Thermo-viscous instability of flow in a weakly heat-conducting channel

TL;DR

This work models thermo-viscous fingering in a narrow, heat-conducting channel by deriving a depth-averaged gap-averaged Darcy-type model that includes Taylor dispersion and wall cooling in the small $Bi$ regime. Linear stability analysis reveals a dispersion relation $\gamma(k; Pe, \Gamma, \beta)$ with a single growth maximum $\gamma_{\max}$ at $k_{\max}$, and the full nonlinear simulations confirm the early-time predictions: fingering emerges from inlet perturbations and selects a characteristic wavelength tied to the base-state heat entry length. In the high-$Pe$ and small-$\beta$ limit, both $\gamma_{\max}$ and $k_{\max}$ scale proportionally to $\Gamma$, with logarithmic dependences on the viscosity contrast, and a critical viscosity ratio $\beta_c$ above which the flow remains stable. The findings offer a framework for understanding thermo-viscous channelization in geophysical contexts (magma through fissures) and in lubrication films, where temperature-dependent mobility governs pattern formation and flow localization.

Abstract

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 Péclet 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 Péclet numbers and viscosity contrasts. From linear stability analysis we find the dispersion relation and quantify how the fastest growth rate $γ_{\max}$ and corresponding wavenumber $k_{\max}$ depend on the global parameters. We further derive analytical expressions for $γ_{\max}$ and $k_{\max}$ in the limit of high Péclet 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.

Figures (13)

• Figure 1: Schematic view of the model setup, with (a) a full three-dimensional view and (b) top view of the gap-averaged channel considered in the present work. Hot, less viscous fluid, shown in red, is injected into a Hele-Shaw cell filled with colder, more viscous fluid, shown in blue.
• Figure 2: Schematic view of the short-time regime (left) and long-time regime (right), showing temperature profiles $T(z)$ and velocity profiles $\mathbf{u}(z)$ across the plate of thickness $L_{\rm p}$ and the gap of half width $h$, as well as the thermal penetration depth $\delta_{\mathrm p}$ in the plate.
• Figure 3: Instability emerging from a sinusoidal perturbation. (a) Temperature field $T(x,y)$ for $t = 1.2\cdot 10^6 \gg t_{\rm pert}$, where different black contours represent different isothermal curves at $T=0.1,0.2,\dots,0.9$. (b) Streamlines of the velocity field $\boldsymbol{u}$, where the local color represents the velocity magnitude $|\boldsymbol{u}|(x,y)$. For this simulation we set $\mathrm{Pe} = 10^3$, $\Gamma = 10^{-5}$, $\beta = 10^{-3}$ and $k = 2\pi/(1.4\cdot10^{5})$. The perturbation amplitude and time are $\epsilon = 10^{-3}$ and $t_{\mathrm{pert}}=10^3$.
• Figure 4: Temporal evolution of the spans in (a) temperature, $T^{\rm span}(x,t)$ (see \ref{['eq:Tspan']}), and (b) streamwise velocity, $u_x^{\rm span}(x,t)$ (see \ref{['eq:uxspan']}), for a sinusoidal inlet velocity perturbation. The global parameters and the perturbation wavelength, amplitude and time are the same of Figure \ref{['fig:Tlevel_and_streamlines_sin']}. Different positions $x$ are represented using a color gradient ranging from light blue ($x=4\cdot10^4$) to dark blue ($x=4\cdot10^5$), with values equally spaced by $4\cdot10^4$. The dashed line indicates the slope of the curves obtained by a best fit procedure in the corresponding time interval. For this simulation, we measured $\gamma \approx 1.61\cdot10^{-5}$.
• Figure 5: Instability emerging from a random perturbation. The temperature field $T(x,y, t)$ is shown at two time instances (a) $t = 1.375\cdot 10^6$ and (b) $t = 1.75\cdot 10^6$. The black contours represent different isothermal curves at $T=0.1,0.2,\dots,0.9$. (b). For this simulation we set $\mathrm{Pe} = 10^3$, $\Gamma = 10^{-5}$, and $\beta = 10^{-3}$. The perturbation amplitude and time are $\epsilon = 10^{-3}$ and $t_{\mathrm{pert}}=10^3$.