On the planar extensional viscosity of mobile liquids

An instrument is described in which steady planar orthogonal stagnation flows of the type proposed by Winter [1] are achieved with lubrication. From pressure readings on steady results acceptable Trouton ratios are achieved for a Newtonian sample and results are obtained for a ‘Boger’ fluid showing high levels of extensional viscosity with Trouton ratios approaching 4 in the limit of zero stretch rate.     

Fluid Sampling From Porous Rock to Determine Filtrate Contamination Profiles Around a Well Bore

Drilling fluid filtrate invades the pores of rock surrounding a well bore during drilling of the well, and contaminates the pore fluid originally within the rock pores. Models for the flow of contaminated pore fluid towards a sampling tool within the well bore are investigated, assuming that the filtrate has the same viscosity as the original pore fluid and that the wellbore radius is small compared to the depth of filtrate invasion. If the filtrate contamination in the fluid withdrawn from the rock is monitored as a function of the volume withdrawn, then it is shown that results can be inverted to give the radial distribution of filtrate around the well bore. A new generation of guarded sampling probes is then considered, and it is shown that the radial distribution of filtrate can be obtained by means of such a probe if the fraction of flow entering the central sampling region of the probe is small compared to that entering the concentric annular guard probe. The effects of dispersion, non-zero wellbore radius and anisotropic hydraulic permeability of the rock are also studied, and numerical simulations are used to give some indication of the effect of the ratio of the filtrate viscosity to that of the original pore fluid.     

Cavity detachment on a hydrofoil with the inclusion of surface tension effects

The non-linear problem of cavity flow past a hydrofoil is considered with taking into account fluid viscosity in the cavity closure region and surface tension, which affect the cavity detachment. The theoretical model is based on the concept of viscous–inviscid interaction between the outer inviscid cavity flow and the inner turbulent separated flow downstream of the cavity. The outer inviscid flow is solved by constructing the complex flow potential, and the wake model is based on the method of integral relationships for separated turbulent flows. The obtained numerical results are compared with experimental data.     

Lattice Boltzmann simulations of viscoplastic fluid flows through complex flow channels

We present the results of lattice Boltzmann (LB) simulations for the planar-flow of viscoplastic fluids through complex flow channels. In this study, the Bingham and Casson model fluids are covered as viscoplastic fluid. The Papanastasiou (modified Bingham) model and the modified Casson model are employed in our LB simulations. The Bingham number is an essential physical parameter when considering viscoplastic fluid flows and the modified Bingham number is proposed for modified viscoplastic models. When the value of the modified Bingham number agrees with that of the “normal” Bingham number, viscoplastic fluid flows formulated by modified viscoplastic models strictly reproduce the flow behavior of the ideal viscoplastic fluids. LB simulations are extensively performed for viscoplastic fluid flows through complex flow channels with rectangular and circular obstacles. It is shown that the LB method (LBM) allows us to successfully compute the flow behavior of viscoplastic fluids in various complicated-flow channels with rectangular and circular obstacles. For even low Re and high Bn numbers corresponding to plastic-property dominant condition, it is clearly manifested that the viscosity for both the viscoplastic fluids is largely decreased around solid obstacles. Also, it is shown that the viscosity profile is quite different between both the viscoplastic fluids due to the inherent nature of the models. The viscosity of the Bingham fluid sharply drops down close to the plastic viscosity, whereas the viscosity of the Casson fluid does not rapidly fall. From this study, it is demonstrated that the LBM can be also an effective methodology for computing viscoplastic fluid flows through complex channels including circular obstacles.     

Who conceived the “complex viscosity”?

In small amplitude oscillatory shear flow, we now universally represent the response of a viscoelastic fluid as a complex quantity, which we call “complex viscosity.” This short piece deepens our understanding of the origins of the complex viscosity and of the man who coined this term, now widely used in rheology.     

