Frank–Kamenetskii Viscosity

• Frank–Kamenetskii viscosity is a linearized, temperature-dependent law derived from the Arrhenius formulation, encapsulating temperature sensitivity in a constant parameter Θ.
• It simplifies convection analysis by enabling clear classification of mobile, sluggish, and stagnant lid regimes through controlled viscosity contrast.
• The law is pivotal for scaling studies, influencing Nusselt–Rayleigh relationships and thermal boundary layer stability in both laboratory and geophysical flows.

Frank–Kamenetskii viscosity is the temperature-dependent viscosity law obtained by linearizing an Arrhenius-type dependence about a reference temperature and then nondimensionalizing with the viscosity at the top surface. In the formulation used for internally heated steady thermal convection, the nondimensional law is

$\eta(T)=\eta_0\exp(-\Theta T),$

with nondimensional temperature $T=\theta_{\mathrm{phys}}/\Delta T$ and nondimensional Frank–Kamenetskii parameter

$$\Theta\equiv \frac{E_a\Delta T}{R\,T_{\mathrm{ref}}^2}.$$

In the cited study, $\Theta$ is denoted by $\gamma$, and the law is used to analyze steady two-dimensional convective solutions, lid-regime classification, viscosity contrast, and Nusselt–Rayleigh scaling under homogeneous internal heating (Okuda et al., 22 Aug 2025).

1. Mathematical formulation

The starting point is an Arrhenius-type temperature dependence,

$$\eta_{\mathrm{phys}}(T_{\mathrm{phys}})=\eta_{\mathrm{ref}}\exp\!\left[\frac{E_a}{R}\left(\frac{1}{T_{\mathrm{phys}}}-\frac{1}{T_{\mathrm{ref}}}\right)\right].$$

Expanding about a reference temperature $T_{\mathrm{ref}}$ and introducing a temperature scale $\Delta T$ produces, to leading order, the Frank–Kamenetskii form

$$\eta_{\mathrm{phys}}\simeq \eta_{\mathrm{ref}}\exp\!\left[-\frac{E_a\Delta T}{R\,T_{\mathrm{ref}}^2}\,\theta_{\mathrm{phys}}\right].$$

Upon nondimensionalization with $\eta_0=\eta$ at the top surface and $T=\theta_{\mathrm{phys}}/\Delta T$0, the viscosity law becomes

$T=\theta_{\mathrm{phys}}/\Delta T$1

where

$T=\theta_{\mathrm{phys}}/\Delta T$2

The paper denotes $T=\theta_{\mathrm{phys}}/\Delta T$3 by $T=\theta_{\mathrm{phys}}/\Delta T$4 (Okuda et al., 22 Aug 2025).

In this representation, the key structural feature is that the temperature sensitivity is controlled by a constant nondimensional parameter. This distinguishes the Frank–Kamenetskii law from the full Arrhenius form, whose effective sensitivity varies with temperature. A plausible implication is that the Frank–Kamenetskii law is especially convenient when regime structure and scaling are the primary targets, because the temperature dependence is compressed into a single constant parameter.

2. Derivation and nondimensionalization

The governing scales are specified as follows: the length scale is the layer thickness $T=\theta_{\mathrm{phys}}/\Delta T$5; the temperature scale is $T=\theta_{\mathrm{phys}}/\Delta T$6 with uniform internal heating $T=\theta_{\mathrm{phys}}/\Delta T$7 and thermal conductivity $T=\theta_{\mathrm{phys}}/\Delta T$8; the time scale is the thermal diffusion time $T=\theta_{\mathrm{phys}}/\Delta T$9 with thermal diffusivity $$\Theta\equiv \frac{E_a\Delta T}{R\,T_{\mathrm{ref}}^2}.$$0; and the viscosity scale is $$\Theta\equiv \frac{E_a\Delta T}{R\,T_{\mathrm{ref}}^2}.$$1, the viscosity at the top surface (Okuda et al., 22 Aug 2025).

Starting from

$$\Theta\equiv \frac{E_a\Delta T}{R\,T_{\mathrm{ref}}^2}.$$2

one writes $$\Theta\equiv \frac{E_a\Delta T}{R\,T_{\mathrm{ref}}^2}.$$3, assumes $$\Theta\equiv \frac{E_a\Delta T}{R\,T_{\mathrm{ref}}^2}.$$4, and expands

$$\Theta\equiv \frac{E_a\Delta T}{R\,T_{\mathrm{ref}}^2}.$$5

This gives

$$\Theta\equiv \frac{E_a\Delta T}{R\,T_{\mathrm{ref}}^2}.$$6

Defining $$\Theta\equiv \frac{E_a\Delta T}{R\,T_{\mathrm{ref}}^2}.$$7 and $$\Theta\equiv \frac{E_a\Delta T}{R\,T_{\mathrm{ref}}^2}.$$8 then yields

$$\Theta\equiv \frac{E_a\Delta T}{R\,T_{\mathrm{ref}}^2}.$$9

The approximation $\Theta$0 is identified as the Frank–Kamenetskii approximation; it neglects higher-order curvature of the true Arrhenius dependence. This fixes the precise mathematical status of the law: it is not an independent constitutive ansatz in the cited treatment, but a leading-order approximation to Arrhenius rheology under a small-perturbation expansion.

