# Cavitation: radius of a bubble for compressible flow

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• JD_PM
In summary, the conversation discusses the governing equations for liquid-vapor mass transfer, specifically the vapor transport equation and the continuity equation. The cavitation model aims to determine the mass transfer rate using the non-conservative form of the continuity equation. The discussion also touches on the expression for the radius of a cavitation bubble, both for incompressible and compressible flow. The question posed is whether the same expression holds for compressible flow.
JD_PM
The liquid-vapor mass transfer (evaporation and condensation) is governed by the vapor transport equation:

$$\frac{\partial}{\partial t} (\alpha_l \rho) + \nabla \cdot (\alpha_l \rho \vec v) = \dot m^{+} + \dot m^{-}$$

In the incompressible flow case (constant density), it reduces to

$$\frac{\partial}{\partial t} (\alpha_l) + \nabla \cdot (\alpha_l \vec v) = \frac{\dot m^{+} + \dot m^{-}}{\rho}$$

The cavitation model of https://www.researchgate.net/profile/Guenter-Schnerr-Professor-Dr-Inghabil/publication/296196752_Physical_and_Numerical_Modeling_of_Unsteady_Cavitation_Dynamics/links/56f6b62308ae81582bf2f940/Physical-and-Numerical-Modeling-of-Unsteady-Cavitation-Dynamics.pdf aims to determine the mass transfer rate linked to the RHS term of transport equation, using the non conservative form of the continuity equation i.e.

\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho v) = 0 \Rightarrow

\Rightarrow \nabla \cdot v = -\frac{1}{\rho}\Big( \frac{\partial \rho}{\partial t} + v \cdot \nabla \rho\Big) = -\frac{1}{\rho} \frac{D \rho}{D t} = \frac{\rho_l - \rho_v}{\rho} \frac{D \alpha_v}{D t}

Where we used ##\rho = (1 - \alpha_v ) \rho_l + \alpha_v \rho_v##

However, for compressible compressible flow

\nabla \cdot v = \frac{\rho_l - \rho_v}{\rho} \frac{D \alpha_v}{D t} + \alpha_l \frac{D \rho_v}{D t} + (1 - \alpha_l) \frac{D \rho_l}{D t}

Where the vapor volume fraction is

\alpha_v = \frac{V_v}{V_l + V_v} = \frac{\frac 4 3 n_0 \pi R^3}{1 + \frac 4 3 n_0 \pi R^3}

The bubble growing and collapsing can be described by the Rayleigh Plesset equation (please see first link). Dropping the second order terms and the viscosity effects between phases in the Plesset equation, the evolution of the radius of the bubble has the expression:

$$\frac{DR}{D t} = \Big( \frac 2 3 \frac{p_b - p}{\rho_l}\Big)^{1/2}$$

It follows that the radius of a bubble is given by

$$R = \Big( \frac{3\alpha_v}{(1-\alpha_v) n_0 4 \pi} \Big)^{1/3}$$

The above radius is obtained assuming incompressible flow.

Does the same expression hold for compressible flow?

There's a couple of typos in the above that are breaking ## \LaTeX ## but in any case you might get more replies if you ask the question like this:

We can derive* the following equation for the radius of a cavitation bubble for flow of an incompressible liquid:

$$R = \Big( \frac{3\alpha_v}{(1-\alpha_v) n_0 4 \pi} \Big)^{1/3}$$

Question: does the same expression hold for compressible flow?

* Derivation:
The liquid-vapor mass transfer (evaporation and condensation) is governed by the vapor transport equation:

$$\frac{\partial}{\partial t} (\alpha_l \rho) + \nabla \cdot (\alpha_l \rho \vec v) = \dot m^{+} + \dot m^{-}$$

In the incompressible flow case (constant density), it reduces to

$$\frac{\partial}{\partial t} (\alpha_l) + \nabla \cdot (\alpha_l \vec v) = \frac{\dot m^{+} + \dot m^{-}}{\rho}$$

The cavitation model of https://www.researchgate.net/profile/Guenter-Schnerr-Professor-Dr-Inghabil/publication/296196752_Physical_and_Numerical_Modeling_of_Unsteady_Cavitation_Dynamics/links/56f6b62308ae81582bf2f940/Physical-and-Numerical-Modeling-of-Unsteady-Cavitation-Dynamics.pdf aims to determine the mass transfer rate linked to the RHS term of transport equation, using the non conservative form of the continuity equation i.e.

\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho v) = 0 \Rightarrow

\Rightarrow \nabla \cdot v = -\frac{1}{\rho}\Big( \frac{\partial \rho}{\partial t} + v \cdot \nabla \rho\Big) = -\frac{1}{\rho} \frac{D \rho}{D t} = \frac{\rho_l - \rho_v}{\rho} \frac{D \alpha_v}{D t}

Where we used ##\rho = (1 - \alpha_v ) \rho_l + \alpha_v \rho_v##

However, for compressible compressible flow

\nabla \cdot v = \frac{\rho_l - \rho_v}{\rho} \frac{D \alpha_v}{D t} + \alpha_l \frac{D \rho_v}{D t} + (1 - \alpha_l) \frac{D \rho_l}{D t}

Where the vapor volume fraction is

\alpha_v = \frac{V_v}{V_l + V_v} = \frac{\frac 4 3 n_0 \pi R^3}{1 + \frac 4 3 n_0 \pi R^3}

The bubble growing and collapsing can be described by the Rayleigh Plesset equation (please see first link). Dropping the second order terms and the viscosity effects between phases in the Plesset equation, the evolution of the radius of the bubble has the expression:

$$\frac{DR}{D t} = \Big( \frac 2 3 \frac{p_b - p}{\rho_l}\Big)^{1/2}$$

It follows that the radius of a bubble is given by

$$R = \Big( \frac{3\alpha_v}{(1-\alpha_v) n_0 4 \pi} \Big)^{1/3}$$

JD_PM

## 1. What is cavitation and how does it occur?

Cavitation is the formation of vapor bubbles in a liquid due to a decrease in pressure. It occurs when the pressure of a liquid drops below its vapor pressure, causing the liquid to vaporize and form bubbles.

## 2. How is the radius of a bubble in cavitation determined for compressible flow?

The radius of a bubble in cavitation for compressible flow is determined by the Rayleigh-Plesset equation, which takes into account factors such as the initial radius, liquid properties, and flow conditions.

## 3. What is the significance of the radius of a bubble in cavitation for compressible flow?

The radius of a bubble in cavitation is significant because it affects the growth and collapse of the bubble, which can have consequences such as erosion and damage to equipment in industrial applications.

## 4. How does the compressibility of a liquid affect cavitation and the radius of a bubble?

The compressibility of a liquid plays a significant role in cavitation as it affects the pressure and density changes within the liquid. This, in turn, affects the size and behavior of the bubbles formed during cavitation.

## 5. Can the radius of a bubble in cavitation be controlled?

Yes, the radius of a bubble in cavitation can be controlled through various methods such as altering the flow conditions, using additives to change the liquid properties, or designing equipment with features to prevent or mitigate cavitation. However, it is a complex phenomenon and complete control may not always be possible.

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