A model long-discontinuous-fiber filled thermoplastic melt in extensional flow

The shear cell model works for dilute fiber filled systems in extensional flow. This research investigates the suitability of the idea for highly aligned fibers in a concentrated suspension. A model fiber-filled polymer system made from nylon fibers in low-density polyethylene provided a means of controlling the material parameters. Two systems, with fiber aspect ratios of 20 and 100, containing 50% 0.5 mm fibers by volume are investigated. The thickness of the polymer layer, i.e. with fibers this size, allows bulk viscosity data to be compared with the data from the filled fluid. A weaving process created the discontinuous fiber/polyethylene preforms with high alignment of the fibers and with control of the fiber to fiber overlap. Testing the polyethylene in simple shear and extending the nylon/polyethylene provided the data needed to check the micro mechanics. A cone and plate rheometer and a capillary instrument produced the viscosity/strain rate data that characterized the specific polyethylene used in the composite. A furnace inset placed in an Instron hydraulic test machine allowed extension of the filled system at strain rates from 0.002 to 0.4 s⁻¹. The shear experiments show that the low-density polyethylene is a simple shear-thinning melt that provides a good model fluid. The extension of the filled systems shows an increase of the apparent extensional viscosity from that of neat polyethylene. Apparent viscosity rises two to three orders of magnitude for the systems investigated. The micromechanics allowed the conversion of the extensional data from the two filled systems to the shear viscosity of the polymer surrounding the fibers. The calculated polyethylene viscosity compares well with the data from the standard rheometers. The shear cell approach may be applied to highly aligned, high fiber-volume-fraction suspensions when the viscosity of the polymer is known at the scale of the film surrounding each fiber.     

A model for the bulk viscosity of a non-Newtonian fluid

A model for the bulk viscosity of a non-Newtonian fluid is presented. An elementary two-phase cell containing an incompressible non-Newtonian fluid and a gas bubble is compared with a one-phase cell consisting of a corresponding compressible non-Newtonian fluid. The rates of work of both cells are set equal to one another. The deformation histories of both cells are represented by the density history. From the extra stress an expression for the bulk viscosity of the compressible non-Newtonian fluid is derived. The material data are expressed in terms of the known properties of the two-phase flow.     

The use of an ultrasonic Doppler velocimeter in turbulent pipe flow

An ultrasonic Doppler velocimeter (UDV) is used for an experimental investigation of turbulent pipe flow at eight different Reynolds numbers (22,300–854,900). The UDV is a multipoint probe in the sense that it takes instantaneous measurements of fluid velocities at different locations simultaneously, with remarkable resolutions in space and time. The performances of the instrument with respect to the properties of the overlap layer of the turbulent flow field are investigated; the experimental results are compared with both already existing and more recently proposed scaling laws, and with other data of experimental nature.     

川藏公路地质环境与整治改建方案的思考

川藏公路由于地质环境复杂、建设标准低、后遗病害多,抗灾能力差,泥石流、滑坡、山崩、雪害、水毁等自然灾害频繁发生,公路阻车断道严重。国家投入巨资进行整治改建,并取得了明显的效果,但由于自然环境特殊、影响因素复杂,许多特大型、大型工程地质病害问题还没有可行、可靠的解决方案。本文通过分析川藏公路沿线的地质环境和灾害特点,总结历年整治改建和经验的教训,提出川藏公路建设的途径、可能达到的目标和应采用的原则。     

Instability analysis of nonlinear surface waves in a circular cylindrical container subjected to a vertical excitation

Singular perturbation theory of two-time scale expansions was developed both in inviscid and weak viscous fluids to investigate the motion of single surface standing wave in a liquid-filled circular cylindrical vessel, which is subject to a vertical periodical oscillation. Firstly, it is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear evolution equation of slowly varying complex amplitude, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from solvability condition of high-order approximation. It shows that when forced frequency is low, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is high, the influence of surface tension is significant, and can not be neglected. This proved that the surface tension has the function, which causes free surface returning to equilibrium location. Theoretical results much close to experimental results when the surface tension is considered. In fact, the damping will appear in actual physical system due to dissipation of viscosity of fluid. Based upon weakly viscous fluids assumption, the fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates damping term and external excitation, was derived from linearized Navier–Stokes equation. The analytical expression of damping coefficient was determined and the relation between damping and other related parameters (such as viscosity, forced amplitude and depth of fluid) was presented. The nonlinear amplitude equation and a dispersion, which had been derived from the inviscid fluid approximation, were modified by adding linear damping. It was found that the modified results much reasonably close to experimental results. Moreover, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent. Finally, instability of the surface wave is analyzed and properties of the solutions of the modified amplitude equation are determined together with phase-plane trajectories. A necessary condition of forming stable surface wave is obtained and unstable regions are illustrated.     

