# 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 However, for cryogenic fluids (such as liquid nitrogen) this assumption is no longer valid.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.