Effective Boundary Conditions Of Robin Problem On A Body Coated By Functionally Graded Material

To protect a heat conductive body from overheating,engineers usually coat the body with a thin layer of thermal insulator.For instance,the space shuttle is usually painted by insulator of nano-materials.To avoid thermal expansion concentrates on the interface of coating and interior body,the coating usually consists of two layers of pure ceramic layer and mixed layer,the mixed layer could be made of functionally graded materials(FGM),which makes the thermal tensor be continuous in the composite media.According to Newton's law of cooling,this problem can be modeled by a heat equation with Robin boundary condition.In order to get the best insulation for the heat conductive body mathematically,we study the problem in the following two directions.1.We consider the effective boundary conditions(EBC)of the heat equation with Robin boundary conditions,that is the asymptotic behaviors of solutions,as the thickness of the coating shrinks.In this part,we construct various special test functions,to describe the limiting equations and boundary conditions.And we find that the EBC depends on the scaling relationship among the thermal conductivity of coating,thermal transport coefficient and the thickness of the coating.Especially,the ones guarantee the EBC being Neumann type are the best,which require that,either thermal transport coefficient tends to 0,or the ratio of thermal conductivity and thickness tends to 0.Finally,by giving an example of linear FGM,we find that the FGM are better than the traditional materials.2.We consider the asymptotic behaviors of Robin principal eigenvalues.According to the method of characteristic expansion,it is easy to see that,to make sure the thermal insulation coating best performance,the principal eigenvalue should be small,and the principal eigenvalue function on the heat conductor should be large.In this part,on the base of variational principle and constructing special test functions,we describe the limiting behaviors of the principal eigenvalue and its eigenvalue function.We find that when the thickness of the coating tends to zero,either thermal transport coefficient tends to 0,or the ratio of thermal conductivity and thickness tends to 0,the heat insulation performance of coating is the best.This conclusion is the same to that of part 1.