\frac{\Delta P + (\frac{\mathbf{R}T}{(V_m - b )} - \frac{a}{V^2})}{\rho})[1 - (\frac{T_0}{T})^3]$

$= \frac{1}{R^2} (\frac{\Delta P + P_{waals})}{\rho})[1 - (\frac{T_0}{T})^3]$

$= \frac{\ddot{R}}{R} + \frac{3}{2}(\frac{\dot{R}}{R})^2 + \frac{4 \nu \dot{R}}{V} + \frac{2S}{\rho V} + \frac{e^2}{6 \pi mV} \frac{\dddot{R}}{R} + \frac{1}{V}(\frac{e}{m})\frac{\partial \dot{U}}{\partial R}$

Distribution of the density and simplifying some terms I get

$\frac{\Delta P + \mathbf{P}}{R^2}[1 - (\frac{T_0}{T})^3]$

$= \frac{\ddot{R}}{R}\rho + \frac{3\rho}{2}(\frac{\dot{R}}{R})^2 + \frac{4 m\nu }{V^2}\dot{R} + \frac{2S}{V} + \frac{e^2}{6 \pi V^2} \frac{\dddot{R}}{R} + \frac{\rho}{V}(\frac{e}{m})\frac{\partial \dot{U}}{\partial R}$

$\Delta P = P - (P_0 - P(t))$

Integrating the volume element we obtain the simplified version of our equations

$\int\ \frac{\Delta P + \mathbf{P}}{R^2}[1 - (\frac{T_0}{T})^3]\ dV$

$= m\frac{\ddot{R}}{R} + \frac{3}{2}m(\frac{\dot{R}}{R})^2 + d\log_V(4 \rho \nu \dot{R} + 2S + \frac{e^2}{6 \pi V}\frac{\dddot{R}}{R} + \rho_q \frac{\partial \dot{U}}{\partial R})$

or

$\frac{F}{\Delta R} = \frac{E}{\Delta A} \equiv \int\ \frac{\Delta P + (\frac{\mathbf{R}T}{(V_m - b )} - \frac{a}{V^2})}{R^2}[1 - (\frac{T_0}{T})^3]\ dV$

$= \int\ [\Delta P + (\frac{\mathbf{R}T}{(V_m - b )} - \frac{a}{V^2})]\gamma\ dR$

$\gamma = [1 - (\frac{T_0}{T})^3]$

[*] SEE NOTES

With $\rho_q$ being a charge density and $c=1$.The dimensions of this equation is force over length or energy over area. It has an ‘’acoustic energy’’ part given by $m\frac{\ddot{R}}{R}$ and a wall velocity term $\frac{3}{2}m(\frac{\dot{R}}{R})^2$. This part $\frac{\Delta P + \mathbf{P}}{R^2}$ can be seen in terms of an ''acoustic intensity'' term. It’s also been known for the surface tension $S$ to have a coefficient of $(1 - \frac{T}{T_C})$ where $T_C$ is the critical temperature (known as the Guggenheim–Katayama formula). As temperature increases the surface tension decreases.

[1] - an alternate simplification from a previous Langrangian of the theory we formalised, requires only the additional binding or repulsive energies from Van der Waals forces

$\mathcal{L} = mR \ddot{R} + \frac{3}{2}m\dot{R}^2 + \frac{4 \nu_L m}{R} \dot{R} + \frac{2\gamma m}{\rho_L R} + \frac{e^2}{6 \pi c^3} \dddot{R} + \frac{1}{2}eV + \frac{\Delta P(t)m + \Delta\mathbf{P}}{\rho_L}$

$= mR \ddot{R} + \frac{3}{2}m\dot{R}^2 + \frac{4 \nu_L m}{R} \dot{R} + \frac{2\gamma m}{\rho_L R} + \frac{e^2}{6 \pi c^3} \dddot{R} + \frac{1}{2}eV + \frac{\Delta P(t)m + (\frac{\mathbf{R}T}{(V_m - b )} - \frac{a}{V^2})}{\rho_L}$

(which is the Langrangian)

The repulsive nature of Van der Waals could temporally explain the expanding of the bubble but it seems more likely related to pressures and temperature.

[2] - Further, there is a part of this equation

$\int\ \frac{\Delta P + \mathbf{P}}{R^2}[1 - (\frac{T_0}{T})^3]\ dV = m\frac{\ddot{R}}{R} + \frac{3}{2}m(\frac{\dot{R}}{R})^2 + 4 \rho \nu \dot{R} + 2S + \frac{e^2}{6 \pi V}\frac{\dddot{R}}{R} + \rho_q \frac{\partial \dot{U}}{\partial R}$

Namely this expression $\frac{e^2}{6 \pi V}$ can be fashioned in a different way:

$\frac{e^2}{2 \epsilon_0} \frac{e^2}{4\epsilon \hbar_0 c}\frac{1}{ \pi R^3}$

This is not too far from the difference of such a potential which actually gives rise to the Lamb shift, a direct consequence itself of the vacuum energy, ie. Casimir effect, par the powers of the fine structure

$<\Delta V>\ = \frac{e^2}{4 \pi \epsilon_0} \frac{e^2}{4 \pi \epsilon_0 \hbar c}(\frac{\hbar}{mc})^2\frac{1}{\pi R^3} \ln \frac{4 \epsilon_0 \hbar}{e^2}$

Notice, we have encountered this kind of notation before in investigating Anandan's difference of geometries which was part of the topic of my paper to the gravitational research foundation.