# Non-isothermal conditions in liquid-gas nitrogen homogeneous mixture

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I am studying the cavitation theory proposed by Schnerr and Sauer (SS) [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] and realized it assumes isothermal conditions within liquid-gas homogeneous mixture as well as incompressible flow. This is alright for, say, water. However, for cryogenic fluids (such as liquid nitrogen) this assumption is no longer valid.

I want to incorporate non-isothermal phenomena in the SS cavitation theory. This is how I am approaching it

To describe the fluid mechanics, we need to work with

1) The continuity equation

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

Mass conservation holds anyway once non-isothermal effects are incorporated.

2) Momentum equation

$$\frac{\partial}{\partial t}(\rho \vec v) + \nabla \cdot (\rho \vec v \vec v) = -\nabla p + \mu \nabla^2 \vec v + \rho \vec g + \vec S$$

Where $\mu$ is the viscosity and $\vec S$ is the surface tension force due to the interface interaction between phases.

Momentum conservation is independent of non-isothermal effects.

Given that the SS model [https://www.afs.enea.it/project/neptunius/docs/fluent/html/th/node343.htm] deals with incompressible, isothermal flow the energy equation is not required. However, we want to incorporate non-isothermal effects. To do so, we need to include it

$$\frac{\partial}{\partial t}(E \rho) + \nabla \cdot (\rho E \vec v) = \rho \dot q - \nabla \cdot (p \vec v ) + \rho (\vec f \cdot \vec v) + \text{viscous terms}$$

Where $\dot q$ is the rate of heat and $\vec f$ refers to external forces.

Given the liquid-gas mixture, the following density and viscosity equations are set to be

$$\rho = \alpha_l \rho_l + (1 - \alpha_l ) \rho_l, \ \ \ \ \mu = \alpha_l \mu_l + (1 - \alpha_l ) \mu_l$$

Where $\alpha_l, \alpha_v$ are the liquid and vapor volume fraction respectively and $\alpha_l = 1$ means there is all liquid and $\alpha_l = 0$ all vapor. It is common practice to use $\alpha_l + \alpha_v = 1$ to eliminate $\alpha_v$.

A new variable, $\alpha_l$, has been introduced. Hence, a new equation must be included. The liquid-vapor mass transfer (evaporation and condensation) is governed by the vapor transport equation:

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

Where $\dot m^{+}, \dot m^{-}$ represent, respectively, the evaporation and condensation mass transfer rates which model the mechanism of cavitation.

Note that for incompressible flow (constant density) the transport equation takes the simple form

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

The above are the fundamental equations. Let's now turn to the SS model

https://ibb.co/GcDXFmY

Where equation 3.5 above is the transport equation.

Questions

1) Does including the energy equation simply incorporate non-isothermal conditions to the LN2 mixture?

2) Regarding incorporating non-isothermal conditions to the SS model: I see that equation 3.7 would take a more complicated final form, given that the density is no longer constant. Is that the only change?

Thank you! :)

Where $E$ is the total energy $E := u + \frac{V^2}{2}$ and $u$ the internal energy.
This is too advanced for me. Have you tried asking on Physics Stack Exchange? It does not look like a homework question so I think it should be acceptable there,
Hi Sammy Gerbil, it's nice to talk to you again. In the meantime I've been reading and now the issue boils down to just thermodynamics. I'll post my next question both here and on PSE.