A high-precision simulation algorithm for gas-liquid two-phase flows on unstructured meshes has been developed to simulate gas entrainment phenomenon in a sodium-cooled fast reactor. In this study, it became clear that unphysical behaviors near gas-liquid interfaces were caused by conventional algorithms. Then, physics-basis considerations were conducted for mechanical balances at gas-liquid interfaces to derive appropriate formulations. By defining momentum and velocity independently and developing the momentum transport equations for both gas and liquid phases, the physically appropriate formulation of momentum transport was derived, which eliminated the unphysical pressure distribution caused by the conventional formulation. In addition, the physically appropriate formulation was derived for the pressure gradient to satisfy the mechanical balances between pressure and surface tension at gas-liquid interfaces. As the validation test, the rising gas bubble in liquid was simulated by the developed simulation algorithm with the physically appropriate formulations, and the simulated terminal bubble shapes on the structured and highly-distorted unstructured meshes coincided with the experimental data under each simulation condition determined by the Morton and Eötvös numbers.