An automated high-pressure,high-temperature,low-frequency viscometer

A computerized apparatus for the measurement of low shear rate viscosity on polymer solutions at elevated pressure and temperature is described. The pressure is variable from 1 bar to 400 bar with an accuracy of ±5 bar. The temperature is variable from 0° to 100°C with an accuracy of ±0.3 K. The instrument is operated in a free-relaxation mode in which the decay of oscillation of the torsion pendulum after an initial displacement is recorded and used to compute the viscosity of the sample. The measurements are performed according to the contents of a user-specified control file, and the oscillation data are stored digitally and later analyzed for parameter estimation. The instrument operates in the very-low frequency range (0.03 to 0.25 Hz) and the accuracy of the measured viscosity is ±0.03 x 10^–3 Ns/m2.     

Laminar length of a non-Newtonian fluid jet

The laminar length of a submerged jet of a non-Newtonian fluid is measured employing a flow visualization technique, which makes use of the birefringent property of the fluid and a circular polariscope. Fluids of two different concentrations are studied. The results indicate that the concentration, hence the non-Newtonian nature of the fluid, has no influence on the laminar length of the jet for 600 〈 Re 〈 1100. In the Reynolds number range of 50 to 200 the laminar length is affected by the viscous properties of the fluid. The results also indicate that there is not much difference in the laminar length of a Newtonian and non-Newtonian jet for 600 〈 Re 〈 1100.The experimental facility is also used as a “falling head” capillary viscometer. The laminar length data and the viscosity data are taken simultaneously. The viscosity obtained from the present test facility is found to agree with that obtained using a standard rotary viscometer in trend only.     

Stability of plane Poiseuille–Couette flows of a piezo-viscous fluid

We examine stability of fully developed isothermal unidirectional plane Poiseuille–Couette flows of an incompressible fluid whose viscosity depends linearly on the pressure as previously considered in Hron et al. [J. Hron, J. Málek, K.R. Rajagopal, Simple flows of fluids with pressure-dependent viscosities, Proc. R. Soc. Lond. A 457 (2001) 1603–1622] and Suslov and Tran [S.A. Suslov, T.D. Tran, Revisiting plane Couette–Poiseuille flows of a piezo-viscous fluid, J. Non-Newtonian Fluid Mech. 154 (2008) 170–178]. Stability results for a piezo-viscous fluid are compared with those for a Newtonian fluid with constant viscosity. We show that piezo-viscous effects generally lead to stabilisation of a primary flow when the applied pressure gradient is increased. We also show that the flow becomes less stable as the pressure and therefore the fluid viscosity decrease downstream. These features drastically distinguish flows of a piezo-viscous fluid from those of its constant-viscosity counterpart. At the same time the increase in the boundary velocity results in a flow stabilisation which is similar to that observed in Newtonian fluids with constant viscosity.     

Flow of a quasi-Newtonian fluid through a planar contraction

Flows through abrupt contractions are dominated by the rapid extension experienced in passing through the contraction. Thus, it is useful to employ a fluid model which considers the extensional viscosity explicitly in its constitutive equation. In this paper, the quasi-Newtonian fluid model, which admits shear thinning and extension thickening of the viscosity depending on the local type of flow as proposed by Schunk and Scriven [P. Schunk, L. Scriven, J, Rheol 34 (1990) 1085], is applied to the numerical simulation of the flow of a dilute polyacrylamide solution through a planar 4 : 1 contraction. In this theory the extra stress tensor does not only depend on the rate of strain tensor but also on the relative rate of rotation of the fluid. The material function – the viscosity function – is allowed to depend on the invariants of these two kinematic tensors yielding a local distinction between extensional, shear or rotation dominated flow. The governing equations are discretized using a finite volume method. Different model parameters are varied and the simulation results are compared with the generalized Newtonian fluid and experimental data.     

The Effects of Fluid Viscosity on the Kinematics and Material Properties of <Emphasis Type="Italic">C. elegans</Emphasis> Swimming at Low Reynolds Number

The effects of fluid viscosity on the kinematics of a small swimmer at low Reynolds numbers are investigated in both experiments and in a simple model. The swimmer is the nematode Caenorhabditis elegans, which is an undulating roundworm approximately 1 mm long. Experiments show that the nematode maintains a highly periodic swimming behavior as the fluid viscosity is varied from 1.0 to 12 mPa s. Surprisingly, the nematode’s swimming speed (~0.35 mm/s) is nearly insensitive to the range of fluid viscosities investigated here. However, the nematode’s beating frequency decreases to an asymptotic value (~1.7 Hz) with increasing fluid viscosity. A simple model is used to estimate the nematode’s Young’s modulus and tissue viscosity. Both material properties increase with increasing fluid viscosity. It is proposed that the increase in Young’s modulus may be associated with muscle contraction in response to larger mechanical loading while the increase in effective tissue viscosity may be associated with the energy necessary to overcome increased fluid drag forces.     

