×

Apparent slip for an upper convected Maxwell fluid. (English) Zbl 1362.76003

Summary: In this study the flow field of a nonlocal, diffusive upper convected Maxwell fluid in a solvent undergoing shearing motion is revisited for pressure driven planar channel flow and the free boundary problem of a liquid layer on a solid substrate is investigated. For large ratios of the zero shear polymer viscosity to the solvent viscosity, channel flows exhibit boundary layers at the channel walls. In addition, for increasing stress diffusion the flow field away from the boundary layers undergoes a transition from a parabolic to a plug flow. Corresponding flow structures and transitions are found for the free boundary problem of a thin layer sheared along a solid substrate. Matched asymptotic expansions are used to first derive sharp-interface models describing the bulk flow with expressions for an apparent slip for the boundary conditions, obtained by matching to the flow in the boundary layers. For a thin-film geometry several asymptotic regimes are identified in terms of the order of magnitude of the stress diffusion, and corresponding new thin-film models with a slip boundary condition are derived.

MSC:

76A05 Non-Newtonian fluids
34E05 Asymptotic expansions of solutions to ordinary differential equations
76A20 Thin fluid films
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Adams, S. M. Fielding, and P. D. Olmsted, {\it The interplay between boundary conditions and flow geometries in shear banding: Hysteresis, band configurations and surface transitions}, J. Non-Newtonian Fluid Mech., 151 (2008), pp. 101-118, . · Zbl 1388.76066
[2] N. P. Adhikari and J. L. Goveas, {\it Effects of slip on the viscosity of polymer melts}, J. Polymer Sci.: Part B: Polymer Physics, 42 (2004), pp. 1888-1904, .
[3] A. Ajdari, {\it Slippage at a polymer/polymer interface: Entanglements and associated friction}, C.R. Acad. Sci., Ser. II, 317 (1993), pp. 1159-1163. · Zbl 0788.76009
[4] A. Ajdari, F. B. Wyart, P. G. de Gennes, L. Leibler, J. Viovy, and M. Rubinstein, {\it Slippage of an entangled polymer melt on a grafted surface}, Physica A, 204 (1994), pp. 17-39, .
[5] P. Ballesta, G. Petekidis, L. Isa, W. C. K. Poon, and R. Besseling, {\it Wall slip and flow of concentrated hard-sphere colloidal suspensions}, J. Rheol., 56 (2012), pp. 1005-1037, .
[6] L. Bécu, S. Manneville, and A. Colin, {\it Spatiotemporal dynamics of wormlike micelles under shear}, Phys. Rev. Lett., 93 (2004), 018301, .
[7] A. Bhardwaj, E. Miller, and J. Rothstein, {\it Filament stretching and capillary breakup extensional rheometry measurements of viscoelastic wormlike micelle solutions}, J. Rheol., 51 (2007), pp. 693-719, .
[8] A. Bhave, R. C. Armstrong, and R. A. Brown, {\it Kinetic theory and rheology of dilute, nonhomogeneous polymer solutions}, J. Chem. Phys., 95 (1991), pp. 2988-3000, .
[9] W. Black and M. D. Graham, {\it Slip, concentration fluctuations, and flow instability in sheared polymer solutions}, Macromolecules, 34 (2001), pp. 5731-5733, .
[10] F. Brochard-Wyart and P. G. de Gennes, {\it Shear-dependent slippage at a polymer/solid interface}, Langmuir, 8 (1992), pp. 3033-3037, .
[11] F. Brochard-Wyart and P. G. de Gennes, {\it Sliding molecules at a polymer/polymer interface}, C. R. Acad. Sci. Ser. II, 327 (1993), pp. 13-17.
[12] M. Cromer, L. P. Cook, and G. H. McKinley, {\it Pressure-driven flow of wormlike micellar solutions in rectilinear microchannels}, J. Non-Newtonian Fluid Mech., 166 (2011), pp. 180-193, . · Zbl 1281.76014
[13] A. W. El-Kareh and L. G. Leal, {\it The existence of solutions for all Deborah numbers for non-Newtonian fluids}, J. Non-Newtonian Fluid Mech., 33 (1989), pp. 257-287, . · Zbl 0679.76004
