A study of simple shearing flows in polar fluids

Summary In the present paper we have made a study of simple shearing flows in three polar fluids — (i) the model ofCondiff andDahler, (ii) the model ofEringen, and (iii) the model ofStokes. We have investigated in detail the components of stress, equations governing the motion and boundary conditions in the above fluids in flows induced by motion of boundaries or a pressure gradient. By means of tables, we have studied the common features of these fluids and related the fluid parameters involved in each case.     

Heat transfer to non-Newtonian flows over a cylinder in cross flow

Over a range of 102<Re^*<5800, 6.5<Pr^*<79, and 0.6<n<1, circumferential wall temperatures for water and aqueous polymer (purely viscous) solution flows over a smooth cylinder were measured experimentally. The cylinder was heated by passing direct electric current through it. Aqueous solutions of Carbopol 934 and EZ1 were used as power-law non-Newtonian fluids. The peripherally averaged heat transfer coefficient for purely viscous non-Newtonian fluids, at any fixed flow rate, decreases with increasing polymer concentration. A new correlation is proposed for predicting the peripherally averaged Nusselt number for power-law fluid flows over a heated cylinder in cross flow.     

Oscillatory rotationally symmetric loss of stability of nonisothermal couette flow

A study is made of the stability of nonisothermal Couette flow — steady flow of a viscous heat conducting fluid between two rotating concentric cylinders heated to different temperatures. The methods of perturbation theory are used to establish conditions sufficient for bifurcation of a neutral curve of oscillatory instability from the neutral curve of monotonic instability. Computer calculations show that for certain values of the parameters of the problem these conditions are realized and there is an oscillatory loss of stability of the nonisothermal Couette flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 76–80, January–February, 1984.I thank V. I. Yudovich for constant interest in the work.     

Onset of unsteady axi-symmetric laminar natural convection in a vertical cylindrical enclosure heated at the wall

In the present study laminar transition to oscillatory convection of fluids having different Prandtl numbers in a laterally heated vertical cylindrical enclosure for different aspect ratios (melt height to crucible radius) of 2–4 is investigated numerically for 0.01 ≤ Pr ≤ 10. Numerical solution to two-dimensional axisymmetric transient Navier Stokes equations and energy equation were solved by finite volume method using SIMPLE algorithm. Numerical results illustrate that there exists a critical Rayleigh number for each Prandtl number beyond which sustained laminar oscillatory flow sets in. The oscillatory regime was characterised by the oscillation of the average kinetic energy and average thermal energy of the melt. For a given aspect ratio, critical Rayleigh number increases with Pr upto 1 and then flattens. It was observed that for low Prandtl number fluids, Pr < 1.0, critical Rayleigh number is found to increase with increase in aspect ratio while for high Prandtl number fluids, Pr ≥ 1.0, it is found to decrease with increase in aspect ratio. The influence of aspect ratio on the transient behaviour of the melt volume below and above the critical Rayleigh number was studied.     

Numerical simulations of negatively buoyant jets in an immiscible fluid using the Particle Finite Element Method

Negatively buoyant jets consist in a dense fluid injected vertically upward into a lighter ambient fluid. The numerical simulation of this kind of buoyancy‐driven flows is challenging as it involves multiple fluids with different physical properties. In the case of immiscible fluids, it requires, in addition, to track the motion of the interface between fluids and accurately represent the discontinuities of the flow variables. In this paper, we investigate numerically the injection of a negatively buoyant jet into a homogenous immiscible ambient fluid using the Particle Finite Element Method and compare the two‐dimensional numerical results with experiments on the injection of a jet of dyed water through a nozzle in the base of a cylindrical tank containing rapeseed oil. In both simulations and experiments, the fountain inlet flow velocity and nozzle diameter have been varied to cover a wide range of Froude Fr and Reynolds Re numbers ( 0.1 < Fr < 30, 8 < Re < 1350), reproducing both weak and strong laminar fountains. The flow behaviors observed for the different numerical simulations fit in the regime map based on the Re and Fr values of the experiments, and the maximum fountain height is in good agreement with the experimental observations, suggesting that particle finite element method is a useful tool for the study of immiscible two‐fluid systems. Copyright © 2011 John Wiley & Sons, Ltd.     

