Theory of creeping gravity currents of a non-Newtonian liquid

Abstract

Recently several experiments on creeping gravity currents have been performed, using highly viscous silicone oils and putties. The interpretation of the experiments relies on the available theoretical results that were obtained by means of the lubrication approximation with the assumption of a Newtonian rheology. Since very viscous fluids are usually non-Newtonian, an extension of the theory to include non-Newtonian effects is needed. We derive the governing equations for unidirectional and axisymmetric creeping gravity currents of a non-Newtonian liquid with a power-law rheology, generalizing the usual lubrication approximation. The equations differ from those for Newtonian liquids, being nonlinear in the spatial derivative of the thickness of the current. Similarity solutions for currents whose volume varies as a power of time are obtained. For the spread of a constant volume of liquid, analytic solutions are found that are in good agreement with experiment. We also derive solutions of the waiting-time type, as well as those describing steady flows from a constant source to a sink. General traveling-wave solutions are given, and analytic formulas for a simple case are derived. A phase plane formalism that allows the systematic derivation of self-similar solutions is introduced. The application of the Boltzmann transform is briefly discussed. All the self-similar solutions obtained here have their counterparts in Newtonian flows, as should be expected because the power-law rheology involves a single-dimensional parameter as the Newtonian constitutive relation. Thus one finds similarity solutions whenever the analogous Newtonian problem is self-similar, but now the spreading relations are rheology-dependent. In most cases this dependence is weak but leads to significant differences easily detected in experiments. The present results may also be of interest for geophysics since the lithosphere deforms according to an average power-law rheology.

The inertial dynamics of thin film flow of non-Newtonian fluids

“…Suppose the fluid has the nonNewtonian, power law, stress-strain relation that the stress ∝ (strain-rate) s for some fixed exponent s: the exponent s = 1 for a Newtonian fluid; s < 1 is shear thinning; and s > 1 is shear thickening. Such a power law is sometimes called Ostwald's or Norton's constitutive relation [1]. Then the systematic analysis developed in this Letter supports the nondimensional model of the flow ( where Re is the nondimensional Reynolds number, c s is the coefficient of proportionality in the nonlinear stress-strain relation, and where g 1 and g 2 are the nondimensional components of gravity along and normal to the flat substrate, respectively.…”

“…The lubrication approximation of very slow flow, negligible Reynolds number, underpins previous theoretical models for non-Newtonian thin fluid films: Perazzo and Gratton [4] and Betelu and Fontelos [3] examined flow with surface tension; this followed experiments comparing travelling waves and similarity solutions by Gratton, Minotti and Mahajan [1]. Gratton et al comment "the differences between Newtonian and nonNewtonian currents are significant and can clearly be observed in experiments".…”

“…Similarly, modulating gravity g 2 in time allows the above model to simulate Faraday waves as previously displayed for Newtonian fluids [2, Section 6.1]. Further, substituting the equilibrium mean velocity (3) into the fluid conservation equation (1), modelling the very slow dynamics at small Re, leads to an accurate lubrication model for nonlinear fluids, one previously approximated by others, e.g. [3,4], and reducing to the classic lubrication model for Newtonian fluids when exponent s = 1.…”

“…This model generalises the model of Newtonian fluids [2]. Fluid is conserved through (1). The momentum equation (2) incorporates effects of inertia,ū t , self-advection, uū x andū 2 η x , bed drag, (ū/η) s , and gravitational forcing, (g 1 − g 2 η x ); the dependence of the coefficients upon s models the subtle effects of the power law rheology.…”

Slumps of viscoplastic fluids on slopes

“…This behaviour close to the front, however, can be anticipated. An intruding free-surface flow of a fluid with power law rheology exhibits h ∼ (x f − x) 1/(n+2) (see, for example [17,26,27]), whereas the arrested state has h ∼ (x f − x) 1/2 . Thus as the flow of a Herschel-Bulkley fluid approaches its arrested state, although the interior becomes close to h ∞ , there exists a diminishing region close to the front within which h ∞ does not provide the dominant shape of the interface.…”