On the Flow of Viscoplastic Fluids in Non-Circular Tubes

International Journal of Heat and Mass Transfer, 2011

The fully developed steady velocity field in pressure gradient driven laminar flow of non-linear viscoelastic fluids with instantaneous elasticity constitutively represented by a class of single mode, non-affine quasilinear constitutive equations is investigated in straight pipes of arbitrary contour @D. A continuous one-to-one mapping is used to obtain arbitrary tube contours from a base tube contour @D 0 . The analytical method presented is capable of predicting the velocity field in tubes with arbitrary cross-section. The base flow is the Newtonian field and is obtained at O(1). Field variables are expanded in asymptotic series in terms of the Weissenberg number Wi. The analysis does not place any restrictions on the smallness of the driving pressure gradients which can be large and applies to dilute and weakly elastic non-linear viscoelastic fluids. The velocity field is investigated up to and including the third order in Wi. The Newtonian field in general arbitrary contours is obtained and longitudinal velocity field components due to shearthinning and to non-linear viscoelastic effects are identified. Third order analysis shows a further contribution to the longitudinal field driven by first normal stress differences. Secondary flows driven by unbalanced second normal stresses in the cross-section manifest themselves as well at this order. Longitudinal equal velocity contours, the secondary flow field structure, the first and the second normal stress differences as well as wall shear stress variations are discussed for several non-circular contours some for the first time.

Conference: ASME IMECE2014 (American Society of Mechanical Engineers - International Mechanical Engineering Conference and Exposition 2014), Montréal, Québec, Canada, November 14-20, 2014, Proc. ASME. 46545; Volume 7: Fluids Engineering Systems and Technologies, paper IMECE2014- 36246, V007T09A033, 2014

ABSTRACT: An analytical method for determining the velocity field, shear stress and energy dissipation in viscoplastic flow in non-circular straight tubes is presented. Bingham constitutive model is used. Flow in tubes is considered. The cross-sectional contours can be arbitrarily chosen through a shape factor imposed on the solution for the longitudinal velocity. The analysis is extended to steady flow in tubes in which the cross-section contour exhibits sharp corners. In these cases three flow zones are distinguished: stagnant, non-zero deformation, and plug zones. The method provides the expressions for determining the boundaries and characteristics of those three zones for a wide variety of cross-section shapes. In particular the dynamics of plug-zones for large values of the yield stress and for contours that markedly differ from circumferences is analyzed. Energy dissipation is determined throughout the entire cross-section, so that the effect of shape on mechanical energy loss is assessed in terms of the yield stress and viscosity of the fluid. Some general expressions that help understand energy dissipation mechanisms are derived by using natural coordinates for the velocity field and related variables. These results draw on several recent works from other researchers and the present authors, which have highlighted the significant difficulty of determining the zones of zero deformation in viscoplastic flow when the related solid boundaries are not elementary.

Journal of Non-Newtonian Fluid Mechanics, 2009

a b s t r a c t Drag flow past circular cylinders concentrically placed in a tube filled with viscoplastic fluids, obeying the Herschel-Bulkley model, is analyzed via numerical simulations with the finite element method. The purpose it to find limiting drag values for cessation of motion of the object in steady flow. Different aspect ratios have been studied ranging from a disk to a long cylinder. For the simulations, the viscoplastic model is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in the flow field in both yielded and practically unyielded regions. The extent and shape of yielded/unyielded regions are determined along with the drag coefficient for a wide range of Bingham numbers. The simulation results are compared with previous experimental values [L. Jossic, A. Magnin, AIChE J. 47 (2001) 2666-2672] for cessation of flow. They show that the values of the drag coefficient are lowest for the disk and highest for the long cylinder. Discrepancies are found and discussed between the simulations and the experiments, with the simulations providing lower values in the limit of very high Bingham numbers.

ISBN: 978-0-7918-4954-5; Proc. ASME. 46545; Volume 7: Fluids Engineering Systems and Technologies, paper IMECE2014- 36246, V007T09A033, 2014

An analytical method for determining the velocity field, shear stress and energy dissipation in viscoplastic flow in non-circular straight tubes is presented. Bingham constitutive model is used. Flow in tubes is considered. The cross-sectional contours can be arbitrarily chosen through a shape factor imposed on the solution for the longitudinal velocity. The analysis is extended to steady flow in tubes in which the cross-section contour exhibits sharp corners. In these cases three flow zones are distinguished: stagnant, non-zero deformation, and plug zones. The method provides the expressions for determining the boundaries and characteristics of those three zones for a wide variety of cross-section shapes. In particular the dynamics of plug-zones for large values of the yield stress and for contours that markedly differ from circumferences is analyzed. Energy dissipation is determined throughout the entire cross-section, so that the effect of shape on mechanical energy loss is asses...

