Slip yield stress effects in start-up Newtonian Poiseuille flows.pdf

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Slip yield stress effects in start-up Newtonian Poiseuille flows.pdf

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Slip yield stress effects in start-up Newtonian Poiseuille flows

Rheol Acta DOI 10.1007/s00397-013-0730-6 ORIGINAL CONTRIBUTION Slip yield stress effects in start-up Newtonian Poiseuille flows George Kaoullas · Georgios C. Georgiou Received: 30 April 2013 / Revised: 9 July 2013 / Accepted: 21 August 2013 ? Springer-Verlag Berlin Heidelberg 2013 Abstract Analytical solutions are derived for various start- up Newtonian Poiseuille flows assuming that slip at the wall occurs when the wall shear stress exceeds a critical value, known as the slip yield stress. Two distinct regimes charac- terise the steady axisymmetric and planar flows, which are defined by a critical value of the pressure gradient. If the imposed pressure gradient is below this critical value, the classical no-slip, start-up solution holds. Otherwise, no-slip flow occurs only initially, for a finite time interval deter- mined by a critical time, after which slip does occur. For the annular case, there is an additional intermediate (steady) flow regime where slip occurs only at the inner wall, and hence, there exist two critical values of the pressure gra- dient. If the applied pressure gradient exceeds both critical values, the velocity evolves initially with no-slip at both walls up to the first critical time, then with slip only along the inner wall up to the second critical time and finally with slip at both walls. Keywords Newtonian fluid · Poiseuille flow · Start-up flow · Navier slip · Slip yield stress G. Kaoullas () Oceanography Centre, University of Cyprus, PO Box 20537, 1678 Nicosia, Cyprus e-mail: g.kaoullas@ G. C. Georgiou Department of Mathematics and Statistics, University of Cyprus, PO Box 20537, 1678 Nicosia, Cyprus e-mail: georgios@ucy.ac.cy Introduction The following are the two main macroscopic boundary con- ditions used at the interface between a fluid and the wall: (a) the classical no-slip condition, where the fluid is assumed to stick at the wall, and (b) the slip condition where the veloc- ity of the fluid relative to that of the wall, known a