ORIGINAL Thermal boundary conditions: an asymptotic analysis D. Moulton Æ J. A. Pelesko Received: 2 July 2006 / Accepted: 11 April 2007 / Published online: 25 July 2007 Ó Springer-Verlag 2007 Abstract A generalized thermal boundary condition is derived for a material region, representing all thermal effects of an adjacent thin layer. The boundary condition is obtained by considering the equations of heat conduction in each region and performing an asymptotic expansion of the temperatures about the ratio of thermal conductivities. From the asymptotic theory, the generalized boundary condition arises naturally for the leading order problem. An example is given to demonstrate the utility of the effective boundary condition. 1 Introduction In a wide variety of applications, we encounter two distinct materials in thermal contact. To describe the thermal behavior in such systems, we can write energy equations in each material coupled with conditions at the interface and any additional boundary conditions. Depending on the complexity of the geometries and the boundary conditions, however, such a coupled system can be quite difficult to analyze and impractical to solve mathematically. A natural simplification of such a model is to transform the condi- tions in one of the regions into an effective boundary condition for the remaining material, so that one must only deal with heat conduction in one region. One such situation that would enable this simplification is if one of the two materials is a thin layer. In such a case, it may be possible to neglect the temperature variance in the transverse direction of the thin layer. Another circumstance which allows for a simplification is if there is a significant disparity in the thermal conductance of the two regions. More precisely, if the thermal conductivity is constant in each region with the property that the ratio of thermal con- ductivities presents a small parameter, we may again be able to neglect temperature variance in the region of high con- ductivity. More promising still are situations with both of these traits in combination; that is, if one of the regions is a thin layer of high conductivity relative to the adjacent region. Problems with these characteristics arise consistently throughout engineering and mathematical application, and have been explored fairly extensively in the literature. For instance, in their classic work on heat conduction in solids, Carslaw and Jaeger [2] explore this very situation of a thin layer that is a good heat conductor in contact with a poorer conductor, and the Carslaw condition, or boundary condition of the fourth kind, is derived, which also takes into account the thermal capacity of the thin layer. The Jaegar condition, or boundary condition of the fifth kind, models a similar situation and also accounts for the interaction of the thin layer with the external environment through convection. More recently, Al-Nimr and Alkam [1] model the same sit- uation and arrive at a generalized effective boundary con- dition that incorporates a variety of conditions on the outer boundary, including the Carslaw and Jaegar conditions. In both the classical approach of Carslaw and Jaeger and the more recent approach of Al-Nimr and Alkam, the pri- mary assertion is that since the thermal conductivity in the thin layer is ‘‘high’’ compared to that of the adjacent domain, the temperature in the transverse direction of the thin layer may be taken to be lumped. An energy balance is then performed across the layer, temperatures are equated at the interface, and the effective boundary condition is D. Moulton (&)  J. A. Pelesko Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA e-mail: moulton@math.udel.edu 123 Heat Mass Transfer (2008) 44:795–803 DOI 10.1007/s00231-007-0277-0