3. Viscosity contrast and parameter interpretation

If the local nondimensional temperature varies between $\Theta$1 at the top and $\Theta$2 at the bottom of the convective layer, the viscosity contrast is

$\Theta$3

In the paper, $\Theta$4 is the mean bottom temperature, and $\Theta$5 measures how many orders of magnitude $\Theta$6 changes over the layer (Okuda et al., 22 Aug 2025).

The parameter $\Theta$7 measures the sensitivity of viscosity to temperature change per unit of the chosen $\Theta$8. Large $\Theta$9 corresponds to strong temperature dependence, and small $\gamma$0 to weak temperature dependence. The reported ranges are $\gamma$1 to $\gamma$2 in lab-fluids and engineering flows, while mantle convection applications are described by $\gamma$3–$\gamma$4, $\gamma$5, and $\gamma$6, giving $\gamma$7–$\gamma$8. These are also the values explored in the paper, with $\gamma$9 from $$\eta_{\mathrm{phys}}(T_{\mathrm{phys}})=\eta_{\mathrm{ref}}\exp\!\left[\frac{E_a}{R}\left(\frac{1}{T_{\mathrm{phys}}}-\frac{1}{T_{\mathrm{ref}}}\right)\right].$$0 up to $$\eta_{\mathrm{phys}}(T_{\mathrm{phys}})=\eta_{\mathrm{ref}}\exp\!\left[\frac{E_a}{R}\left(\frac{1}{T_{\mathrm{phys}}}-\frac{1}{T_{\mathrm{ref}}}\right)\right].$$1 (Okuda et al., 22 Aug 2025).

This parameterization makes viscosity contrast an exponentially amplified function of the thermal state. A plausible implication is that relatively modest changes in $$\eta_{\mathrm{phys}}(T_{\mathrm{phys}})=\eta_{\mathrm{ref}}\exp\!\left[\frac{E_a}{R}\left(\frac{1}{T_{\mathrm{phys}}}-\frac{1}{T_{\mathrm{ref}}}\right)\right].$$2 or in bottom temperature can reorganize the convective regime even when other control parameters remain fixed.

4. Relation to the Arrhenius-law formulation

The paper also implements the full Arrhenius law in nondimensional form as

$$\eta_{\mathrm{phys}}(T_{\mathrm{phys}})=\eta_{\mathrm{ref}}\exp\!\left[\frac{E_a}{R}\left(\frac{1}{T_{\mathrm{phys}}}-\frac{1}{T_{\mathrm{ref}}}\right)\right].$$3

with $$\eta_{\mathrm{phys}}(T_{\mathrm{phys}})=\eta_{\mathrm{ref}}\exp\!\left[\frac{E_a}{R}\left(\frac{1}{T_{\mathrm{phys}}}-\frac{1}{T_{\mathrm{ref}}}\right)\right].$$4 the offset and $$\eta_{\mathrm{phys}}(T_{\mathrm{phys}})=\eta_{\mathrm{ref}}\exp\!\left[\frac{E_a}{R}\left(\frac{1}{T_{\mathrm{phys}}}-\frac{1}{T_{\mathrm{ref}}}\right)\right].$$5 the dimensionless activation energy (Okuda et al., 22 Aug 2025).

A local Frank–Kamenetskii-parameter analog may be defined as

$$\eta_{\mathrm{phys}}(T_{\mathrm{phys}})=\eta_{\mathrm{ref}}\exp\!\left[\frac{E_a}{R}\left(\frac{1}{T_{\mathrm{phys}}}-\frac{1}{T_{\mathrm{ref}}}\right)\right].$$6

