Non-isothermal flows

Heat Transfer Models

Since the fluid phase and the grains are tracked separately, so is their temperature. In the fluid, it obeys a standard advection–diffusion balance,

while each grain, tracked individually, carries its own temperature through

What differs from one configuration to the next is how heat crosses an interface — and the framework distinguishes two fundamentally different situations.

• Fluid-fluid and fluid-ambient interfaces are resolved. When the interface itself is tracked by the mesh — a free surface losing heat to the atmosphere, or the boundary of a droplet — heat transfer is imposed directly on the energy equation as a boundary condition, either a prescribed temperature or a prescribed flux:

  $$ T = \hat{T} \qquad \text{or} \qquad \mathbf{q}\cdot\mathbf{n} = h_w\big(\hat{T}_w - T\big). $$

• Solid-fluid interfaces are unresolved. The boundary layer around each grain is thinner than the fluid mesh and cannot be captured directly, so the exchange has to be reconstructed through a convective correlation instead of computed directly. What that correlation actually looks like depends heavily on the underlying physics and on what the "grain" represents — a sand particle in a dense packed bed and a liquid droplet do not exchange heat with their surroundings the same way, so the correlation is chosen case by case, typically built from a Reynolds–Colburn-type analogy with the drag law rather than reused off the shelf. This convective exchange is currently the group's only mechanism at this interface; extending it to other pathways, such as conduction through the granular contact network (the q_cond term above), is ongoing work.

Thermo-rheological Coupling

In non-isothermal flows, the mechanical response of the material is generally inseparable from its thermal history. This shows up directly in the momentum balance, where the viscosity — and, where buoyancy matters, the density — entering the stress tensor are no longer constants but fields that follow the local temperature:

The specific form of $\eta(T)$ is chosen to match the material at hand. Cooling lava, for instance, is captured by a Vogel–Fulcher–Tammann law, whose steep growth as $T$ drops is what turns a cooled surface layer into a crust orders of magnitude stiffer than the fluid core:

This momentum balance is solved together with the energy equation at every time step, so that thermal and mechanical evolution feed back into one another self-consistently rather than being treated as separate, sequential processes.

Modelling of Phase Change

Many non-isothermal flows involve a change of state, whether vaporisation, solidification, or crystallisation, which introduces an additional layer of coupling between the thermal and mechanical fields. Depending on the application, this can take the form of a heat flux entirely consumed by latent heat at a moving interface, or of a progressive rigidification of the material as cohesive bonds form between solidifying particles, potentially followed by fracture once local stresses exceed a given threshold. So far, each mechanism has been implemented for the specific case study that needed it — vaporisation for the digging droplet, a numerical viscosity cap standing in for solidification on the lava crust. Turning phase change into a first-class, reusable part of the model, rather than a bespoke treatment rebuilt for every new application, is active, ongoing work within this axis, and a prerequisite for reproducing emergent behaviours such as localised fluidisation, chimney formation, or crust fracturing directly from the underlying physics.