Geophysics Problem of the day: one-dimensional flow with
varying viscosity and shear stress

I still have vivid and occasionally lucid dreams even 25+
years after my college roommate and I stopped telling each other our dreams
every morning. In a recent dream, I was trying to set up this problem on the
blackboard, but my legs kept buckling out from under me.

Here’s the picture I was trying to draw (in my dream).

Couette Flow Cartoon |

Here’s today’s Geophysics Problem of the Day:

Part 1. Remind yourself how to derive Couette Flow
(1-dimensional flow between plates). For the first part, assume that the
viscosity of the fluid is constant. Wikepedia is a good help on this. Derive
formulas for velocity as a function of depth and shear stress as a function of
depth.

Part 2. Using your derived relationships, determine velocity
and shear stress as a function of depth for the Earth’s upper mantle, assuming
that the lower mantle is fixed. Also calculate accumulated strain over a ~100
million year timescale as a function of depth.Use typical plate velocities
(1-10 cm/yr) and mantle viscosities (~10

^{19}-10^{20}Pa s)
In real materials, viscosity is strongly dependent on
temperature, with the following general relationship:

where T

_{Hom}is the “homologous” temperature: the ratio of the actual temperature to the melting point. Using the homologous temperature is important here, because is shows that for many solids the viscosity—resistance to flow—decreases as the melting temperature is approached. Below the lithosphere, the temperature increases with depth through the mantle. But likely the melting temperature also increases with depth. So calculating homologous temperature as a function of depth is easy to say, but hard to do.
Part 3. Rederive Couette-style flow using a depth-dependent
viscosity, and use these relationships to calculate velocity, shear stress, and
accumulated strain as a function of depth for Earth’s upper mantle and
transition zone using a depth-dependent viscosity. Try a depth-dependent viscosity
model such as Peltier,

*Science*1996.
Part 3. Rederive Couette-style flow using a depth-dependent
viscosity, and use these relationships to calculate velocity, shear stress, and
accumulated strain as a function of depth for Earth’s upper mantle and
transition zone using a depth-dependent viscosity. Try a depth-dependent viscosity
model such as Peltier,

*Science*1996.Mantle Couette Cartoon. |

Part 4. Viscosity describes a material's ability to
support a shear stress over a long timescale. During a solid-state phase
transformation, the ability of a material to support a shear stress is expected
to approach zero at equilibrium (=long time scales). In the Earth, the top and
bottom of the transition zone correspond to major mineralogical phase
transitions. What if these transformations significantly reduce the viscosity
in thin shells at the top and bottom of the transition zones? To examine the
potential effects of this, calculate the depth-dependent velocity, shear stress,
and accumulated strain for a constant viscosity mantle with two sharp
lower-viscosity zones, corresponding to the top and bottom of the transition
zone, as shown in the picture at left. You might want to do this for different values
of low-viscosities. How high does it need to be to avoid an effective
"free-slip" boundary condition, thereby ruining the flow for
everything underneath?

Please send me your solutions!

Note 1: If this problem is too boring, then try it in a spherical
coordinate system, with the outside surface of the sphere forced to rotate with
an angular velocity, and a fixed small interior volume.

Note 2: If you need additional information or values such as
constants or material properties don’t be afraid to look them up and/or make
them up.

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