A note on start-up and large amplitude oscillatory shear flow of multimode viscoelastic fluids

Start up of plane Couette flow and large amplitude oscillatory shear flow of single and multimode Maxwell fluids as well as Oldroyd-B fluids have been analyzed by analytical or semi-analytical procedures. The result of our analysis indicates that if a single or a multimode Maxwell fluid has a relaxation time comparable or smaller than the rate of change of force imparted on the fluid, then the fluid response is not singular as Elasticity Number (E ). However, if this is not the case, as E , perturbations of single and multimode Maxwell fluids give rise to highly oscillatory velocity and stress fields. Hence, their behavior is singular in this limit. Moreover, we have observed that transients in velocity and stresses that are caused by propagation of shear waves in Maxwell fluids are damped much more quickly in the presence of faster and faster relaxing modes. In addition, we have shown that the Oldroyd-B model gives rise to results quantitatively similar to multimode Maxwell fluids at times larger than the fastest relaxation time of the multimode Maxwell fluid. This suggests that the effect of fast relaxing modes is equivalent to viscous effects at times larger than the fastest relaxation time of the fluid. Moreover, the analysis of shear wave propagation in multimode Maxwell fluids clearly show that the dynamics of wave propagation are governed by an effective relaxation and viscosity spectra. Finally, no quasi-periodic or chaotic flows were observed as a result of interaction of shear waves in large amplitude oscillatory shear flows for any combination of frequency and amplitudes.     

Rheological characterization of complex fluids in electro-magnetic fields

The paper is focused on the experimental investigations and rheological characterisations in magnetic and electric fields of liquids based on water in crude oils emulsions, added with ferrofluids (two types of crude oils are used in experiments: asphaltic and paraffinic, respectively). The final samples disclose weakly effects in the presence of magnetic field (saturated magnetization: M_n < 300 [G]) and behave almost as isolators in electric field (conductivity: σ < 10⁻⁵ [S/m]). The main goal of the study is to explore to what extent rheometry of complex fluids in electric and magnetic fields is able to offer value information about the internal structure of the samples. The experimental results prove that anomalous rheological behaviour (thixotropy, non-monotonic flow curve or viscosity function) of a complex fluid (in our case, emulsions based on paraffinic oil) generate also thixotropic properties and non-monotonic answers in the presence ferrofluids, under low magnetic and/or electric fields intensity. Our prospective study suggests that novel experimental procedures based on interaction: electro-magnetic field–complex fluids can be developed, in order to determine indirectly some relevant rheological properties of the complex fluids with internal network structure.     

Unsteady flow of an elastico-viscous fluid in tubes of uniform cross-section

In this paper, the uniqueness of solution for internal bounded unsteady flows of a shortmemory fluid is first established. Closed-form solutions are then obtained for the equations characterizing flows of such fluids in circular and rectangular tubes of uniform cross-section under an arbitrary pressure gradient. Special cases including the oscillatory flow between two parallel plates are discussed.     

On corner flows of Oldroyd-B fluids

Exact series solutions for planar creeping flows of Oldroyd-B fluids in the neighbourhood of sharp corners are presented and discussed. Both reentrant and non-reentrant sectors are considered. For reentrant sectors it is shown that more than one type of series solution can exist formally, one type exhibiting Newtonian-like asymptotic behaviour at the corner, away from walls, and another type exhibiting the same kind of asymptotics as an Upper Convected Maxwell (UCM) fluid. The solutions which are Newtonian-like away from walls are shown to develop non-integrable stress singularities at the walls when the no-slip velocity boundary condition is imposed. These mathematical solutions are therefore inadmissible from the physical viewpoint under no-slip conditions. An inadmissible solution, with stress singularities which are not everywhere integrable, is identified among the solutions of UCM-type. For a 270° reentrant sector the radial behaviour of the normal stress is everywhere r^−0.613. In the viscometric region near a wall, the radial normal stress σ^rr behaves like (rε)^−0.613, where ε is the angle made with the wall. In addition σ^rθ is infinite (not integrable) at the wall even when r is non-zero. Another UCM-type solution has a normal stress behaviour away from walls which is r^−0.985 for 270° sector. Again, this solution has a non-integrable stress singularity and is therefore inadmissible. Finally, for non-reentrant sectors it is shown that the flow is always Newtonian-like away from walls.     

