Heat Conduction in a 1-D Rod with an Oscillating Boundary

Classical studies explore the conduction of heat in uniform materials or boundaries of fixed boundaries - where equilibration proceeds monotonically toward a static steady state [9]. This paper explores the effect of oscillating boundaries on 1-dimensional rods for achieving equilibration during thermal conduction. Specifically, the rod’s length oscillates sinusoidally with time which fundamentally alters the diffusion process. By applying a coordinate transformation from the moving domain to a fixed reference domain, this paper derives a modified partial differential equation that includes an additional advective term reflecting the stretching and compression of the rod. Because the resulting problem is analytically intractable for general oscillations, this paper implements a finite difference method of lines combined with implicit time integration to compute numerical solutions. The simulations demonstrate that oscillatory boundary motion prevents convergence to a static equilibrium: instead, the system approaches a periodic steady state where the temperature distribution fluctuates in synchrony with the boundary oscillations. The amplitude and frequency of the oscillations determine the degree of modulation, with larger or faster oscillations producing more pronounced departures from the classical static case. Thus, our results show that introducing oscillatory geometry transforms the nature of equilibration in diffusion problems, replacing the standard monotonic approach to equilibrium with a dynamically sustained periodic state. These insights are relevant for microscale oscillating devices and thermally stressed structures where dynamic boundary conditions arise.