A homogeneous relaxation low mach number model

In the present paper, we investigate a new homogeneous relaxation model describing behaviour of two-phase fluid flow in low Mach number regime, which can be obtained as approximation well-known HRM. For this specific model, derive an equation state to describe thermodynamics fluid. We prove some theoretical properties satisfied by solutions and provide well-balanced scheme. To go further, instantaneous formal convergence towards HEM. An asymptotic-preserving scheme is introduced allow numerical simulations coupling between spatial regions with different characteristic times.