and Applied Mechanics
56, 4, pp. 1205-1216, Warsaw 2018
DOI: 10.15632/jtam-pl.56.4.1205
Implications of inertia for hydroelastic instability of Herschel-Bulkley fluids in plane Poiseuille flow
-driven Herschel-Bulkley fluid passing through a two-dimensional channel lined with a po-
lymeric coating. The no-viscous hyperelastic polymeric coating is assumed to follow the
two-constant Mooney-Rivlin model. In this work, analytical basic solutions are determined
for both the polymeric gel and the fluid at very low Reynolds numbers. Next, the basic so-
lutions are subjected to infinitesimally-small, normal-mode perturbations. After eliminating
the nonlinear terms, two 4-th order differential equations are obtained. The equations with
appropriate boundary conditions are then numerically solved using the shooting method.
The results of the solution show that the inertia terms in the perturbed equations destabili-
ze the pressure-driven Herschel-Bulkley fluid flow. The investigation reveals that the elastic
parameter has a stabilizing effect on the flow. Also, based on the obtained results, the yield
stress, depending on the power-law index, has a stabilizing or destabilizing effect on the
flow. Since in this work the inertia terms are included in the pertinent governing equations,
therefore, the results of this study are much more realistic and reliable than previous works
in which inertia terms were absent. In addition, unlike the previous works, the present study
considers both the shear-thinning and shear-thickening types of fluids. Hence, the results of
this work embrace all the fluids which obey the Herschel-Bulkley model.
References
Babenko V.V., Kozlov L.F., 1972, Experimental investigation of hydrodynamic stability on
rigid and elastic damping surfaces, Journal of Hydraulic Research, 10, 383-408
Bird R.B., Armstrong R.C., Hassager O., 1987, Dynamics of Polymeric Liquids, 1, John
Wiley, New York
Chien W.L., Rising H., Ottino J.M., 1986, Laminar and chaotic mixing in several cavity flows,
Journal of Fluid Mechanics, 170, 355-377
Davies C., Carpenter P.W., 1997, Instabilities in a plane channel flow between compliant walls,
Journal of Fluid Mechanics, 352, 205-243
Drazin P.G., Reid W.H., 2004, Hydrodynamic Stability, 2nd edit., Cambridge University Press
Franjione J.G., Ottino J.M., 1992, Symmetry concepts for the geometric analysis of mixing
flows, Philosophical Transactions of the Royal Society A, 338, 301-323
Fu T.S., Joseph D.D., 1970, Linear stability of asymmetric flow in channels, Physics of Fluids, 13, 217-222
Gad-el-Hak M., 2002, Compliant coatings for drag reduction, Progress in Aerospace Sciences, 38, 77-99
Gkanis V., Kumar S., 2005, Stability of pressure-driven creeping flows in channels lined with a
nonlinear elastic solid, Journal of Fluid Mechanics, 524, 357-375
Jafargholinejad S., 2015, Hydroelastic instability of Herschel-Bulkley fluids in channel flows,
Ph.D. dissertation, Islamic Azad University
Jafargholinejad S., Najafi M., Sadeghy K., 2015, Hydroelastic instability of viscoplastic
fluids in planar channel flow, Journal of the Society of Rheology, Japan, 43, 5, 157-164
Jensen K.F., 1999, Micromechanical systems: status, challenges and opportunities, AIChE Journal, 45, 2051-2054
Kramer M.O., 1960, Boundary-layer stabilization by distributed damping, Journal of the Aerospace
Sciences, 27, 1, 69-69
Kramer M.O., 1960, Boundary layer stabilization by distributing damping, Journal of the American
Society for Naval Engineers, 72, 25-33
Kandlikar S.G., Willistein D.A., Borrelli J., 2005, Experimental evaluation of pressure
drop elements and fabricated nucleation sites for stabilizing flow boiling in microchannels, Third
International Conference on Microchannels and Minichannels, ASME Paper, ICMM2005-75197,
Toronto, Canada
Lai W.M., Rubin D., Krempl E., 2010, Introduction to Continuum Mechanics, 4th Ed., Elsevier
Lee K.C., Finlayson B.A., 1986, Stability of plane Poiseuille and Couette flow of a Maxwell
fluid, Journal of Non-Newtonian Fluid Mechanics, 21, 1, 65-78
Muralikrishnan R., Kumaran V., 2002, Experimental study of the instability of the viscous
flow past a flexible surface, Physics of Fluids, 14, 2, 775-780
Ottino J.M., 1989, The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge
University Press
Pourjafar M., Hamedi H., Sadeghy K., 2015, Stability of power-law fluids in creeping plane
Poiseuille: the effect of wall compliance, Journal of Non-Newtonian Fluid Mechanics, 216, 22-30