Thermo-viscous instability of flow in a weakly heat-conducting channel
An instability may arise when a hot viscous fluid enters a thin gap and cools through heat transfer to a colder surrounding environment. Fluids whose viscosity increases strongly upon cooling create a positive feedback in which warmer regions flow faster and cool more slowly, leading to the formation of thermo-viscous "fingers". Here we investigate this mechanism in the long time, small Biot number regime, where cooling through the plates is weak but acts over sufficiently long times that the temperature becomes nearly uniform across the gap heat. This asymptotic limit enables a depth-averaged description that incorporates both thermal diffusion and hydrodynamic (Taylor) dispersion, allowing us to analyze the dependence of the instability on the P\'eclet number, viscosity contrast, and wall cooling rate. Using numerical simulations of temperature-dependent viscous flow in a Hele-Shaw geometry, we show that fingering instabilities emerge in response to small inlet perturbations within a range of P\'eclet numbers and viscosity contrasts. From linear stability analysis we find the dispersion relation and quantify how the fastest growth rate $\gamma_{\max}$ and corresponding wavenumber $k_{\max}$ depend on the global parameters. We further derive analytical expressions for $\gamma_{\max}$ and $k_{\max}$ in the limit of high P\'eclet number and large viscosity contrast, revealing the scaling behavior that controls pattern selection. These results clarify the physical mechanisms driving thermo-viscous fingering in the small Biot number regime and have implications for systems in which temperature-dependent viscous fluids are confined within narrow gaps, such as lubrication flows in mechanical components and magma invasion in small scale fissures.