In recent years, the interior ocean of an icy moon has been identified as one of the most promising candidates as a location for extraterrestrial life. Some icy moons show evidence of the existence of internal oceans, such as the plumes of water vapor on Europa and Enceladus. Because liquid water is essential to the origin of life, it is important to understand the development of the inner ocean, especially the temperature distribution and evolution within these moons. To achieve this, we aim to simulate tidal heating an internal ocean of an icy moon by 3-dimensional numerical fluid calculations using Smoothed Particle Hydrodynamics (SPH) method, a particle-based method for simulating fluid dynamics that is commonly used for problems involving fracture or large deformation. We added viscosity and thermal conduction terms into the governing equations of SPH. However, it was found that standard SPH cannot solve rigid rotating systems such as satellites when viscosity is taken into account. (1) The viscosity in the SPH method acts as an unphysical force that stops the rotation of the system with shear. (2) Standard SPH has the problem of not being able to calculate properly on contact discontinuities and free surfaces, and therefore cannot correctly solve the energy distribution near the satellite's surface.
To address these challenges, we modified the formulation of viscosity and introduced Density Independent SPH (DISPH) to improve the behavior of discontinuous surfaces. In addition, radiative cooling was introduced by using an algorithm that defines the surface of the fluid using the particle method. We also introduced an equation of state that takes into account phase transitions. Through these modifications, we have developed an SPH method that incorporates all the necessary physical processes for accurately simulating the evolution of icy moons with internal oceans.