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Rheol Acta (2012) 51:511526DOI 10.1007/s00397-011-0610-x

ORIGINAL CONTRIBUTION

Laminar axisymmetric flow of a weakly compressibleviscoelastic fluid

Kostas D. Housiadas Georgios C. Georgiou Ioannis G. Mamoutos

Received: 13 August 2011 / Revised: 2 November 2011 / Accepted: 30 November 2011 / Published online: 24 December 2011 Springer-Verlag 2011

Abstract The combined effects of weak compressibilityand viscoelasticity in steady, isothermal, laminar ax-isymmetric Poiseuille flow are investigated. Viscoelas-ticity is taken into account by employing the Oldroyd-Bconstitutive model. The fluid is assumed to be weaklycompressible with a density that varies linearly withpressure. The flow problem is solved using a regu-lar perturbation scheme in terms of the dimensionlessisothermal compressibility parameter. The sequence ofpartial differential equations resulting from the pertur-bation procedure is solved analytically up to secondorder. The two-dimensional solution reveals the effectsof compressibility and the other dimensionless numbersand parameters in the flow. Expressions for the averagepressure drop, the volumetric flow rate, the total axialstress, as well as for the skin friction factor are alsoderived and discussed. The validity of other techniquesused to obtain approximate solutions of weakly com-pressible flows is also discussed in conjunction with thepresent results.

K. D. Housiadas (B) I. G. MamoutosDepartment of Mathematics, University of the Aegean,Karlovassi, Samos, 83200, Greecee-mail: [email protected]

G. C. GeorgiouDepartment of Mathematics and Statistics,University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus

Present Address:I. G. MamoutosDepartment of Marine Sciences, University of the Aegean,Mytilene, Lesvos, Greece

Keywords Pressure drop Compressibility Two-dimensional flow Oldroyd-B fluid Flow curve Pipe flow

Introduction

The importance of compressibility in non-Newtonianviscous flows has been underlined in many studiesduring the last decades. A measure of the fluid com-pressibility is the Mach number, Ma, which is definedas the ratio of the characteristic speed of the fluid tothe speed of sound in that fluid. A zero Mach numbercorresponds to incompressible flow, whereas for lowMach numbers (Ma

512 Rheol Acta (2012) 51:511526

on cessation of the piston movement can be almostentirely attributed to the compressibility of the melt.

Numerical simulations of viscous compressible flowshave been reported by various researchers in the pasttwo decades. Georgiou and Crochet (1994a, b) per-formed numerical simulations of the time-dependentNewtonian extrudate-swell problem with slip at thewall in order to verify the compressibility/slip mech-anism for the stickslip instability, i.e., that the com-bination of compressibility and nonlinear slip leads toself-sustained oscillations of the pressure drop and ofthe mass flow rate in the unstable regime. They alsopointed out that a very low fluid compressibility maynot have an effect on steady-state solutions but canchange dramatically flow dynamics. Taliadorou et al.(2007) presented similar simulations for a Carreau fluidand included the barrel region where the polymer meltis compressed and decompressed periodically. Guo andWu (1997, 1998) presented certain numerical resultsfor the non-isothermal flow of a compressible gas ina microtube by utilizing a simplified form of the gov-erning equations. They used a finite difference forwardmarching procedure and found that fluid compressibil-ity increases the skin friction coefficient. Valette et al.(2006) simulated time-dependent pressure driven flowsfor a polymer melt flowing within an entry and exit slitgeometry using the Rolie-Poly constitutive model andreported that their simulation gave an accurate descrip-tion of the experimental data. Moreover, the simula-tion predicted an initially unexpected time-dependentvariation of the absolute pressure. Taliadorou et al.(2008) simulated the extrusion of strongly compress-ible Newtonian liquids and found that compressibil-ity can lead to oscillatory steady-state free surfaces.Webster et al. (2004) introduced numerical algorithmsfor solving weakly compressible, highly viscous laminarNewtonian flows at low Mach numbers. They appliedtheir methods to the driven cavity and the contractionflow problems. Subsequently, Keshtiban et al. (2004,2005) and Belblidia et al. (2006) simulated the flowof weakly compressible Oldroyd-B fluids in entryexitflows in high-pressure-drop cases.

