Corner Flow of Power Law Fluids

Abstract

SynopsisA local analysis of the flow of power law fluids near corners is performed. The equation for the stream function is shown to allow separated solutions in plane polar coordinates. The radial behavior is shown to be algebraic and results are given for the exponent for different values of corner angle and power law exponent. In addition, the critical angle for the onset of an eddy structure is found as function of the power law exponent.

The flow of an Oldroyd fluid around a sharp corner

“…In this solution the stress behaves like r -".4555 near to the corner. For power law fluids with a viscosity p = y" -* and for a 270" corner, Henricksen and Hassager [3] found CI = 0.37 for the thinning n = 0.5 (stress r -".31), and a = 0.640 for the thickening n = 1.5 (stress i" -".540). For a suspension of rigid rods aligned with the flow, Keiller and Hinch [4] found a = 0.58 (stress r -".42) at an effective concentration of the rods (p = 5, and a = 0.62 (stress Y -".38) for 4 = 20.…”

The determination of apparent viscosity using a wide gap, double concentric cylinder

“…Griffiths [8] cites work in which it is shown how, for a different singular problem in which there is an interior angle of 2π rather than two interior angles of 3π/2, the numerical error in the solution behaves, and why it is not simply given by the analytic estimate. Finally, it is noted that Henriksen and Hassager [13], have carried out an analysis of the nature of the flow of a power law fluid past a re-entrant corner, of the sort that occurs in a planar contraction. However, we know of no similar analysis for the current problem.…”

Formation of ice eddies in subglacial mountain valleys

“…Specifically, Moffatt determined the critical angle of the corner required to form an eddy. Fenner [] and Henriksen and Hassager [] extended Moffatt's calculation to non‐Newtonian fluids and determined that the critical angle as a function of the rheological parameters.…”

“…In glaciology, n can vary from n = 1 to 4 but is typically chosen to be 3 [ Durham et al , ; Goldsby and Kohlstedt , ; Cuffey and Paterson , ]. In a power law fluid, the critical angle α c for the formation of Moffatt eddies depends on the rheological exponent n [ Fenner , ; Henriksen and Hassager , ].…”

“…For a fluid with a power law rheology, we aim to determine how the critical angle α c depends on the rheological exponent n . To this end, we derive a similarity solution for corner eddies that form in slow viscous power law fluids [ Fenner , ; Henriksen and Hassager , ]. The momentum and mass conservation equations for a two‐dimensional incompressible flow are given as Taking the curl of the momentum equation, using mass conservation, and writing the result in polar coordinates, we obtain Using Glen's law from equation , we can relate the deviatoric stress to the strain rates as The strain rates are derivatives of the velocity (equation , and, by definition, the velocity can be written in terms of the stream function as Because each term in equation scales as 1/ r , we can write the stream function as …”