[14] R. Fetzer, K. Jacobs, A. Münch, B. Wagner, and T. P. Witelski, {\it New slip regimes and the shape of dewetting thin liquid films}, Phys. Rev. Lett., 95 (2005), 127801, .
[15] S. M. Fielding, {\it Complex dynamics of shear banded flows}, Soft Matter, 3 (2007), pp. 1262-1279, .
[16] H. P. Greenspan, {\it On the motion of a small viscous droplet that wets a surface}, J. Fluid Mech., 84 (1978), pp. 125-143, . · Zbl 0373.76040
[17] C.-M. Ho and Y.-C. Tai, {\it Micro-electro-mechanical systems (MEMS) and fluid flows}, Annu. Rev. Fluid Mech., 30 (1998), pp. 579-612, .
[18] C. Huh and L. Scriven, {\it Hydrodynamic model of steady movement of a solid/liquid/fluid contact line}, J. Colloid Interface Sci., 35 (1971), pp. 85-101, .
[19] S. Jachalski, A. Münch, and B. Wagner, {\it Thin-film models for viscoelastic liquid bi-layers}, WIAS Preprint 2187, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany, 2015.
[20] E. Lauga, M. P. Brenner, and H. A. Stone, {\it Microfluidics: The no-slip boundary condition}, in Handbook of Experimental Fluid Dynamics, Springer, Berlin, Heidelberg, 2007, pp. 1219-1240, .
[21] L. Léger, {\it Friction mechanisms and interfacial slip at fluid-solid interfaces}, J. Phys.: Condensed Matter, 15 (2003), pp. S19-S29.
[22] M. P. Lettinga and S. Manneville, {\it Competition between shear banding and wall slip in wormlike micelles}, Phys. Rev. Lett., 103 (2009), 248302, .
[23] C. Masselon, A. Colin, and P. D. Olmsted, {\it Influence of boundary conditions and confinement on nonlocal effects in flows of wormlike micellar systems}, Phys. Rev. E, 81 (2010), 021502, .
[24] V. G. Mavrantzas and A. N. Beris, {\it Theoretical study of wall effects on rheology of dilute polymer solutions}, J. Rheol., 36 (1992), pp. 175-213, .
[25] E. Miller and J. P. Rothstein, {\it Transient evolution of shear-banding wormlike micellar solutions}, J. Non-Newtonian Fluid Mechanics, 143 (2007), pp. 22-37, .
[26] A. Münch and B. Wagner, {\it Contact-line instability of dewetting thin films}, Phys. D, 209 (2005), pp. 178-190, . · Zbl 1077.76027
[27] A. Münch, B. Wagner, M. Rauscher, and R. Blossey, {\it A thin-film model for corotational Jeffreys fluids under strong slip}, Euro. Phys. J. E, 20 (2006), pp. 365-368, .
[28] A. Münch, B. Wagner, and T. P. Witelski, {\it Lubrication models with small to large slip lengths}, J. Eng. Math., 53 (2005), pp. 359-383, . · Zbl 1158.76321
[29] P. Neogi and C. A. Miller, {\it Spreading kinetics of a drop on a rough solid surface}, J. Colloid Interface Sci., 92 (1983), pp. 338-349, .
[30] P. D. Olmsted, {\it Perspectives on shear banding in complex fluids}, Rheol. Acta, 47 (2008), pp. 283-300, .
[31] P. D. Olmsted, O. Radulescu, and C. Y. D. Lu, {\it Johnson-Segalman model with a diffusion term in cylindrical Couette flow}, J. Rheol., 44 (2000), pp. 257-275, .
[32] S. E. Orchard, {\it On surface leveling in viscous liquids and gels}, Appl. Sci. Res. A, 11 (1962), pp. 451-464, . · Zbl 0112.41601
[33] H. A. Stone, A. D. Stroock, and A. Ajdari, {\it Engineering flows in small devices: Microfluidics toward a lab-on-a-chip}, Annu. Rev. Fluid Mech., 36 (2004), pp. 381-411, . · Zbl 1076.76076
[34] P. A. Vasquez, G. H. McKinley, and L. P. Cook, {\it A network scission model for wormlike micellar solutions}, J. Non-Newtonian Fluid Mech., 144 (2007), pp. 122-139, . · Zbl 1196.76021
[35] L. Zhou, G. H. McKinley, and L. P. Cook, {\it Wormlike micellar solutions: III. VCM model predictions in steady and transient shearing flows}, J. Non-Newtonian Fluid Mech., 211 (2014), pp. 70-83, .
[36] L. Zhou, P. A. Vasquez, L. Cook, and G. H. McKinley, {\it Modeling the inhomogeneous response and formation of shear bands in steady and transient flows of entangled fluids}, J. Rheol., 32 (2008), pp. 591-623, .
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.