Thermal response of a Maxwell fluid in oscillatory boundary layer flow over a wavy wall

The thermal behavior of a viscoelastic fluid, modeled using the linear Maxwell model, is analyzed in oscillatory boundary layer flow over a wavy wall. Based on a previously derived velocity field (Cuevas et al. J. Non-Newton. Fluid Mech. 321 (2023) 105125) that assumes both a small oscillation amplitude and a Stokes layer thickness much smaller than the wall wavelength, the heat transfer equation is solved using a perturbation method. A time-harmonic temperature is prescribed at the wall, and a constant temperature is imposed at the outer edge of the boundary layer. The first order solution corresponds to the thermal analogue of the Stokes' second problem and is independent of the viscoelasticity of the fluid. At second order, convective heat transport gives rise to a temperature field composed of a time-periodic component and a steady distribution, analogous to the steady streaming observed in the corresponding flow problem. The non-vanishing steady temperature at the edge of the inner thermal layer leads to a generalized form of the Rayleigh's law of streaming for the thermal viscoelastic case and, consequently, to the formation of an outer thermal layer whose thickness is estimated.