A new model for the analytical prediction of wall shear stress distributions at the base of orbitally shaken shallow fluid layers is developed. This model is a generalisation of the classical extended Stokes solution and will be referred to as the potential theory-Stokes model. The model is validated using a large set of numerical simulations covering a wide range of flow regimes representative of those used in laboratory experiments. It is demonstrated that the model is in much better agreement with the simulation data than the classical Stokes solution, improving the prediction in 63% of the studied cases. The central assumption of the model—which is to link the wall shear stress with the surface velocity—is shown to hold remarkably well over all regimes covered.
REFERENCES
1.
P.
Alpresa
, S.
Sherwin
, P. D.
Weinberg
, and M.
van Reeuwijk
, “Orbitally shaken shallow fluid layers. I. Regime classification
,” Phys. Fluids
30
, 032107
(2018
).2.
R. E.
Berson
, M. R.
Purcell
, and M. K.
Sharp
, “Computationally determined shear on cells grown in orbiting culture dishes
,” Adv. Exp. Med. Biol.
614
, 189
–198
(2008
).3.
M. M.
Salek
, P.
Sattari
, and R. J.
Martinuzzi
, “Analysis of fluid flow and wall shear stress patterns inside partially filled agitated culture well plates
,” Ann. Biomed. Eng.
40
, 707
–728
(2012
).4.
N.
Filipovic
, K.
Ghimire
, I.
Saveljic
, Z.
Milosevic
, and C.
Ruegg
, “Computational modeling of shear forces and experimental validation of endothelial cell responses in an orbital well shaker system
,” Comput. Methods Biomech. Biomed. Eng.
19
, 581
–590
(2015
).5.
A.
Dardik
, L.
Chen
, J.
Frattini
, H.
Asada
, F.
Aziz
, F. A.
Kudo
, B. E.
Sumpio
, N.
Haven
, and W.
Haven
, “Differential effects of orbital and laminar shear stress on endothelial cells
,” J. Vasc. Surg.
41
, 869
–880
(2005
).6.
K.
Ley
, E.
Lundgren
, E.
Berger
, and K. E.
Arfors
, “Shear-dependent inhibition of granulocyte adhesion to cultured endothelium by dextran sulfate
,” Blood
73
, 1324
–1330
(1989
).7.
L. W.
Kraiss
, A. S.
Weyrich
, N. M.
Alto
, D. A.
Dixon
, T. M.
Ennis
, V.
Modur
, T. M.
McIntyre
, S. M.
Prescott
, and G. A.
Zimmerman
, “Fluid flow activates a regulator of translation, p70/p85 S6 kinase, in human endothelial cells
,” Am. J. Physiol.: Heart Circ. Physiol.
278
, H1537
–H1544
(2000
).8.
S.
Yun
, A.
Dardik
, M.
Haga
, A.
Yamashita
, S.
Yamaguchi
, Y.
Koh
, J. A.
Madri
, and B. E.
Sumpio
, “Transcription factor Sp1 phosphorylation induced by shear stress inhibits membrane type 1-matrix metalloproteinase expression in endothelium
,” J. Biol. Chem.
277
, 34808
–34814
(2002
).9.
H.
Asada
, J.
Paszkowiak
, D.
Teso
, K.
Alvi
, A.
Thorisson
, J. C.
Frattini
, F. A.
Kudo
, B. E.
Sumpio
, and A.
Dardik
, “Sustained orbital shear stress stimulates smooth muscle cell proliferation via the extracellular signal-regulated protein kinase 1/2 pathway
,” J. Vasc. Surg.
42
, 772
–780
(2005
).10.
H.
Kim
, “Different effects of orbital shear stress on vascular endothelial cells: Comparison with the results of in vivo study with rats
,” Vasc. Spec. Int.
31
, 33
–40
(2015
).11.
F. S.
Sherman
, Viscous Flow
, 1st ed., McGraw-Hill Series in Mechanical Engineering (McGraw-Hill
, New York
, 1990
).12.
J. M. D.
Thomas
, A.
Chakraborty
, M. K.
Sharp
, and R. E.
Berson
, “Spatial and temporal resolution of shear in an orbiting petri dish
,” Biotechnol. Prog.
27
, 460
–465
(2011
).13.
A.
Chakraborty
, S.
Chakraborty
, V. R.
Jala
, B.
Haribabu
, M. K.
Sharp
, and R. E.
Berson
, “Effects of biaxial oscillatory shear stress on endothelial cell proliferation and morphology
,” Biotechnol. Bioeng.
109
, 695
–707
(2012
).14.
J. M. D.
Thomas
, A.
Chakraborty
, R. E.
Berson
, M.
Shakeri
, and M. K.
Sharp
, “Validation of a CFD model of an orbiting culture dish with PIV and analytical solutions
,” AIChE J.
63
, 4233
(2017
).15.
A.
Chakraborty
, S.
Chakraborty
, V. R.
Jala
, J. M.
Thomas
, M. K.
Sharp
, R. E.
Berson
, and B.
Haribabu
, “Impact of bi-axial shear on atherogenic gene expression by endothelial cells
,” Ann. Biomed. Eng.
44
, 3032
(2016
).16.
G. K.
Batchelor
, An Introduction to Fluid Dynamics Book
(Cambridge University Press
, 1967
), p. 631
.17.
G. K.
Vallis
, Atmospheric and Oceanic Fluid Dynamics
(Cambridge University Press
, 2006
), p. 773
.18.
G.
Piñeiro
, S.
Perelman
, J. P.
Guerschman
, and J. M.
Paruelo
, “How to evaluate models: Observed vs. predicted or predicted vs. observed?
,” Ecol. Modell.
216
, 316
–322
(2008
).19.
M.
Reclari
, M.
Dreyer
, S.
Tissot
, D.
Obreschkow
, F. M.
Wurm
, and M.
Farhat
, “Surface wave dynamics in orbital shaken cylindrical containers
,” Phys. Fluids
26
, 052104
(2014
).20.
A.
Royon-Lebeaud
, E. J.
Hopfinger
, and A.
Cartellier
, “Liquid sloshing and wave breaking in circular and square-base cylindrical containers
,” J. Fluid Mech.
577
, 467
(2007
).21.
R. A.
Ibrahim
, Liquid Sloshing Dynamics
(Cambridge University Press
, 2005
).22.
D.
Joseph
, T.
Funada
, and J.
Wang
, Journal of Chemical Information and Modeling
(Cambridge University Press
, 2008
), Vol. 53.23.
H. M.
Kim
and J. P.
Kizito
, “Stirring free surface flows due to horizontal circulatory oscillation of a partially filled container
,” Chem. Eng. Commun.
196
, 1300
–1321
(2009
).24.
W.
Weheliye
, M.
Yianneskis
, and A.
Ducci
, “On the fluid dynamics of shaken bioreactors-flow characterization and transition
,” AIChE J.
59
, 334
–344
(2013
).© 2018 Author(s).
2018
Author(s)
You do not currently have access to this content.