Viscosity measurements with a magnetoviscometer in the zero-shear and transition region of polypropylenes

Streaming of a non-Newtonian fluid around a sphere is of special importance not only for measuring viscosities with falling spheres, but also for many problems connected with polymer processing. Using the mentioned measuring principle, attention has to be paid to the following points: The sphere is moving in a fluid (melt) of finite extension which requires the application of wall and perhaps end corrections. These are possibly not the same for Newtonian and non-Newtonian fluids. To calculate the viscosity with the help of Stokes law the steady-state velocity is necessary, and it is essential, how long it takes the sphere to reach it. To compare our results with other data, an average shear rate has to be calculated, since there is no uniform shear rate around the sphere. Velocities being very low in our experiments result in very small Reynolds numbers (Re < 10^–3), which allows the application of Stokes law practically without corrections.The experiments were performed at zero shear and in the transition region above. It turned out, that it is usually not possible to extrapolate from shear-dependent viscosity data to zero-shear viscosity.Dedicated to Prof. A. Neckel on the occasion of his 60th birthday     

Computational analysis of binary collisions of shear-thinning droplets

Scale-reduced models of transport processes and reactive mixing in sprays require improved closure laws, taking into account the characteristic features of elementary spray processes. The present paper investigates binary droplet collisions as such an elementary process. In the case of shear-thinning liquids considered here, this requires a profound understanding of the influence of the non-Newtonian fluid rheology on the flow inside the colliding drops and the collision complex dynamics. We employ direct numerical simulations based on the Volume-of-Fluid method to study these collisions. The results give a quantitative prediction of the resulting droplet collision diameter as well as a qualitative prediction of the complete time evolution. During collisions, extremely thin fluid lamellae appear inside the expanding complex. These have to be accounted for in a physically sound simulation and we apply a stabilization of the lamella to keep it from rupturing. The simulations show that in all considered cases an effective constant viscosity can be found a posteriori which leads to the same collision dynamics. But this effective viscosity is neither the mean nor the minimum viscosity.     

On the Mathematical Modelling of a Compressible Viscoelastic Fluid

Thermodynamical considerations have largely been avoided in the modelling of complex fluids by invoking the assumption of incompressibility. This approximation allows pressure to be defined as a Lagrange multiplier, and therefore its natural connection with other thermodynamic variables such as density and temperature is irretrievably lost. Relaxing this condition to allow more realistic modelling involves much more than prescribing an equation of state. Even for a simple isothermal viscoelastic model, as explored in this paper, the transition to a compressible model is non-trivial. This paper shows that pressure enters the governing equations in a non-intuitive way. Furthermore, a fluid volume element, which is no longer constant, radically changes the way the basic element of the constitutive equations is viewed—stress is no longer the fundamental constitutive link between the momentum equations and velocity. The importance of geometry in fluid modelling is emphasised through the use of the Lie derivative, which is of a more fundamental character than the “upper” and “lower” convected derivatives prevalent in the literature and which are found to be almost redundant for a compressible fluid. There is now a strong body of non-equilibrium thermodynamics theory for flowing systems, which proves indispensible for this development. These fundamental principles are described herein using methodology and examples, that are sometimes conflicting, from the literature. The main conflict arises from the relationship between thermodynamic pressure and the trace of Cauchy stress, where the current preferred choice is (up to a constant) to set them equal—this is shown to be incorrect. Other issues such as the dependence of viscosity on density, bulk viscosity, integral modelling, the principle of objectivity and convected derivatives, are also clarified and resolved.     

Free convection effects on a vertical cone with variable viscosity and thermal conductivity

The present paper deals with a flow of a viscous incompressible fluid along a heated vertical cone, with due allowance for variations of viscosity and thermal diffusivity with temperature. The fluid viscosity is assumed to be an exponential function of temperature, and the thermal diffusivity is assumed to be a linear function of temperature. The governing equations for laminar free convection of the fluid are transformed into dimensionless partial differential equations, which are solved by a finite difference method with the Crank–Nicolson implicit scheme. Dependences of the flow parameters on the fluid viscosity and thermal conductivity are obtained.