International Journal of Heat and Mass Transfer 55 (2012) 2731–2745, 2012

The fully developed steady velocity field in pressure gradient driven laminar flow of non-linear viscoelas-tic fluids with instantaneous elasticity constitutively represented by a class of single mode, non-affine quasilinear constitutive equations is investigated in straight pipes of arbitrary contour. A continuous one-to-one mapping is used to obtain arbitrary tube contours from a base tube contour. The analytical method presented is capable of predicting the velocity field in tubes with arbitrary cross-section. The base flow is the Newtonian field and is obtained at O(1). Field variables are expanded in asymptotic series in terms of the Weissenberg number Wi. The analysis does not place any restrictions on the smallness of the driving pressure gradients which can be large and applies to dilute and weakly elastic non-linear vis-coelastic fluids. The velocity field is investigated up to and including the third order in Wi. The Newtonian field in general arbitrary contours is obtained and longitudinal velocity field components due to shear-thinning and to non-linear viscoelastic effects are identified. Third order analysis shows a further contribution to the longitudinal field driven by first normal stress differences. Secondary flows driven by unbalanced second normal stresses in the cross-section manifest themselves as well at this order. Longitudinal equal velocity contours, the secondary flow field structure, the first and the second normal stress differences as well as wall shear stress variations are discussed for several non-circular contours some for the first time.

2003

The unsteady flow of the Green–Rivlin fluids in straight tubes of arbitrary cross-section driven by a pulsating pressure gradient is investigated. The non-linear constitutive structure defined by a series of nested integrals over semi-infinite time domains is perturbed simultaneously with the boundary of the base flow through a novel approach to domain mapping. The dominant primary component of the flow, the longitudinal field, and the much weaker transversal field arise at the first and the second orders of the analysis, respectively. The secondary field is driven by first order terms stemming from the linearly viscoelastic longitudinal flow at the first order. The domain mapping technique employed yields a continuous spectrum of unconventional closed cross-sectional shapes. We present longitudinal velocity profiles and transversal time-averaged, mean secondary flow streamline patterns for a specific fluid and for representative cross-sectional shapes in the spectrum the triangular, square and hexagonal shapes.

Advances in the Flow & Rheology of Non-Newtonian Fluids Edition: First Publisher: Elsevier Science BV, Amsterdam; Editors: Dennis A. Siginer and Daniel DeKee, 1999

Conduit flow is a common occurrence in many industrial and biological systems. It is also, in some cases, a convenient approximate model for studying fluid motion through porous media, filters, tissues, and other slow fluid motion in complex solid matrices. urrent technological advances in many fields require a good understanding of the dynamics of fluids other than Newtonian in conduit flow. This knowledge is necessary in order to estimate energy loss, transport properties, and many other variables of industrial interest. Viscoelastic fluids constitute an important class among non-Newtonian fluids, the study of which is rendered more difficult by several properties and phenomena exhibited by these fluids such as stress relaxation, strain recovery, die swell, normal stress differences, drag reduction, and flow enhancement. The flow of non-linear viscoelastic fluids in non-circular pipes may lead to the occurrence of secondary flows, a phenomenon not well covered in the technical literature. Secondary flows have a significant influence on important industrial phenomena, such as transport, and energy loss. A large number of competing viscoelastic constitutive models exist to predict flow phenomena. Integral models seem to predict experimental data better. In this chapter, the simple fluid of multiple integral-type models with fading memory is considered. Secondary flows are determined in the case of laminar longitudinal flows in approximately triangular and square conduits, when the flow is driven by small-amplitude oscillatory pressure gradients. The chapter is organized as follows. The mathematical background is developed and the summary of a novel analytical method devised by the first author and co-authors for determining the velocity field of laminar Newtonian unsteady flow in non-circular pipes is presented in Section 2. This is followed by an analysis in Section 3 of the pulsating flow in circular pipes of a viscoelastic fading memory fluid of the multiple integral type. Results of these sections are combined in the next section, where a mathematical expression for the axial velocity is developed for flow in non-circular pipes driven by a pressure gradient oscillating around a non-zero mean. The chapter closes with Section 5 where analytical steps that lead to the determination of the transversal velocity field are developed. Plots that depict the main features of axial and secondary flow fields are also presented.