Orientation effects in extensional and squeezing flows of certain anisotropic fluids

The Ericksen-Leslie continuum theory of anisotropic fluids is here used to examine the behaviour of the orientation pattern within the bulk of a fluid that is undergoing an extensional-type of flow. The flow (which is irrotational and generally unsteady) is considered to be generated by application of prescribed normal stresses, and the orientation pattern (represented by a director field n) is taken to be spatially homogeneous but time-dependent. By means of a phase-plane analysis it is shown that, in stretching flows, the director eventually aligns parallel to the direction of imposed stretch, whereas in squeezing-type flows it eventually lies in the plane normal to the direction of squeezing. In both cases the lateral components of n may vary non-monotonically in time, before approaching their asymptotic values; also the lateral components of velocity may change sign during flow.A two-parameter classification is given of all possible modes of behaviour of these model fluids in these flows. Also analytical solutions are obtained for certain special cases, such as axisymmetric flow.     

Uniqueness of free-boundary flows under gravity

A class of steady potential flows of an ideal fluid is considered in which the fluid flows between fixed boundaries and then emerges as a jet with one free boundary. Gravity acts on the fluid perpendicularly to the direction of the jet at infinity downstream. An inverse Froude number α is defined in terms of the flux Q and the depth d of the fluid at the separation point. It is proved that for each α>0 there is at most one flow which reaches to a supercritical uniform stream depth at infinity downstream. Monotonicity properties are proved for various flow parameters, and the behaviour of the flow as α → 0 is described.     

Two-dimensional unsteady laminar flow of a power law fluid across a square cylinder

The two-dimensional and unsteady free stream flow of power law fluids past a long square cylinder has been investigated numerically in the range of conditions 60≤Re≤160 and 0.5≤n≤2.0. Over this range of Reynolds numbers, the flow is periodic in time. A semi-explicit finite volume method has been used on a non-uniform collocated grid arrangement to solve the governing equations. The global quantities such as drag coefficients, Strouhal number and the detailed kinematic variables like stream function, vorticity and so on, have been obtained for the above range of conditions. While, over this range of Reynolds number, the flow is known to be periodic in time for Newtonian fluids, a pseudo-periodic flow regime displaying more than one dominant frequency in the lift is observed for shear-thinning fluids. This seems to occur at Reynolds numbers of 120 and 140 for n=0.5 and 0.6, respectively. Broadly speaking, the smaller the value of the power law index, lower is the Reynolds number of the onset of the pseudo-periodic regime. This work is concerned only with the fully periodic regime and, therefore, the range of Reynolds numbers studied varies with the value of the power law index. Not withstanding this aspect, in particular here, the effects of Reynolds number and of the power law index have been elucidated in the unsteady laminar flow regime. The leading edge separation in shear-thinning fluids produces an increase in drag values with the increasing Reynolds number, while shear-thickening fluid behaviour delays this separation and shows the lowering of the drag coefficient with the Reynolds number. Also, the preliminary results suggest the transition from the steady to unsteady flow conditions to occur at lower Reynolds numbers in shear-thinning fluids than that in Newtonian fluids.     

