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The viscoplastic Stokes layer. (English) Zbl 1274.76012
Summary: A theoretical and experimental study is presented of the viscoplastic version of the Stokes problem, in which a oscillating wall sets an overlying fluid layer into one-dimensional motion. For the theory, the fluid is taken to be described by the Herschel-Bulkley constitutive law, and the flow problem is analogous to an unusual type of Stefan problem. In the theory, when the driving oscillations are relatively weak, the overlying viscoplastic layer moves rigidly with the plate. For sufficiently strong oscillations, the fluid yields and numerical solutions illustrate how localized plug regions coexist with sheared regions and migrate vertically through the fluid layer. For the experiments, a layer of kaolin slurry in a rectangular tank is driven sinusoidally back and forth. The experiments confirm the threshold for shearing flow, equivalent to a balance between inertia and yield-stress. However, although kaolin is well described by a Herschel-Bulkley rheology, the layer dynamics notably differs between theory and experiments, revealing rheological behaviour not captured by the steady flow rule.

76A05 Non-Newtonian fluids
76D07 Stokes and related (Oseen, etc.) flows
76-05 Experimental work for problems pertaining to fluid mechanics
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[1] Batchelor, G. K.: Fluid dynamics, (1967) · Zbl 0152.44402
[2] Davis, S. H.: The stability of time-periodic flows, Ann. rev. Fluid mech. 8, 57-74 (1976)
[3] Yakhot, V.; Colosqui, C.: Stokes second flow problem in a high-frequency limit: application to nanomechanical resonators, J. fluid mech. 586, 249-258 (2007) · Zbl 1119.76052 · doi:10.1017/S0022112007007148
[4] Rajagopal, K. R.: A note on unsteady unidirectional flows of a non-Newtonian fluid, Int. J. Nonlinear mech. 17, 369-373 (1982) · Zbl 0527.76003 · doi:10.1016/0020-7462(82)90006-3
[5] Hayat, T.; Asghar, S.; Siddiqui, A. M.: Stoke’s second problem for a Johnson – Segalman fluid, Appl. math. Comput. 148, 697-706 (2004) · Zbl 1035.76003 · doi:10.1016/S0096-3003(02)00928-1
[6] Aumaitre, S.; Puls, C.; Macelwaine, J. N.; Gollub, J. P.: Comparing flow thresholds and dynamics for oscillating and inclined granular layers, Phys. rev. E 75, 061307 (2007)
[7] Comparini, E.: A one-dimensional Bingham flow, J. math. Anal. appl. 167, 129-139 (1992) · Zbl 0763.76003
[8] Comparini, E.; De Angelis, E.: Flow of a Bingham fluid in a concentric cylinder viscometer, Adv. math. Sci. appl. 6, 97-116 (1996) · Zbl 0853.76004
[9] P. Coussot, Mudflow rheology and dynamics, IAHR Monograph Series, Balkema, 1997. · Zbl 0949.76001
[10] Barnes, H. A.: Thixotropy-a review, J. non-Newtonian fluid mech. 70, 1-33 (1997)
[11] Pignon, F.; Magnin, A.; Piau, J. -M.: Butterfly light scattering pattern and rheology of a sheared thixotropic Clay gel, Phys. rev. Lett. 79, 4689-4692 (1997)
[12] Roussel, N.; Le Roy, R.; Coussot, P.: Thixotropy modelling at local and macroscopic scales, J. non-Newtonian fluid mech. 117, 85-95 (2004) · Zbl 1130.76310 · doi:10.1016/j.jnnfm.2004.01.001
[13] Billigham, J.; Feguson, J. W. J.: Laminar, unidirectional flow of a thixotropic fluid in a circular pipe, J. non-Newtonian fluid mech. 47, 21-55 (1993) · Zbl 0781.76003 · doi:10.1016/0377-0257(93)80043-B
[14] Blom, J. G.; Zegeling, P. A.: Algorithm 731: a moving-grid interface for systems of one-dimensional time-dependent partial differential equations, ACM trans. Math. softw. 20, 194-214 (1994) · Zbl 0889.65099 · doi:10.1145/178365.178391 · www.acm.org
[15] L. Petzold, A description of DASSL: a differential/algebraic systems solver, in IMACS Trans. Sci. Comp., vol. 1, R.S. Steplman (Ed.), North-Holland, Amsterdam, 1983.
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