which depends on $$\eta_{\mathrm{phys}}(T_{\mathrm{phys}})=\eta_{\mathrm{ref}}\exp\!\left[\frac{E_a}{R}\left(\frac{1}{T_{\mathrm{phys}}}-\frac{1}{T_{\mathrm{ref}}}\right)\right].$$7, whereas the Frank–Kamenetskii version uses constant $$\eta_{\mathrm{phys}}(T_{\mathrm{phys}})=\eta_{\mathrm{ref}}\exp\!\left[\frac{E_a}{R}\left(\frac{1}{T_{\mathrm{phys}}}-\frac{1}{T_{\mathrm{ref}}}\right)\right].$$8. In practice, the Frank–Kamenetskii law, $$\eta_{\mathrm{phys}}(T_{\mathrm{phys}})=\eta_{\mathrm{ref}}\exp\!\left[\frac{E_a}{R}\left(\frac{1}{T_{\mathrm{phys}}}-\frac{1}{T_{\mathrm{ref}}}\right)\right].$$9, is the linearization of the Arrhenius law about $T_{\mathrm{ref}}$0. The paper reports that regime boundaries between mobile, sluggish, and stagnant lid are only mildly shifted under the full Arrhenius law with modest $T_{\mathrm{ref}}$1, and that mapping $T_{\mathrm{ref}}$2 brings the two formulations into almost quantitative agreement (Okuda et al., 22 Aug 2025).

The comparison indicates that constant-$T_{\mathrm{ref}}$3 rheology captures the essential three-regime behavior while simplifying the temperature dependence. This suggests that the main discrepancy between the two formulations lies not in the existence of the regimes themselves, but in slight shifts of regime-boundary location.

5. Role in convective regime classification

In the cited study, two-dimensional steady-state convective solutions with the Frank–Kamenetskii viscosity are obtained by the Newton method for different values of the Rayleigh number and of the strength of the dependence of the viscosity on temperature. Lid-regime classification is based on the surface-velocity mobility $T_{\mathrm{ref}}$4, with the thresholds

• $T_{\mathrm{ref}}$5: mobile lid,
• $T_{\mathrm{ref}}$6: sluggish lid,
• $T_{\mathrm{ref}}$7: stagnant lid (Okuda et al., 22 Aug 2025).

As $T_{\mathrm{ref}}$8 increases, or as the basal Rayleigh number $T_{\mathrm{ref}}$9 decreases, the sequence is mobile regime $\Delta T$0 sluggish lid $\Delta T$1 stagnant lid. The sluggish lid regime is found between the mobile and stagnant lid regimes. It is characterized by a large viscosity contrast through the boundary layer below the conductive lid, rapid changes of lid thickness and Nusselt number with $\Delta T$2 at almost constant $\Delta T$3, and a peak in the effective viscosity contrast across the boundary layer. The summary further states that this regime is promoted by the stress-free top boundary and disappears under a rigid-top condition (Okuda et al., 22 Aug 2025).

Within this classification, Frank–Kamenetskii viscosity acts as the mechanism that sets how abruptly rheological resistance increases toward the cold upper boundary. A plausible implication is that the sluggish lid regime is not merely intermediate by definition, but is associated with a distinct boundary-layer structure rather than a smooth interpolation between mobile and stagnant limits.

6. Scaling behavior and instability

The Nusselt–Rayleigh scaling depends on both $\Delta T$4 and $\Delta T$5. At moderate $\Delta T$6 and moderate $\Delta T$7, corresponding to a weak lid, classical scaling $\Delta T$8 emerges, approximately the isoviscous limit. At high $\Delta T$9 or strong viscosity contrast, thin strong plumes dominate and one finds $$\eta_{\mathrm{phys}}\simeq \eta_{\mathrm{ref}}\exp\!\left[-\frac{E_a\Delta T}{R\,T_{\mathrm{ref}}^2}\,\theta_{\mathrm{phys}}\right].$$0 in both mobile and stagnant-lid regimes when rescaled by the convective-layer thickness. For most solutions in the mobile and stagnant lid regimes, the Nusselt number is proportional to the $$\eta_{\mathrm{phys}}\simeq \eta_{\mathrm{ref}}\exp\!\left[-\frac{E_a\Delta T}{R\,T_{\mathrm{ref}}^2}\,\theta_{\mathrm{phys}}\right].$$1 power of the Rayleigh number, and the paper states that this can be derived by taking into account the effect of thin and strong downwelling plumes (Okuda et al., 22 Aug 2025).

Time-evolution calculations show that the steady solutions become unstable for large Rayleigh numbers, where additional downward plumes grow on the background convective flows. The stated explanation is that the timescale of the Rayleigh–Taylor instability for the thermal boundary layer between the conductive lid and the convective core becomes shorter than the timescale of horizontal advection by the background flow (Okuda et al., 22 Aug 2025).

These results place Frank–Kamenetskii viscosity at the center of both steady scaling and loss of steadiness. In the steady problem it determines whether boundary layers remain thick and symmetric or become thin with focused plumes; in the unsteady transition it conditions the thermal-boundary-layer structure on which plume-forming instability develops.