Convection in a variable-viscosity fluid: Newtonian versus power-law rheology

doi:10.1016/0012-821X(83)90060-2

Earth and Planetary Science Letters

Volume 64, Issue 1, July 1983, Pages 153-162

https://doi.org/10.1016/0012-821X(83)90060-2 Get rights and content

Abstract

A number of finite-element calculations of convection in a variable-viscosity fluid have been carried out to investigate the effects of non-Newtonian flow when rheology is also subject to a strong temperature and pressure influence. A variety of cases has been studied in the range of effective Rayleigh numbers between 10⁴ and 10⁶, including different modes of heating and a range of values for activation energy and activation volume. Power-law creep with a stress exponent of 3 turns out to lead to considerably different flow pattern and heat transfer properties than Newtonian rheology. In general, the effect is to reduce viscosity contrasts imposed by p,T dependence, which can lead in some circumstances to the mobilisation of otherwise stagnant regions within the cell. The properties of non-Newtonian flow can be closely imitated by a Newtonian fluid with a reduced value of the activation enthalpy bH^* with b≅0.3–0.5. It appears possible that non-Newtonian rheology plays a key role in determining the convective style in a planetary mantle.

Reference (22)

PostR.L.
High temperature creep of Mt. Burnet dunite
Tectonophysics
(1977)
DurhamW.B. et al.
Transient and steady state creep of pure forsterite at low stress
Phys. Earth Planet. Inter.
(1979)
DeBremaeckerJ.-Cl.
Convection in the Earth's mantle
Tectonophysics
(1977)
CserepesL.
Numerical studies of non-Newtonian mantle convection
Phys. Earth Planet. Inter.
(1982)
KaratoS.
Rheology of the lower mantle
Phys. Earth Planet. Inter.
(1981)
JacobyW.R. et al.
On the effects of the lithosphere on mantle convection and evolution
Phys. Earth Planet. Inter.
(1982)
KohlstedtD.L. et al.
Low stress high temperature creep in olivine single crystals
J. Geophys. Res.
(1974)
WeertmanJ. et al.
High temperature creep of rock and mantle viscosity
Annu. Rev. Earth Planet. Sci.
(1975)
StockerR.L. et al.
On the rheology of the upper mantle
Rev. Geophys. Space Phys.
(1973)
ParmentierE.M. et al.
Studies of finite amplitude non-Newtonian thermal convection with application to convection in the Earth's mantle
J. Geophys. Res.
(1976)

ParmentierE.M. et al.

Thermal convection in non-Newtonian fluids: volumetric heating and boundary layer scaling

J. Geophys. Res.

(1982)

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The recovery behaviors of casein hydrogels after wear testing were fit to both power law (Table 2) and exponential models (Table 3). In rheology, power law models are typically used for characterizing shear rate dependency of non-Newtonian fluids [40], whereas exponential models are used to characterize stress relaxation and creep behaviors [12,13,34,41]. Generally, the R2 values for the power law model were ~0.84 and the RMSE values were ~0.1 mm (Table 2).
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Seismic tomography can be combined with constraints from geoid, topography and other surface observations to gain information about mantle structure and dynamics. This approach has been taken with much success for the Earth mantle, and here it is, for the first time, applied to the Moon. Lunar tomography has much lower resolution as for the Earth and is mostly restricted to the near side, nevertheless we can assess under what assumptions the fit between predicted geoid (based on a tomography model) and observed geoid is best. Among the models tested, we find the most similar pattern (correlation about 0.5) if we only consider tomography below 225 km depth, if density anomalies cause little or no dynamic topography and if we compare to the geoid with the flattening ( $l = 2, m = 0$ ) term removed. This could mean that (a) like for the Earth, seismic anomalies shallower than 225 km are caused by a combination of thermal and compositional effects and therefore cannot be simply converted to density anomalies; (b) the lithosphere is sufficiently thick to prevent dynamic topography more than a small fraction of total topography; and (c) flattening is a “fossil” bulge unrelated to present-day mantle anomalies. However, we have to be cautious with interpreting our results, because for models with a comparatively higher correlation and a conversion from seismic velocity to density anomalies similar to the Earth’s upper mantle, the amplitude of the predicted geoid is much lower than observed. This could either mean that the tomography model is strongly damped, or that the geoid is mostly due to shallow causes such as crustal thickness variations, with only a small part coming from the deeper mantle.

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Convection in a variable-viscosity fluid: Newtonian versus power-law rheology

Abstract

Tectonophysics

Phys. Earth Planet. Inter.

Tectonophysics

Phys. Earth Planet. Inter.

Phys. Earth Planet. Inter.

Phys. Earth Planet. Inter.

Low stress high temperature creep in olivine single crystals

J. Geophys. Res.

High temperature creep of rock and mantle viscosity

Annu. Rev. Earth Planet. Sci.

On the rheology of the upper mantle

Rev. Geophys. Space Phys.

Studies of finite amplitude non-Newtonian thermal convection with application to convection in the Earth's mantle

J. Geophys. Res.

Thermal convection in non-Newtonian fluids: volumetric heating and boundary layer scaling

J. Geophys. Res.