Relationship between rheological and dynamic parameters

for a power-law lava flowA. Piombo and M. Dragoni

Dipartimento di Fisica, Alma Mater Studiorum - Università di Bologna, Italy

During the cooling lava behaves as non-Newtonian, pseudoplastic fluid.

Laboratory experiments on lava samples suggest that a power-law constitutive equation may be appropriate.

We consider a constitutive equation where shear stress is proportional to strain rate raised to an integer power n greater than or equal to 1.

Formulae are obtained relating the lava rheology to surface velocity, thickness, width, and flow rate, which can be measured in the field.

We consider a horizontally unbounded layer of lava, flowing down a slope driven by the gravity force.We assume that the lava is isothermal. This assumption is justified by the fact that the temperature decrease along a lava flow is very slow.

No quantities depend on the coordinates x and y, so that the problem is one-dimensional.

x

y

z

where: - ρ is the density of lava, - α is the slope angle, - g is the gravity acceleration.

1. Free-surface boundary condition at z = h, where h is the flow thickness

2. Vanishing velocity at z = 0

The Cauchy equation for steady-state flow

Boundary conditions

For the constitutive equation, we assume this power law

where eij is the strain, σij is the viscous stress, andC (Pa-n s-1) is a positive constant, and n ≥ 1.

If n = 1 the equation reduces to the constitutive equation for a Newtonian fluid and C is the inverse of viscosity.

Constitutive equation

Temperature dependence of C

Constitutive equation

Qualitative plot of the constitutive equation and comparison with Bingham case (τ is the yield stress of Bingham fluid)

Stress and strain rate measurements from laboratory experiments where a power-law model seems to be the

most suitable one.

(e.g. Hardee, H. C., and J. C. Dunn (1981), Convective heat transfer in magmas near the liquidus, J. Volcanol. Geotherm. Res., 10, 195-207.Sonder, I., B. Zimanowski, and R. Büttner (2006), Non-Newtonian viscosity of basaltic magma, Geophys. Res. Lett., 33, L02303, doi:10.1029/2005GL024240)

Results

Velocity profile and comparison with Bingham rheology

V is the surface velocity, τ is the yield stress of Bingham fluid, vp is the plug velocity

The flow thickness h as a function of n and q/V in comparison with Bingham rheology.

n as a function of the volume flow rate Q

We introduce the quantity β which is a function of the lava flow parameters (flow rate Q, surface velocity V, thickness h, width L); we imagine that their values can be measured in the field.

If β = 2/3, the fluid is Newtonian (n = 1).

Conclusions• This model shows that a power-law constitutive equation

can describe for n > 1 the presence of a nearly undeformed part of a lava flow, similar to the plug of Bingham rheology.

• Under certain conditions, this model allows constraining lava rheology through formulae which relate the rheology to surface velocity, thickness, width, and flow rate of a lava flow; these quantities can be measured in the field.

• For any choice of thickness, width and surface velocity of lava flow, the flow rate assumes values in a narrow range independently of the value of n.

• The model shows the importance of taking measurements of flow rate, surface velocity and geometry of lava flow with a good degree of accuracy.