On the conjugate interface conditions and Galilean invariance

In the referred paper("H. Karani, C. Huber, Physical Review E, 91(2)(2015) 023304"), a total heat flux continuity condition for conjugate heat transfer problems with moving interfaces was proposed. The authors asserted both conductive and advective heat fluxes are conserved simultaneously in their formulation. This condition had been cited by many subsequent studies. However, it is found that the total heat flux continuity condition violates Galilean invariance. The original diffusion heat flux continuity condition is reasonable for both stationary and moving interfaces.