Meccanica Vol. 53, Issue 1–2, pp 161–173, 2017

Flow of Bingham plastics through straight, long tubes with non-circular cross-sections is studied by means of an analytical method that allows to model a wide spectrum of tube geometries. Shear stress normal to equal velocity lines, velocity field and plug zones are explored, in particular in a tube with equilateral triangular cross-section for small values of the Bingham number Bi, and they are compared with corresponding numerical solutions. We show that a circular plug is present at the center of the triangular tube cross-section, consistent with numerical simulations as well as with previous results in the literature, if calculations up to and including first order in the shape perturbation parameter are taken into account. However with the inclusion of the second order terms in the algorithm this structure is no longer present and no plug zone is predicted for the same pressure drop. We find that in that case normal shear stresses are always greater than the yield stress of the fluid. As a result, the central region becomes a pseudo-plug since it presents small but non-vanishing relative deformations and does not move as a rigid core. The energy dissipation function for the Bingham fluid flow is written in terms of natural coordinates. Its distribution depends only on the normal shear stress at any point with Bingham number as a parameter.

Developments in the Flow of Complex Fluids in Tubes, 2015

This monograph together with its complimentary volume [Siginer, D. A., Stability of Non-linear Constitutive Formulations for Viscoelastic Fluids, Springer, New York, 2014] in this series is an attempt to give an overall comprehensive view of a complex field, only 60 or so years old, still far from being settled on firm grounds, that of the dynamics of viscoelastic fluid flow and suspension flow in tubes. The monograph on “Stability of Non-linear Constitutive Formulations for Viscoelastic Fluid Media” covers the development of constitutive equation formulations for viscoelastic fluids in their historical context together with the latest progress made, and this volume covers the state-of-the-art knowledge in predicting the flow of viscoelastic fluids and suspensions in tubes highlighting the historical as well as the most recent findings. Most if not all viscoelastic fluids in industrial manufacturing processes flow in laminar regime through tubes, which are not necessarily circular, at one time or another during the processing of the material. Laminar regime is by far the predominant flow mode for viscoelastic fluids encountered in manufacturing processes, and it is extensively covered in this monograph. Turbulent flow of dilute viscoelastic solutions is a topic which has not received much attention except when related to drag reduction. For particle-laden flows there are very interesting developments in both laminar and turbulent regime, and they are duly covered. It is critically important that the flow of non-linear viscoelastic fluids and suspensions in tubes can be predicted on a sound basis, thus the raison d’eˆtre of this volume. As flow behavior predictions are directly related to the constitutive formulations used, this volume relies heavily on the volume on [Siginer, D. A., Stability of Non-linear Constitutive Formulations for Viscoelastic Fluids, Springer, New York, 2014]. The science of rheology defined as the study of the deformation and flow of matter was virtually single-handedly founded and the name invented by Professor Bingham of Lafayette College in the late 1920s. Rheology is a wide encompassing science which covers the study of the deformation and flow of diverse materials such as polymers, suspensions, asphalt, lubricants, paints, plastics, rubber, and biofluids, all of which display non-Newtonian behavior when subjected to external stimuli and as a result deform and flow in a manner not predictable by Newtonian mechanics. The development of rheology, which had gotten to a slow start, took a boost during WWII as materials used in various applications, in flame throwers, for instance, were found to be viscoelastic. As Truesdell and Noll famously wrote [Truesdell, C. and Noll, W., the Non-Linear Field Theories of Mechanics, 2nd ed., Springer, Berlin, 1992] “By 1949 all work on the foundations of Rheology done before 1945 had been rendered obsolete.” In the years following WWII, the emergence and rapid growth of the synthetic fiber and polymer processing industries, appearance of liquid detergents, multigrade oils, non-drip paints, and contact adhesives, and developments in pharmaceutical and food industries and biotechnology spurred the development of rheology. All these examples clearly illustrate the relevance of rheological studies to life and industry. The reliance of all these fields on rheological studies is at the very basis of many if not all of the amazing developments and success stories ending up with many of the products used by the public at large in everyday life. Non-Newtonian fluid mechanics, which is an integral part of rheology, really made big strides only after WWII and has been developing at a rapid rate ever since. The development of reliable constitutive formulations to predict the behavior of flowing substances with non-linear stress–strain relationships is quite a difficult proposition by comparison with Newtonian fluid mechanics with linear stress–strain relationship. The latter does enjoy a head start of two centuries tracing back its inception to Newton and luminaries like Euler and Bernoulli. With the former the non-linear structure does not allow the merging of the constitutive