Only a few approximate analytical solutions for com-pressible viscous flows in capillaries/tubes and channelshave been reported in the literature. These have beenobtained following three basic techniques. The first ap-proach is the one-dimensional approximation in whichcross-sectional averaged quantities and equations areconsidered (Shapiro 1953). Since non-linear terms areaveraged, a closure of the resulting equations is nec-essary, which, however, introduces errors leading toerroneous predictions even at the leading order of thecompressibility (Schwartz 1987).

The second technique is the lubrication approxima-tion (Prudhomme et al. 1986; van den Berg et al. 1993;Harley et al. 1995; Zohar et al. 2002), valid for slowflows or flows in very long tubes and channels, so thatboth the velocity component and the pressure gradientin the transverse direction can be assumed to be zero.As will be demonstrated in the present work, theseassumptions introduce errors similar to those of theone-dimensional technique.

The third technique is a regular perturbation pro-cedure according to which the dependent flow vari-ables are expanded as series solutions in terms of asmall parameter related to the fluid compressibility(Schwartz 1987; Venerus 2006; Taliadorou et al. 2009a;Venerus and Bugajsky 2010; Housiadas and Georgiou2011). The perturbation technique involves fewer as-sumptions than the other techniques and leads to two-dimensional expressions for the axial velocity and pres-sure in Poiseuille flow and to non-zero radial velocity.

The limitations of the one-dimensional and lubrica-tion approximations for studying compressible New-tonian Poiseuille flows were pointed out by Schwartz(1987), who solved the weakly compressible Newtonianflow in a channel by using a fourth-order perturbationscheme based on the principle of slow variation. Healso assumed a zero bulk viscosity and that the massdensity of the fluid was proportional to the pressure(thermally perfect gas). The first two-dimensional as-ymptotic solution for weakly compressible Newtonianflow in a capillary was presented by Venerus (2006),who used a streamfunction/vorticity formulation and aregular perturbation scheme with the small parameterbeing the dimensionless isothermal compressibility pa-rameter to obtain a solution up to second order. Morerecently, Taliadorou et al. (2009a) proposed an anal-ogous perturbation scheme, but in a velocity/pressureformulation, and obtained up to second order the solu-tions for the compressible Poiseuille flow in both tubesand channels. The solution of the latter problem wasalso derived by Venerus and Bugajsky (2010) using thestream function/vorticity formulation.

All the above studies concerned only Newtonianflows. Viscoelasticity was taken into account only veryrecently by Housiadas and Georgiou (2011) who con-sidered the plane Poiseuille flow of an Oldroyd-Bfluid and extended the primary-variable perturbationscheme of Taliadorou et al. (2009a). The viscoelasticextra-stress tensor was an extra-field that was per-turbed. In the present work, we derive the second-order regular perturbation solution of the axisymmetricPoiseuille flow of a liquid following the Oldroyd-Bmodel. The limitations of this model are well known;however, we need to stress here that it is not possible

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to obtain analytical solutions with other, more realistic,constitutive equations, like the Phan-Thien and Tannerand Giesekus models.

Along with the analytical solution, we also offerthe resulting expressions for the average viscometricproperties of the fluid, the pressure drop, the volu-metric flow rate, and the difference of the total stressin the main flow direction between the exit and theentrance of the tube. Most importantly, the skin frictionfactor is also derived and discussed, extending thus theNewtonian results of Venerus (2006).

The rest of the paper is organized as follows. InGoverning equations, the conservation equations,the constitutive model, and the equation of stateare presented in both dimensional and dimension-less forms. The perturbation procedure is described inPerturbation solution and the analytical solution upto second order in terms of the compressibility parame-ter presented. In Results and discussion, criteria forthe validity of the perturbation solution are providedand important features of the solution are underlined.The main conclusions of this work are summarized inthe last section.

Governing equations

We consider the isothermal, steady, pressure-drivenflow of a weakly compressible viscoelastic fluid in a cir-cular tube with constant radius R and length L; notethat throughout the extra-star denotes a dimensionalquantity. Cylindrical coordinates is the natural choicefor describing the flow. For isothermal, steady flow,and neglecting gravity, the continuity and momentumequations are:

(u) = 0 (1)

u u = T (2)where is the mass density of the fluid, u is thevelocity vector, and T is the total stress tensor:

T = p I + s + (3)In Eq. 3, p is the total pressure, s is the constant zeroshear-rate (Newtonian) viscosity of the pure solvent,

is the augmented shear-rate tensor, is the additionalextra-stress tensor introduced due to the presence ofthe polymer, and I is the unit tensor. For a compres