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Temperature-Dependent Transport

Source: PHY653B Ch. 9

Intuition

Real fluids do not have constant viscosity. Air's \(\nu\) roughly doubles between 0 °C and 100 °C; water's falls by a factor of six. Every derivation in Chapters 7–8 assumed a constant \(\tau\). The last chapter asks the question that turns a solver into a research tool: does any of it survive when \(\nu = \nu(T)\)?

Does the derivation survive?

In LBM, making viscosity temperature-dependent is trivially easy to implement — set

\[\tau(\mathbf{x},t) = \frac{\nu(T(\mathbf{x},t))}{c_s^2} + \frac12\]

and use the local \(\tau\) in the collision. Two lines of code. The question is whether the Chapman–Enskog result still holds.

Largely yes, with a caveat that matters. The expansion is local: it relates \(\nu\) to \(\tau\) at a point, and nothing in it requires \(\tau\) to be uniform. But it does assume \(\tau\) varies slowly compared with the mean free path. Gradients in \(\tau\) introduce additional terms at \(O(\varepsilon^2)\), so a sharp viscosity front is not correctly captured.

The practical rule: \(\tau\) may vary by a large factor across the domain provided it varies smoothly. This is the ordinary situation in convection, and the ease of implementation is a genuine advantage of LBM over pressure-projection Navier–Stokes solvers, where variable viscosity complicates the elliptic solve considerably.

Stability becomes the binding constraint

With constant \(\tau\) you choose one value and check it exceeds \(\tfrac12\). With \(\tau(T)\) you must guarantee it everywhere, at all times, including in transients. The cold corner of a convection cell can drive \(\tau\) toward the floor while the bulk is comfortable, and the run dies in a single cell.

Worse, the failure is often not a graceful blow-up but a local instability that pollutes the solution and then decays, leaving a run that finishes and is wrong. Monitor \(\min\tau\) over the whole domain every step; it is cheap and it is the single most useful diagnostic in this chapter.

How \(\nu(T)\) reshapes convection

The physics is genuinely different, not merely quantitatively shifted. In Rayleigh–Bénard convection with temperature-dependent viscosity:

  • Symmetry is broken. With constant \(\nu\), hot rising plumes and cold sinking plumes are mirror images. With \(\nu(T)\) they are not — one is more viscous than the other, so the flow becomes top–bottom asymmetric.
  • The critical Rayleigh number shifts, and the relevant \(Ra\) must be defined with respect to some reference viscosity, which is a matter of convention that must be stated.
  • Boundary layers become asymmetric, changing the Nusselt-number scaling.

This is the regime of mantle convection, where viscosity varies by many orders of magnitude with temperature and the asymmetry is the dominant effect — the reason plate tectonics looks the way it does rather than being a symmetric roll pattern.

Validating a model with no exact solution

The hardest and most valuable part of the chapter. Once \(\nu = \nu(T)\) there is no analytic solution and no tabulated benchmark. What remains:

  1. Recover the constant-\(\nu\) limit. Set \(d\nu/dT \to 0\) and reproduce de Vahl Davis and \(Ra_c\) exactly. Non-negotiable.
  2. Manufactured solutions. These do not care that the problem is unphysical: manufacture a \(T\) field and a \(\nu(T)\) law, derive the source, and measure the observed order. This is the main verification tool when benchmarks run out.
  3. Convergence under refinement. The answer must stop moving.
  4. Physical consistency checks. Energy balance, Nusselt number computed two independent ways (from the wall flux and from the volume average) agreeing.
  5. Limiting-case behaviour. Small \(d\nu/dT\) must perturb the constant-\(\nu\) answer smoothly; a discontinuous change signals a bug.

The general lesson, and where the semester ends: when you leave the benchmarked regime, the burden of evidence shifts entirely onto you. No one else has computed the answer. Verification becomes the whole of your credibility, which is why Chapter 1's apparatus is the foundation of the course rather than a preliminary.

Common mistakes

  • Checking \(\tau > \tfrac12\) only at the initial condition. Transients probe extremes the steady state never reaches.
  • Not reporting the reference viscosity used to define \(Ra\). The number is meaningless without it, and papers disagree on the convention.
  • Assuming a finished run is a valid run. A local instability can pollute and decay.

Knowledge graph position

Prerequisites: thermal LBM, Chapman–Enskog, MMS. Leads to: research-level LBM; mantle and magma convection; the end of the taught material.

Quiz

Q1 (conceptual). Does the Chapman–Enskog result survive making \(\tau\) a function of position?

Answer

Essentially yes — the expansion is local, so \(\nu = c_s^2(\tau-\frac12)\) holds pointwise. The caveat is that it assumes \(\tau\) varies slowly on the mean-free-path scale; sharp gradients in \(\tau\) add terms at second order that the derivation does not capture.

Q2 (conceptual). Why does temperature-dependent viscosity break the symmetry of Rayleigh–Bénard convection?

Answer

With constant \(\nu\), rising hot plumes and sinking cold plumes are mirror images. With \(\nu(T)\) the hot fluid and cold fluid have different viscosities, so the two plume types experience different drag. The cell becomes top–bottom asymmetric, boundary layers differ, and the Nusselt scaling changes — the dominant effect in mantle convection.

Q3 (MCQ). With no analytic solution and no tabulated benchmark available, the primary verification tool is:

  • (a) comparing against another code
  • (b) the method of manufactured solutions, plus recovery of the constant-viscosity limit
  • (c) checking that the run completes
  • (d) refining until the picture looks smooth
Answer

(b). MMS supplies an exact solution by construction regardless of how unphysical the manufactured problem is, and recovering the benchmarked constant-\(\nu\) limit proves you have not broken what already worked.