equations for the stress components with the linear momentum equation as it is the case with Newtonian fluids ending up with the Navier–Stokes equations. Thus, the practitioner ends up with six additional scalar equations to be solved in three dimensions for the six independent components of the symmetric stress tensor. The difficulties in solving in tandem this set of non-linear field equations, which may involve both inertial and constitutive non-linearities, cannot be underestimated. Perhaps equally importantly at this point in time in the unfolding development of the science, we are not fortunate enough to have developed a single constitutive formulation for viscoelastic fluids, which may lend itself to most applications and yield reasonably accurate predictions together with the field balance equations. The field is littered with a plethora of equations, some of which may yield reasonable predictions in some flows and utterly unacceptable predictions in others. Thus, we end up with classes of equations for viscoelastic fluids that would apply to classes of flows and fluids, an ad hoc concept at best that hopefully will give way one day to a universal equation, which may apply to all fluids in all motions. In addition the stability of these equations is a very important issue. Any given constitutive equation should be stable in the Hadamard and dissipative sense and should not violate the basic principles of thermodynamics. Dynamics of tube flow of non-Brownian suspensions and its underpinning field turbulent motion of linear (Newtonian) fluids shows interesting similarities with the flow of viscoelastic fluids in that the secondary flows of viscoelastic fluids in laminar flow driven by unbalanced normal stresses have a counterpart in the turbulent motion of linear fluids in straight tubes of non-circular cross section and in the laminar motion of particle-laden linear fluids. The latter secondary flows are driven by normal stresses due to shear-induced migration of particles. This is a new topic of hot research thrust given its implications in applications. The direction of these normal stresses is opposite of those present in the flow field of a viscoelastic fluid. In fact the tying thread among these seemingly different motions is that all are driven by normal stresses. The turbulent flow of linear fluids is known to have a transversal field due to the anisotropy of the Reynolds stress tensor in non-circular cross sections which entails unbalanced normal Reynolds stresses in the cross section perpendicular to the axial direction. Secondary field also exists in the turbulent flow of linear fluids in circular cross sections if the symmetry is somehow broken due, for example, to unevenly distributed roughness on the boundary, which would again trigger anisotropy of the Reynolds stress tensor. It is not possible to develop a good understanding of the mechanics of the secondary field both in laminar and turbulent motion of particle-laden fluids without a clear grasp of the underlying mechanics of the turbulent secondary field of homogeneous linear fluids. Thus a complete review of both is presented including interesting constitutive similarities with viscoelastic fluids which do arise when certain non-linear closure approximations are made for the anisotropic part of the Reynolds stress tensor. The impact of the secondary flows on engineering calculations is particularly important as turbulent flows in ducts of non-circular cross section are often encountered in engineering practice. Some examples are flows in heat exchangers, ventilation and air-conditioning systems, nuclear reactors, impellers, blade passages, aircraft intakes, and turbomachinery. If neglected significant errors may be introduced in the design as secondary flows lead to additional friction losses and can shift the location of the maximum momentum transport from the duct centerline. The secondary velocity depends on cross-sectional coordinates alone and therefore is independent of end effects. It is only of the order of 1–3 % of the streamwise bulk velocity, but by transporting high-momentum fluid toward the corners, it distorts substantially the cross-sectional equal axial velocity lines; specifically it causes a bulging of the velocity contours toward the corners with important consequences such as considerable friction losses. The need for turbulence models that can reliably predict the secondary flows that may occur in engineering applications is of paramount importance. Efforts have not been spared to be thorough in the presentation with commentaries about the successes and failures of each theory and the reasons behind them. The link between..... Palapye, Botswana and Santiago, Chile Dennis A. Siginer

International Journal of Non-Linear Mechanics, 2005

The fully developed pipe flow of a class of non-linear viscoelastic fluids is investigated. Analytical expressions are derived for the stress components, the friction factor and the velocity field. The friction factor which depends on the Deborah and Reynolds numbers is substantially smaller than the corresponding value for the Newtonian flow field with implications concerning the volume flow rate. We show that non-affine models in the class of constitutive equations considered such as Johnson–Segalman and some versions of the Phan–Thien–Tanner models are not representative of physically realistic flow fields for all Deborah numbers. For a fixed value of the slippage factor they predict physically admissible flow fields only for a limited range of Deborah numbers smaller than a critical Deborah number. The latter is a function of the slippage.