Macroscopic equations for microscopic transport processes

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Typically, engineering materials consist of about 10^23 atoms or molecules, and the behavior of a material results from the interplay of all its particles.  Despite this huge number of individual particles, a wide variety of flow and heat transfer problems can be described by a rather low number of partial differential equations, e.g. the well-known equations of Navier-Stokes and Fourier for fluid and gas flows, or Fick's law of diffusion. Due to the intense interaction between the particles (e.g. collisions of gas molecules), disturbances are effectively distributed between the particles, so that in many applications the particles behave not as individuals, but as a continuum.      Nevertheless, the response of the continuum results from processes on the microscopic scale, and this gives an important reason to study the relation between the microscopic and the macroscopic behavior.  Macroscopic models for advanced materials--e.g. PEM membranes for fuel cells--require many transport coefficients, and understanding of the microscopic behavior can inform the modelling of the latter.      Under more extreme conditions--e.g. rarefied gas flows, microflows, non-equilibrium evaporation and condensation and other interface processes--the standard engineering laws break down, and must be replaced by advanced models, which account for more details of the processes on the atomic scale.      Modern engineering design relies on numerical simulations of devices, and the processes therein. Good and accurate mathematical models are indispensable for this task. The proposed research considers the relation between the micro- and the macroworld in order to derive, and validate, mathematical transport models for rarefied gases, polymer membranes for use in fuel cells, and evaporation and condensation processes. The resulting models will describe the processes on a macroscopic level, i.e. as a continuum, but incorporate knowledge from the microscopic material behavior. Thus, they will go beyond the standard engineering laws and open the door for advanced simulations of these complicated and important processes.