Three-dimensional flow of Newtonian and Boger fluids in square–square contractions

The flow of a Newtonian fluid and a Boger fluid through sudden square–square contractions was investigated experimentally aiming to characterize the flow and provide quantitative data for benchmarking in a complex three-dimensional flow. Visualizations of the flow patterns were undertaken using streak-line photography, detailed velocity field measurements were conducted using particle image velocimetry (PIV) and pressure drop measurements were performed in various geometries with different contraction ratios. For the Newtonian fluid, the experimental results are compared with numerical simulations performed using a finite volume method, and excellent agreement is found for the range of Reynolds number tested (Re₂ ≤ 23). For the viscoelastic case, recirculations are still present upstream of the contraction but we also observe other complex flow patterns that are dependent on contraction ratio (CR) and Deborah number (De₂) for the range of conditions studied: CR = 2.4, 4, 8, 12 and De₂ ≤ 150. For low contraction ratios strong divergent flow is observed upstream of the contraction, whereas for high contraction ratios there is no upstream divergent flow, except in the vicinity of the re-entrant corner where a localized atypical divergent flow is observed. For all contraction ratios studied, at sufficiently high Deborah numbers, strong elastic vortex enhancement upstream of the contraction is observed, which leads to the onset of a periodic complex flow at higher flow rates. The vortices observed under steady flow are not closed, and fluid elasticity was found to modify the flow direction within the recirculations as compared to that found for Newtonian fluids. The entry pressure drop, quantified using a Couette correction, was found to increase with the Deborah number for the higher contraction ratios.     

Stability of flows of the Hamel type in channels with permeable walls

The stability of nonparallel flows of a viscous incompressible fluid in an expanding channel with permeable walls is studied. The fluid is supplied to the channel through the walls with a constant velocity v₀ and through the entrance cross section, where a Hamel velocity profile is assigned. The resulting flow in the channel depends on the ratio of flow rates of the mixing streams. This flow was studied through the solution of the Navier—Stokes equations by the finite-difference method. It is shown that for strong enough injection of fluid through the permeable walls and at a distance from the initial cross section of the channel the flow approaches the vortical flow of an ideal fluid studied in [1]. The steady-state solutions obtained were studied for stability in a linear approximation using a modified Orr—Sommerfeld equation in which the nonparallel nature of the flow and of the channel walls were taken into account. Such an approach to the study of the stability of nonparallel flows was used in [2] for self-similar Berman flow in a channel and in [3] for non-self-similar flows obtained through a numerical solution of the Navier—Stokes equations. The critical parameters *, R*, and Cr* at the point of loss of stability are presented as functions of the Reynolds number R₀, characterizing the injection of fluid through the walls, and the parameter , characterizing the type of Hamel flow. A comparison is made with the results of [4] on the stability of Hamel flows with R₀ = 0.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 125–129, November–December, 1977.The author thanks G.I. Petrov for a discussion of the results of the work at a seminar at the Institute of Mechanics of Moscow State University.     

Development and instability of steady convective flows in a square cavity heated from below and a field of vertically directed vibrational forces

The present paper is devoted to numerical investigation of the spatial structure and stability of secondary vibrational convective flows resulting from instability of the equilibrium of a fluid heated from below. Vibrations parallel to the vector of the gravitational force (vertical vibrations) are considered. As in earlier work [7–9], a region of finite size is used — a square cavity heated from below. It is shown that enhancement of the vibrational disturbance of the natural convective flow may either stabilize or destabilize flows with different spatial structures; it may also stabilize certain solutions of the system of convection equations that are unstable in the absence of vibrational forces. In addition, increase of the vibrational Rayleigh number can lead to a change of the mechanisms responsible for equilibrium instability and oscillatory instability of the secondary steady flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 9–18, March–April, 1991.I thank G. Z. Gershuni for assistance and extremely fruitful discussions of the results of the paper.     

Shear flows of a thixotropic fluid

A thixotropic fluid with a viscosity dependent on a structural parameter, which satisfies a very simple kinetic equation, is examined. The stability and evolution of shear flows of such a fluid are investigated. Some classes of problems for which approximate solutions can be obtained are considered. Solutions are obtained for problems of changes in structure in oscillatory Couette flows. The apparent viscoelasticity of thixotropic fluids is analyzed. Some aspects of the thixotropic behavior of blood are discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp.3–12, May–June, 1978.In conclusion, I thank S. A. Regirer for discussion of the work and useful comments.     

