Isothermal Tube Flow of Non-Linear Viscoelastic Fluids, Part I: Constitutive Instabilities and the Longitudinal Field (original) (raw)
2012, International Journal of Engineering Science 56:111-126, 2012
Theories and attendant methodologies developed independently of thermodynamic considerations and set within a thermodynamic framework to derive rheological constitutive equations for viscoelastic fluids have been reviewed in their historical context. The stability of Maxwell-like differential and single integral type constitutive formulations in current use and their relationship to experimentally observed physical instabilities are reviewed in particular in the light of inherent Hadamard and dissipative type of instabilities they may be subject to as a consequence of defective constitutive formulations. The state of the art in predicting the longitudinal field, the pressure drop and the friction factors for the flow of generalized Newtonian and viscoelastic fluids in circular and non-circular straight tubes is reviewed.
Related papers
Isothermal tube flow of non-linear viscoelastic fluids. Part II: Transversal field
International Journal of Engineering Science, 2011
Theories and attendant methodologies developed independently of thermodynamic considerations and set within a thermodynamic framework to derive rheological constitutive equations for viscoelastic fluids have been reviewed in their historical context. The stability of Maxwell-like differential and single integral type constitutive formulations in current use and their relationship to experimentally observed physical instabilities are reviewed in particular in the light of inherent Hadamard and dissipative type of instabilities they may be subject to as a consequence of defective constitutive formulations. The state of the art in predicting the longitudinal field, the pressure drop and the friction factors for the flow of generalized Newtonian and viscoelastic fluids in circular and non-circular straight tubes is reviewed.
On the fully developed tube flow of a class of non-linear viscoelastic fluids
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.
2001
Previous experimental measurements and linear stability analyses of curvilinear shearing flows of viscoelastic fluids have shown that the combination of streamwise curvature and elastic normal stresses can lead to flow destabilization. Torsional shear flows of highly elastic fluids with closed streamlines can also accumulate heat from viscous dissipation resulting in nonuniformity in the temperature profile within the flow and nonlinearity in the viscometric properties of the fluid. Recently, it has been shown by Al-Mubaiyedh et al. ͓Phys. Fluids 11, 3217 ͑1999͔͒ that the inclusion of energetics in the linear stability analysis of viscoelastic Taylor-Couette flow can change the dominant mode of the purely elastic instability from a nonaxisymmetric and time-dependent secondary flow to an axisymmetric stationary Taylor-type toroidal vortex that more closely agrees with the stability characteristics observed experimentally. In this work, we present a detailed experimental study of the effect of viscous heating on the torsional steady shearing of elastic fluids between a rotating cone and plate and between two rotating coaxial parallel plates. Elastic effects in the flow are characterized by the Deborah number, De, while the magnitude of the viscous heating is characterized by the Nahme-Griffith number, Na. We show that the relative importance of these two competing effects can be quantified by a new dimensionless thermoelastic parameter, ⌰ϭNa 1/2 /De, which is a material property of a given viscoelastic fluid independent of the rate of deformation. By utilizing this thermoelastic number, experimental observations of viscoelastic flow stability in three different fluids and two different geometries over a range of temperatures can be rationalized and the critical conditions unified into a single flow stability diagram. The thermoelastic number is a function of the molecular weight of the polymer, the flow geometry, and the temperature of the test fluid. The experiments presented here were performed using test fluids consisting of three different high molecular weight monodisperse polystyrene solutions in various flow geometries and over a large range of temperatures. By systematically varying the temperature of the test fluid or the configuration of the test geometry, the thermoelastic number can be adjusted appreciably. When the characteristic time scale for viscous heating is much longer than the relaxation time of the test fluid ͑⌰Ӷ1͒ the critical conditions for the onset of the elastic instability are in good agreement with the predictions of isothermal linear stability analyses. As the thermoelastic number approaches a critical value, the strong temperature gradients induced by viscous heating reduce the elasticity of the test fluid and delay the onset of the instability. At even larger values of the thermoelastic parameter, viscous heating stabilizes the flow completely.
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.
CONSTITUTIVE-RELATED INSTABILITIES IN PIPE FLOW OF A VISCOELASTIC FLUID
2000
The analytical solution for the steady-state flow in a pipe of viscoelastic fluids obeying the complete Phan-Thien--Tanner constitutive equation with a linear stress coefficient is derived. The results include the radial profiles of all relevant stresses, of the axial velocity and of the viscosity. The pipe flow is found to be unstable when the pressure gradient exceeds a critical value
Isothermal Tube Fow of Non-linear Viscoelastic Fluids. Part II: Transversal Field
International Journal of Engineering Science 49(6):443-465, 2011
Constitutive criteria for the existence of secondary flows, similarities and analogies, developments in the history of transversal flows, recent research on secondary flows of dilute solutions in rotating pipes and channels and related drag reduction are reviewed in depth as well as research concerning fundamental aspects of transversal flows and industrial applications relevant to secondary flows.
Laminar flow of non-linear viscoelastic fluids in straight tubes of arbitrary contour
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.
Proceedings of the Eurotherm 2008 - 5th European Thermal-Sciences Conference, 18-22 May 2008, Eindhoven, the Netherlands, 2008
Heat transfer enhancement in steady pressure gradient driven laminar flow of a class of non-linear viscoelastic fluids in straight tubes of non-circular cross-section at constant temperature is discussed together with the flow structure, and the physics is clarified. The variation of the average Nusselt number Nu with the Weissenberg Wi and Reynolds Re numbers in cross-sections with n axes of symmetry is analysed. 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 and temperature fields in tubes with arbitrary cross-section. The base flow is the Newtonian field and is obtained at the leading order. Heat transfer enhancements represented by average Nusselt numbers of an order of magnitude larger as compared to their Newtonian counterparts are predicted as a function of the Reynolds and Weissenberg numbers even for slightly non-Newtonian dilute fluids. The asymptotic independence of Nu = f(Pe,Wi) → Nu= f(Pe) with increasing Wi is shown analytically for the first time. The implications on the heat transfer enhancement of the change of type of the vorticity equation is discussed in particular for slight deviations from Newtonian behaviour where a rapid rise in enhancement seems to occur as opposed to the behaviour for larger values of the Weissenberg number where the rate of increase is much slower. The coupling between viscoelastic and inertial nonlinearities is crucial to enhancement. Fluid vorticity will change type when the velocity in the centre of the tube is larger than a critical value defined by the propagation of the shear waves. The asymptotic independence of Nu from elasticity with increasing Wi is related to the thickness of the supercritical region around the tube axis controlled by the interaction of the viscoelastic Mach number M and the Elasticity number E. The physics of the interaction of the effects of the Elasticity E, Viscoelastic Mach M, Reynolds Re and Weissenberg Wi numbers on generating the heat transfer enhancement is discussed.
Secondary Flows of Viscoelastic Fluids in Tubes
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.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.