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On the Wall Shear Stress Gradient in Fluid Dynamics

Published online by Cambridge University Press:  24 March 2015

C. Cherubini*
Nonlinear Physics and Mathematical Modeling Laboratory International Center for Relativistic Astrophysics – I.C.R.A., University Campus Bio-Medico of Rome, Via A. del Portillo 21, I-00128 Rome, Italy
S. Filippi
Nonlinear Physics and Mathematical Modeling Laboratory International Center for Relativistic Astrophysics – I.C.R.A., University Campus Bio-Medico of Rome, Via A. del Portillo 21, I-00128 Rome, Italy
A. Gizzi
Nonlinear Physics and Mathematical Modeling Laboratory
M. G. C. Nestola
Nonlinear Physics and Mathematical Modeling Laboratory
*Corresponding author. Email addresses: (C. Cherubini), (S. Filippi), (A. Gizzi), (M. G. C. Nestola)
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The gradient of the fluid stresses exerted on curved boundaries, conventionally computed in terms of directional derivatives of a tensor, is here analyzed by using the notion of intrinsic derivative which represents the geometrically appropriate tool for measuring tensor variations projected on curved surfaces. Relevant differences in the two approaches are found by using the classical Stokes analytical solution for the slow motion of a fluid over a fixed sphere and a numerically generated three dimensional dynamical scenario. Implications for theoretical fluid dynamics and for applied sciences are finally discussed.

