Magnetosensitive elastomers respond to external magnetic fields by changing their stiffness and shape. These effects result from interactions among magnetized inclusions that are embedded within an elastic matrix. Strong external magnetic fields induce internal restructuring, for example the formation of chain-like aggregates. However, such reconfigurations affect not only the overall mechanical properties of the elastomers but also the transport through such systems. We concentrate here on the transport of heat, that is thermal conductivity. For flat, thin model systems representing thin films or membranes and modeled by bead-spring discretizations, we evaluate the internal restructuring in response to magnetization of the particles. For each resulting configuration, we evaluate the associated thermal conductivity. We analyze the changes in heat transport as a function of the strength of magnetization, particle number, density of magnetizable particles (at fixed overall particle number), and aspect ratio of the system. We observe that varying any one of these parameters can induce pronounced changes in the bulk thermal conductivity. Our results motivate future experimental and theoretical studies of systems with magnetically tunable thermal but also electric conductivity---both of which have only rarely been addressed so far.

Thin fluid or elastic films and membranes are found in nature and technology, for instance, as confinements of living cells or in loudspeakers. When applying a net force, resulting flows in an unbounded two-dimensional incompressible low-Reynolds-number fluid or displacements in a twodimensional linearly elastic solid seem to diverge logarithmically with the distance from the force center, which has led to some debate. Recently, we have demonstrated that such divergences cancel when the total (net) force vanishes. Here, we illustrate that, if a net force is present, the boundaries play a prominent role. Already a single no-slip boundary regulates the flow and displacement fields and leads to their decay to leading order inversely in distance from a force center and the boundary. In other words, it is the boundary that stabilizes the system in this situation, unlike the threedimensional case, where an unbounded medium by itself is able to absorb a net force. We quantify the mobility and displaceability of an inclusion as a function of the distance from the boundary, as well as interactions between different inclusions. In the case of free-slip boundary conditions, a kinked boundary is necessary to achieve stabilization.