A pseudo‐viscoelastic approach for transient polymeric flow exiting a channel

The influence of elasticity of a fluid exiting a channel is examined on transient coating downstream. A hybrid spectral/boundary element approach is proposed to solve the problem. The flow inside the channel is assumed to be fully developed. A viscoelastic instability of one‐dimensional plane Couette flow is first determined for a large class of Oldroyd fluids with added viscosity, which typically represent polymer solutions composed of a Newtonian solvent and a polymeric solute. The Johnson–Segalman equation is used as the constitutive model. The velocity profile inside the channel is taken as the exit profile for the emerging free‐surface flow. The flow is assumed to be Newtonian as it emerges from the channel. An estimate of the magnitude of the rate‐of‐strain tensor components in the free‐surface region reveals that they are generally smaller than the shear rate inside the channel. The evolution of the flow front is simulated using the boundary element method. For the channel flow, the problem is reduced to a nonlinear dynamical system using the Galerkin projection method. Stability analysis indicates that the channel velocity may be linear or non‐linear depending on the range of the Weissenberg number. The evolution of the coating flow at the exit is examined for steady as well as transient (monotonic and oscillatory) channel flow. It is found that adverse flow can exist as a result of fluid elasticity, which can hinder the process of blade coating. Copyright © 2003 John Wiley & Sons, Ltd.     

Transient free‐surface flow of a viscoelastic fluid in a narrow channel

The interplay between inertia and elasticity is examined for transient free‐surface flow inside a narrow channel. The lubrication theory is extended for the flow of viscoelastic fluids of the Oldroyd‐B type (consisting of a Newtonian solvent and a polymeric solute). While the general formulation accounts for non‐linearities stemming from inertia effects in the momentum conservation equation, and the upper‐convected terms in the constitutive equation, only the front movement contributes to non‐linear coupling for a flow inside a straight channel. In this case, it is possible to implement a spectral representation in the depthwise direction for the velocity and stress. The evolution of the flow field is obtained locally, but the front movement is captured only in the mean sense. The influence of inertia, elasticity and viscosity ratio is examined for pressure‐induced flow. The front appears to progress monotonically with time. However, the velocity and stress exhibit typically a strong overshoot upon inception, accompanied by a plug‐flow behaviour in the channel core. The flow intensity eventually diminishes with time, tending asymptotically to Poiseuille conditions. For highly elastic liquids the front movement becomes oscillatory, experiencing strong deceleration periodically. A multiple‐scale solution is obtained for fluids with no inertia and small elasticity. Comparison with the exact (numerical) solution indicates a wide range of validity for the analytical result. Copyright © 2004 John Wiley & Sons, Ltd.     

Homotopy Analysis Method to Walter’s B fluid in a vertical channel with porous wall

In this research the steady three-dimensional flow of a Walter’s B fluid in a vertical channel with porous wall, through which the fluid is injected uniformly into the channel through one side of the channel, is studied analytically using Homotopy Analysis Method (HAM). The channel is assumed to be infinite and uniform. The effects of the elasticity of the fluid on the flow and heat transfer on the walls of the channel are discussed.     

3D particle image velocimetry of the flow field around a sphere sedimenting near a wall: Part 1. Effects of Weissenberg number

The flow fields surrounding a sphere sedimenting through a liquid near a vertical wall are characterized using 3D stereoscopic particle-image velocimetry (PIV) experiments. Three different fluids, a Newtonian reference fluid, a constant (shear) viscosity Boger fluid, and a shear-thinning elastic fluid, are used to determine the effects of both elasticity and shear-thinning on the flow field. All three fluids have similar zero shear viscosities. The Weissenberg number is manipulated by varying the diameter and the composition of the ball. Significant differences are found for the different types of fluid, demonstrating both the influence of elasticity and shear-thinning on the velocity fields. In addition, the impact of the wall on the flow field is qualitatively different for each fluid. We find that the flow behind the sphere is strongly dependent on the fluid properties as well as the elasticity. Also, the presence of a negative wake is found for the shear-thinning fluid at high Weissenberg number (Wi > 1).     

Fluid flow and heat transfer in a two-dimensional wavy channel

A numerical study is presented for the laminar fully developed flow and heat transfer in a two-dimensional wavy channel. The effects of the geometry, Reynolds and Prandtl number on the flow field and heat transfer are investigated. The channel is characterized by a wavy wall, heated at uniform heat flux, and an opposite wall, being plane and adiabatic. The extent of the wall waviness and the distance between the channel walls are found to significantly affect the streamlines contours as well as the heat transfer coefficients. Comparisons with the straight channel, in the same flow rate and heat transfer conditions, have been performed. Pressure drop of the wavy channel is found to be always larger than the value characteristic of a straight channel, while heat transfer performance decreases or increases depending on the values of the parameters (geometry, Reynolds and Prandtl numbers).     