Super-stable parallel flows of multiple visco-plastic fluids

We consider the stability of a multi-layer plane Poiseuille flow of two Bingham fluids. It is shown that this two-fluid flow is frequently more stable than the equivalent flow of either fluid alone. This phenomenon of super-stability results only when the yield stress of the fluid next to the channel wall is larger than that of the fluid in the centre of the channel, which need not have a yield stress. Our result is in direct contrast to the stability of analogous flows of purely viscous generalised Newtonian fluids, for which short wavelength interfacial instabilities can be found at relatively low Reynolds numbers. The results imply the existence of parameter regimes where visco-plastic lubrication is possible, permitting transport of an inelastic generalised Newtonian fluid in the centre of a channel, lubricated at the walls by a visco-plastic fluid, travelling in a stable laminar flow at higher flow rates than would be possible for the single fluid alone.     

Investigations of wake flows of a flat plate in steady,oscillatory and combined flows

Numerical study on near wake flows of a flat plate in three kinds of oncoming flows is made by using the discrete vortex model and improved vorticity creation method. For steady oncoming flow, both gross and detailed features of the wake flow are calculated and discussed. Then, in harmonic oscillatory oncoming flow two different wake flow patterns withK _c=2,4 and 10 are obtained respectively. Our results present a new wake flow pattern for lowKc numbers (Kc<5) describing vortex shedding, pairing and moving in a period of the oscillatory flow starting from rest. The calculated drag and inertia force coefficients are closer to experimental data from the U-tube than the previous results of vortex simulation. For in-line combined oncoming flow the vortex lock-in and dynamic characteristics are simulated. The results are shown to be in good agreement with experiments. The project supported by National Natural Science Fundation of China and LNM of Institute of Mechanics. CAS     

Near critical phenomena in laminar boundary layers

Asymptotic analysis of boundary layer separation in the limit of large Reynolds number Re→∞ has shown that in a number of cases which are of importance from a practical point of view solutions of the resulting interaction equations describing two-dimensional (2-D) steady flows exist up to a limiting value Γ_c of the relevant controlling parameter Γ only while two branches of solutions exist in a regime Γ<Γ_c. The present study aims at a better understanding of near critical flows |Γ-Γ_c|→0 and in particular the changes of the flow behaviour associated with the passage of Γ through Γ_c.     

Boundary layer structure of oscillatory open-channel shallow flows over smooth and rough beds

The boundary layer structure of oscillatory shallow open channel flows has been studied in a wide flume. Fluorescence solution was released at a porous rough bed through a diffuser covered by gravel of 0.5 cm grain size. A planar laser-induced fluorescence (PLIF) system was used to visualise the dye plumes in both vertical and horizontal planes for a qualitative understanding of the roles of large-scale flow structures in mass transport. A variety of tests were conducted for a range of oscillatory periods (30–240 s), water depths (3–16 cm) and velocity amplitudes (0.027–0.325 m/s), which cover a wide range of oscillatory flows with Reynolds numbers Re _a varied from 0.3 × 10⁴ (laminar) to 2.1 × 10⁶ (fully turbulent). For quantitative investigation, a novel technique, namely combined laser-induced fluorescence (LIF) and 2D laser Doppler velocimetry (LDV) (LIF/LDV), was developed and used to measure the velocity and solute concentration simultaneously in a vertical plane over 50 cycles. From the dye plumes revealed by the PLIF in transitional flows, there are different patterns of flow structure and solute transport with three representative stages of acceleration, deceleration and flow reversal. In the acceleration stage, turbulence was suppressed with dye layers adhering to the surface with little vertical mass transport. In the deceleration stage, flame-like turbulent structures occurred when turbulence generation was prominent. This was investigated quantitatively by recording the percentage occurrence of the adhered smooth layers per cycle. For those smooth bed cases with Re _a < 1.8 × 10⁵, the adhered smooth dye layers type of boundary layer occupied 100% of the oscillation period. Over a sufficiently high Re _a , a rough bed can generate fully turbulent oscillatory flows without the appearance of adhering dye layers. Between these two extremes, a transitional flow regime occurs in a wide range of flow conditions: Re _a > 2.7 × 10⁴ over the rough bed and Re _a > 8.3 × 10⁶ over a smooth bed.