# Interface conditions for heat transfer

I have two different scenarios for heat transfer I need some help with understanding.

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?

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For two solids, continuity exists both in temperature and heat flux. In one dimension,
k1dt/dn = k2dt/dn at n=interface. The temperature condition you have already noted.

For fluid flowing over a solid, the situation is linked by using a film coefficient such that the heat flux at the solid equals a film coefficient multiplied by the difference in bulk fluid temperature and the surface temperature of the object. It is generally considered a boundary condition (commonly called the mixed condition). That is how it is mathematically handled.

There are formulas that provide the film coefficient based on whether the flow is laminar or turbulent based on fluid properties such as density, viscosity, specific heat, and fluid conductivity. Velocity is also important.

If, per chance, you are trying to solve the Fourier equation for transient heat transfer in one dimension by separation of variables and your domain in space consists of two different solids with differing thermal properties, you have to use what is called discontinuous weighting functions to maintain orthogonality of eigenfunctions.

Thank you for the reply! It was along the lines I had expected. Is it Newton's law of cooling you refer to in the fluid/solid problem? I was considering a situation when the solid is not surrounded by a fluid domain big enough such that the heat transfer can be modeled by a boundary condition.

Thank you for the reply! It was along the lines I had expected. Is it Newton's law of cooling you refer to in the fluid/solid problem? I was considering a situation when the solid is not surrounded by a fluid domain big enough such that the heat transfer can be modeled by a boundary condition.
Yes, the mixed bc is Newton's law of cooling.

"I was considering a situation when the solid is not surrounded by a fluid domain big enough such that the heat transfer can be modeled by a boundary condition."

Can you be more specific? I'm not sure what you are alluding to here.