Local similarity solution for the flow of a “second-grade” viscoelastic fluid above a moving plate

The boundary layer flow of a viscoelastic fluid of the second-grade type over a rigid continuous plate moving through an otherwise quiescent fluid with constant velocity U is studied. Assuming the flow to be laminar and two-dimensional, local similarity solution is found with fluid's elasticity and plate's withdrawal speed as the main variables. Results are presented for velocity profiles, boundary layer thickness, wall skin friction coefficient and fluid entrainment in terms of the local Deborah number. A marked formation of boundary layer is predicted, even at low Reynolds numbers, provided the Deborah number is sufficiently large. The boundary layer thickness and the wall skin friction coefficient are found to scale with fluid's elasticity—both decreasing the higher the fluid's elasticity. The amount of fluid entrained is also predicted to decrease whenever a fluid exhibits elastic behavior.     

Stagnation-point flow of a viscoelastic fluid towards a stretching surface

An analysis is made of the steady two-dimensional stagnation-point flow of an incompressible viscoelastic fluid over a flat deformable surface when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. It is shown that for a viscoelastic fluid of short memory (obeying Walters’ B′ model), a boundary layer is formed when the stretching velocity of the surface is less than the inviscid free-stream velocity and velocity at a point increases with increase in the elasticity of the fluid. On the other hand, an inverted boundary layer is formed when the surface stretching velocity exceeds the velocity of the free stream and the velocity decreases with increase in the elasticity of the fluid. A novel result of the analysis is that the flow near the stretching surface is that corresponding to an inviscid stagnation-point flow when the surface stretching velocity is equal to the velocity of the free stream. Temperature distribution in the boundary layer is found when the surface is held at constant temperature and surface heat flux is determined. It is found that temperature at a point decreases with increase in the elasticity of the fluid.     

Rheology of a suspension of particles in viscoelastic fluids

In this paper we study the bulk stress of a suspension of rigid particles in viscoelastic fluids. We first apply the theoretical framework provided by Batchelor [J. Fluid Mech. 41 (1970) 545] to derive an analytical expression for the bulk stress of a suspension of rigid particles in a second-order fluid under the limit of dilute and creeping flow conditions. The application of the suspension balance model using this analytical expression leads to the prediction of the migration of particles towards the centerline of the channel in pressure-driven flows. This is in agreement with experimental observations. We next examine the effects of inertia (or flow Reynolds number) on the rheology of dilute suspensions in Oldroyd-B fluids by two-dimensional direct numerical simulations. Simulation results are verified by comparing them with the analytical expression in the creeping flow limit. It is seen that the particle contribution to the first normal stress difference is positive and increases with the elasticity of the fluid and the Reynolds number. The ratio of the first normal stress coefficient of the suspension and the suspending fluid decreases as the Reynolds number is increased. The effective viscosity of the suspension shows a shear-thinning behavior (in spite of a non-shear-thinning suspending fluid) which becomes more pronounced as the fluid elasticity increases.     

Stabilization of small perturbations of Poiseuille flow in a channel with elastic walls

Arbitrary three-dimensional perturbations are considered. It is established that as the compliance of the walls increases, oblique waves become the most dangerous, which essentially differentiates the system in question from Poiseuille flow in a rigid channel. The flow stability is analyzed over a broad interval of values of the elasticity parameter overlapping the values for real materials.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 67–72, May–June, 1988.     

On the interplay between inertial and viscoelastic effects for the flow in weakly modulated channels

The flow inside a spatially modulated channel is examined for viscoelastic fluids of the Oldroyd‐B type. The lower wall is flat and the upper wall is sinusoidally modulated. The modulation amplitude is assumed to be small. Thus, a regular perturbation expansion of the flow field coupled to a variable‐step finite‐difference scheme is used to solve the problem. Convergence and accuracy assessment against earlier experimental results indicate that there is a significant range of validity of the perturbation approach. The influences of wall geometry, inertia and viscoelasticity on the flow kinematics and stresses are investigated systematically. In particular, the interplay between the flow and fluid parameters effects on the conditions for the onset of backflow, number of vortices, their size and location is revealed. The distance between the flow separation and reattachment locations identifies the vortex size. Non‐monotonic dependence of the vortex size on elasticity is reported. The critical conditions for the onset of negative elasticity effects on vortex size are identified. The critical Reynolds number for the onset of backflow initially decreases then levels off or even increases as elasticity increases. For highly elastic fluid and large enough Reynolds number, more than one vortex appear near the lower wall. Copyright © 2005 John Wiley & Sons, Ltd.     

