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1.) Given two different materials in perfect contact with each other with no thermal resistance at the interface, what are the interface conditions at the interface? Considering two heat equations with variables T1 and T2, the possible conditions would be

I.) T1 = T2 and \kappa_1 T1_x = \kappa_2 T2_x,

II.) T1 = T2 and \alpha_1 T1_x = \alpha_2 T2_x

where \kappa_{1,2} is the thermal conductivity and \alpha_{1,2} is the thermal diffusivity. From a mathematical point of view, both give a well-posed coupling but only one should represent real physics.

2.) Same situation, but with a fluid flowing over a solid?