Research Article
Copyright © Global-Science Press 2015 

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[1]Alcubierre, M., Introduction to 3+1 Numerical Relativity, Oxford University Press, New York (2008).Google Scholar
[2]Anthony, T.R. and Cline, H. E., Heat treating and melting material with a scanning laser or electron beam, J Applied Physics, 48 (2008), 38883900.Google Scholar
[3]Aris, R., Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Dover Publications (1990).Google Scholar
[4]Bifano, T.G., Johnson, H.T., Bierden, P. and Mali, R.K., Elimination of stress-induced curvature in thin-film structures, J. Microelectromech. Syst., 5 (2002), 592597.Google Scholar
[5]Bird, R.B., Stewart, W.E. and Lingthfoot, E.N., Transport Phenomena, 2nd ed., John Wiley and Sons (2007).Google Scholar
[6]Buchanan, J.R., Kleinstreurer, C., Truskey, G.A. and Lei, M., Relation between non-uniform hemodynamics and sites of altered permeability and lesion growth at the rabbit aorto-celiac junction, Atehrosclerosis, 143 (1999), 2740.Google Scholar
[7]Chaichana, T., Sun, Z. and Jewkes, J., Computation of hemodynamics in the left coronary artery with variable angulations, J. biomech., 44 (2011), 18691878.Google Scholar
[8]Chakravarthy, S.S. and Curtin, W.A., Origin of plasticity length-scale effects in fracture, Phys. Rev. Lett., 105 (2010), 115502.Google Scholar
[9]Chandrasekhar, S., The Mathematical Theory of Black Holes, Clarendon Press, Oxford (2000).Google Scholar
[10]Chatzizisis, Y.S.Coskun, A.U., Jonas, M., Edelman, E.R., Feldman, C.L. and Stone, P.H., Role of endothelial shear stress in the natural history of coronary atherosclerosis and vascular remodeling: molecular, cellular, and vascular behavior, J. Am. Coll. Cardiol., 49 (2007), 23792393.Google Scholar
[11]Chen, H.Y., Moussa, I.D., Davidson, C. and Kassab, G.S., Impact of main branch stenting on endothelial shear stress: role of side branch diameter, angle and lesion, J.R. Soc. Interface, 9 (2012), 11871193.Google Scholar
[12]Clouet, E., Dislocation core field. I. Modeling in anisotropic linear elasticity theory, Phys. Rev. B, 84 (2011), 224111.Google Scholar
[13]Cornelius, T.L., Biomechanical Systems: Techniques and Applications, Volume II: Cardiovascular Techniques, CRC Press (2000).Google Scholar
[14]Das, A.J., Tensors: The Mathematics of Relativity Theory and Continuum Mechanics, Springer, Berlin (2007).Google Scholar
[15]DePaola, N., Gimbrone, M.A., Davies, P.F., and Dewey, C.F., Vascular endothelium responds to fluid shear stress gradients, Arterioscler. Thromb. Vasc. Biol., 12 (1992), 12541257.Google Scholar
[16]Dolan, J.M., Kolega, J. and Meng, H., High fluid shear stress and spatial shear stress gradients affect endothelial proliferation, survival, and alignment, Ann. Biomed. Eng., 41 (2013), 14111427.Google Scholar
[17]Finol, E.A., Amon, C.H., Flow-induced wall shear stress in abdominal aortic aneurysms: Part ii-pulsatile flow hemodynamics. Computer Methods in Biomechanics, Biomed. Eng., (4) (2002), 319328.Google Scholar
[18]Ivanov, D.S. and Zhigilei, L.V., Combined atomistic-continuum modeling of short-pulse laser melting and disintegration of metal films, Phys. Rev. B, 68 (2003), 064114.Google Scholar
[19]Kleinert, H., Multivalued Fields in Condensed Matter, Electromagnetism and Gravitation, World Scientific, Singapore (2008).Google Scholar
[20]LaDisa, J.F. Jr., Olson, L.E., Guler, I., Hettrick, D.A., Kersten, J.R., Warltier, D.C. and Pagel, P.S., Circumferential vascular deformation after stent implantation alters wall shear stress evaluated with time-dependent 3D computational fluid dynamics models, J. Appl. Physiol., 98 (2005), 947957.Google Scholar
[21]Landau, L.D. and Lifshitz, E.M., The Classical Theory of Fields, 4th Ed., Elsevier (1975).Google Scholar
[22]Landau, L.D. and Lifshitz, E.M., Fluid Mechanics, 2nd ed., Elsevier (2004).Google Scholar
[23]Lei, M., , M., Giddens, D. P., Jones, S.A., Loth, F., and Bassiouny, H.J., Pulsatile flow in an end-to-side vascular graft model: comparison of computations with experimental data, Biomec. Eng., 123 (2001), 8087.CrossRefGoogle Scholar
[24]Li, M. and Selinger, R.L.B., Molecular dynamics simulations of dislocation instability in a stress gradient, Phys. Rev. B, 67 (2003), 134108.Google Scholar
[25]Meng, H., Tutino, V.M., Xiang, J. and Siddiqui, A., High WSS or Low WSS? Complex Interactions of Hemodynamics with Intracranial Aneurysm Initiation, Growth, and Rupture: Toward a Unifying Hypothesis, Am. J. Neuroradiol., 35, (2014), 12541262.Google Scholar
[26]Murphy, J. and Boyle, F., Predicting neointimal hyperplasia in stented arteries using time-dependant computational fluid dynamics: a review, Comput. Biol. Med., 40 (2010), 408418.Google Scholar
[27]Nagel, T., Resnick, N., Dewey, C.F. Jr. and Gimbrone, M.A. Jr., Vascular endothelial cells respond to spatial gradients in fluid shear stress by enhanced activation of transcription factors, Arterioscler. Thromb. Vasc. Biol., 19 (1999), 18251834.Google Scholar
[28]Pauchard, L., Adda-Bedia, M., Allain, C. and Couder, Y., Morphologies resulting from the directional propagation of fractures, Phys. Rev. E, 67 (2003), 027103.Google Scholar
[29]Peng, J., Ji, V., Seiler, W., Tomescu, A., Levesque, A. and Bouteville, A., Residual stress gradient analysis by the GIXRD method on CVD tantalum thin films, Surf. Coat. Technol., 200 (2006), 27382743.Google Scholar
[30]Poisson, E., A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics, Cambridge University Press (2004).Google Scholar
[31]Rikhtegar, F., Wyss, C., Stok, K.S., Poulikakos, D., Mller, R. and Kurtcuoglu, V., Hemodynamics in coronary arteries with overlapping stents, Biomech, J.., 47 (2014), 505511.Google Scholar
[32]Salsac, A.V., Sparks, S.R., Chomaz, J.M. and Lasheras, J.C., Evolution of the wall shear stresses during the progressive enlargement of symmetric abdominal aortic aneurysms, J. Fluid Mech., 560 (2006), 1951.Google Scholar
[33]Sankaran, S., Moghadam, M.E., Kahn, A.M., Tseng, E.E., Guccione, J.M. and Mardsen, A.L., Patient-specific multiscale modeling of blood flow for coronary artery bypass graft surgery, Ann. Biomed. Eng., 40 (2012), 22282242.Google Scholar
[34]Swaminathan, T.N., Mukundakrishnan, K., Ayyaswamy, P.S. and Eckmann, D.M., Effect of a soluble surfactant on a finite-sized bubble motion in a blood vessel, J. Fluid Mech., 642 (2010), 509539.Google Scholar
[35]Tardy, Y., Resnick, N., Nagel, T., Gimbrone, M.A. Jr and Dewey, C.F., Shear stress gradients remodel endothelial monolayers in vitro via a cell proliferation-migration-loss cycle, Thromb. Vasc. Biol., 17 (1997), 31023106.Google Scholar
[36]Wells, D.R., Archie, J.P. Jr and Kleinstreuer, C., Effect of carotid artery geometry on the magnitude and distribution of wall shear stress gradients, J. Vasc. Surg., 23 (1996), 667678.Google Scholar
[37]Womersley, J.R., Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, J. Physiol., 127 (1955), 553563.Google Scholar