Controlling thermal waves with transformation complex thermotics

•Convection can be regarded as a complex form of conduction.•The theory of transformation complex thermotics is established.•Three devices are designed to cloak, concentrate, and rotate thermal waves.•The effective complex thermal conductivity of porous medium is derived. Conduction and convection are two fundamental methods of heat transfer, which are generally considered independent. We manage to unify them by coining a complex thermal conductivity whose real and imaginary parts are related to conduction and convection, respectively. Accordingly, the conduction-convection process with thermal waves is dominated by the complex conduction equation, thus called complex thermotics herein. To go further, we establish the theory of transformation complex thermotics by proving the form-invariance of the complex conduction equation and deriving the transformation principle of complex thermal conductivities. As model applications, we design three devices with functions of cloaking, concentrating, and rotating thermal waves. Moreover, experimental demonstration is suggested with a porous medium whose effective complex thermal conductivity is calculated with the method of weighted average. These findings could broaden the fundamental knowledge of conduction and convection, and have potential applications in designing thermal metamaterials.