Hydrodynamically induced cross stream migration of dissolved macromolecules (modelled as nonlinearly elastic dumbbells)

The hydrodynamically induced cross-stream migration of a nonlinear elastic dumbbell in nonhomogeneous flows is considered. A generally valid expression for the migration velocity is derived and the result applied to viscometric flows. Effects of elasticity and size of the particle, as well as the strength of the flow field, are studied in detail for channel flow and for circular Couette flow. Estimates of the dynamics of developments of non-uniform concentration profiles are discussed.     

Motion of a circular cylinder in a viscous liquid between parallel plates

The two-dimensional motion of a cylinder in a viscous fluid between two parallel walls of a vertical channel is studied. It is found that when the cylinder moves very closely along one of the channel walls, it always rotates in the direction opposite to that of contact rolling along the nearest wall. When the cylinder is away from the walls, its rotation depends on the Reynolds number of the flow. In this study two numerical methods were used. One is for the unsteady motion of a sedimenting cylinder initially released from a position close to one of the channel walls, where the Navier-Stokes equations are solved for the fluid and Newton's equations of motion are solved for the rigid cylinder. The other method is for the steady flow in which a cylinder is fixed in a uniform flow field where the channel walls are sliding past the cylinder at the speed of the approaching flow, or equivalently a cylinder is moving with a constant velocity in a quiescent fluid. The flow field, the drag, the side force (lift), and the torque experienced by the cylinder are studied in detail. The effects of the cylinder location in the channel, the size of the channel relative to the cylinder diameter, and the Reynolds number of the flow are examined. In the limit when the cylinder is translating very closely along one of the walls, the flow in the gap between the cylinder and the wall is solved analytically using lubrication theory, and the numerical solution in the other region is used to piece together the whole flow field.This research was supported by NSF DMR91-20668 through the Laboratory for Research on the Structure of Matter at the University of Pennsylvania and from the Research Foundation of the University of Pennsylvania.     

An experimental investigation of the flow structure over a corrugated waveform in a transpired air collector

An experimental investigation of the flow dynamics in a channel with a corrugated surface is presented. Particle image velocimetry was used to obtain two-dimensional velocity fields at three different locations along the channel length, over a range of Reynolds numbers. The results show a significant impact of the corrugation waveform on the mean and turbulent flow structure inside the channel. Strong bursting flow originating from the trough, sweeping flow from the bulk region and the vortex shedding off the crest were observed. Their interactions created a complex three-dimensional flow structure extended over almost the entire channel. The mean velocity profiles indicate a strong diffusion of shear. The profiles of various turbulent properties show the enhancement of turbulence in the vicinity of the waveform. It was found that the turbulence in the channel was almost entirely produced in this region above the corrugation trough. Significant momentum transfer from the corrugation wall by the turbulent velocity field was also observed. The mean and turbulent flow behaviour was found to be periodic with respect to the waveform over most of the channel length. The results show the presence of strong turbulence even at the Reynolds number that falls within the conventional laminar range.     

DNS of the Turbulent Channel Flow of a Dilute Polymer Solution

A direct numerical simulation of the turbulent channel flow of a dilute polymer solution has been performed in order to compare its turbulence statistics with those obtained in a Newtonian channel flow. The viscoelastic flow has been simulated by solving the whole set of continuity, momentum and constitutive equations for the six independent components of the extra-stress tensor induced by polymer addition. The Finitely Extensible Nonlinear Elastic dumbbell model was adopted in order to simulate a non-linear modulus of elasticity and a finite extendibility of the polymer macromolecules. Simulations were carried out under the narrow channel assumption at a Reynolds number of 169 based on the channel half height and on the friction velocity; they showed a significant reduction in drag, dependent on the influence of the elastic properties of the chains. A qualitative comparison with experiments at a higher Reynolds number has shown that the model here adopted is capable of reproducing all the main features of the polymer solution flow. Analysis of the turbulence statistics suggests that a dilute polymer solution can affect the intensity of the streamwise vortices, leading to an increase in the spacing between low speed streaks and eventually to a turbulent shear stress reduction.     

