Although the experimental capability to design and fabricate specific microstructures in materials has advanced significantly in the past few decades, a description of their effective thermal behavior on the macroscale has not kept pace. The length scales of the fundamental processes (approximately 100 nm–10 micron, or microscale) are typically several orders of magnitude smaller than the scales dictated by a given application (1 mm–1 cm, or macroscale). We are interested in extending classical asymptotic approaches to allow for the spatial pattern wavenumber to vary on the macroscale variables and to find how changes in microstructure geometry affect macroscopic properties and transport. To this end, we consider here the thermal transport of a binary fluid (coolant) through nonuniformly spaced laminates, as a simple model for heat sinks in electronics. Power is continuously being generated by the laminates, and the local rates of heat transport depend on the composition of the binary mixture. However, segregation of the mixture can occur under favorable temperature gradients, which could potentially affect the local heat transfer from the laminate to the coolant. We derive sets of effective equations that describe the evolution of the temperature field, the local coolant composition, the fluid velocity and the pressure over the macroscale that are driven by the net effects of the microscale processes. The evolution of these quantities is driven by the local heat generation within the laminates, lamellar geometry gradients, the prescribed flow rate of coolant entering the system, along with the fundamental material properties of the laminates and the coolants. Microscale values of all of these quantities are known in terms of the solutions to these effective eqautions. We are especially interested in geometries in the laminate spacing which allow for better thermal transport by the coolant for a prescribed power distribution.

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