Warm up of an automotive catalyst substrate by pulsating flow: a single channel modelling approach

Rapid warm up of an automotive catalyst substrate is important for early light off. This work considers the results from a model of warm up in a single channel. The mass flow is pulsating with high amplitude, about 75% of mean flow, but without flow reversal. The flow regime is laminar within the channel. Pulsations occur with frequency in the range 16–100 Hz, and are important in close-coupled systems where the catalyst is located near to the engine and where the rate of rise of gas inlet temperature with time is rapid, about 15 K/s. The use of a single channel model with conjugate heat transfer enables the heat transfer coefficient to be evaluated and compared with results from steady flow simulations. The value of the augmentation factor based on heat flux is found to be less than unity. The value of the augmentation factor based on heat transfer coefficient depends on the method for calculating the mean heat transfer coefficient, but is generally less than unity. The changes caused by pulsations will be small in practical systems. Changes in wall temperature found in the simulations are the result of the cumulative effect of changes in the mass flow rate.     

Kinematic Parameters of a Two-Phase Flow in a Rectangular Channel

A kinematic two-phase flow pattern formed in a rectangular channel due to the interaction of a gas flow with an initially stationary or moving water layer is investigated. Using laser diagnostics and hot-wire methods, the velocity distributions in the water and the air are found for a stratified flow regime.     

Two-dimensional study of drop deformation under simple shear for Oldroyd-B liquids

The effect of viscoelasticity on the deformation of a circular drop suspended in a second liquid in shear is investigated with direct numerical simulations. A numerical algorithm based on the volume-of-fluid method for interface tracking is implemented in two dimensions with the Oldroyd-B constitutive model for viscoelastic liquids. The code is verified against a normal mode analysis for the stability of two-layer flow in a channel; theoretical growth rates are reproduced for the interface height, velocity and stress components. Drop simulations are performed for drop and matrix liquids of different viscosities and elasticities. A new feature is found for the case of equal viscosity, when the matrix liquid is highly elastic and surface tension is low; hook-like structures form at the drop tips. This is due to the growth of first normal stress differences that occur slightly above the front tip and below the back tip as the matrix elasticity increases above a threshold value.     

Non-Local Interactions and Feedback Instability in a High Reynolds Number Flow

The question of non-locality is considered for a model supersonic flow at high Reynolds number in a channel formed between two parallel plates of different length, using the channel length as a control parameter. Examples are given of time-periodic stable and unstable flows forced by a disturbance positioned in the middle of the channel. It is shown that in certain parameter ranges the flow in a channel of ever increasing length is not approximated by the solutions obtained for infinitely long channels. This is interpreted in terms of a feedback interaction between the flow near the channel ends and the disturbance source. Feedback is shown to result from a slow upstream decay of disturbances coupled with a relatively fast downstream growth of instability waves. For a free (non-forced) flow, the feedback is found to lead to a form of global or resonant instability. Examples of growth rate calculations for the feedback modes are given.     

Darcy–Brinkman Flow Through a Corrugated Channel

A perturbation analysis is carried out to the second order to give effective equations for Darcy–Brinkman flow through a porous channel with slightly corrugated walls. The flow is either parallel or normal to the corrugations, and the corrugations of the two walls are either in phase or half-period out of phase. The present study is based on the assumptions that the corrugations are periodic sinusoidal waves of small amplitude, and the channel is filled with a sparse porous medium so that the flow can be described by the Darcy–Brinkman model, which approaches the Darcian or Stokes flow limits for small or large permeability of the medium. The Reynolds number is also assumed to be so low that the nonlinear inertia can be ignored. The effects of the corrugations on the flow are examined, quantitatively and qualitatively, as functions of the flow direction, the phase difference, and the wavelength of the corrugations, as well as the permeability of the channel. It is found that the corrugations will have greater effects when it is nearer the Stokes’ flow limit than the Darcian flow limit, and when the wavelength is shorter. For the same wavelength and phase difference, cross flow is more affected than longitudinal flow by the corrugations. Opposite effects can result from 180° out-of-phase corrugations, depending on the flow direction, the wavelength, as well as the permeability.     

Gravity-flow structure in a miscible fluid

Steady-state density flows in a horizontal channel are studied based on a two-layer shallow water model, developed by the author, with allowance for the mixing between the layers. The structure of a gravity flow and the intensity of mixing in the flow head are shown to depend significantly on the channel depth. Conditions behind the flow front, which determine the basic characteristics of a gravity flow, are found. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 79–85, May–June, 1998.