A study of conformation-dependent friction in a dumbbell model for dilute solutions

We consider predictions of rheological behavior in a variety of shear and extensional flows for an elastic dumbbell model with a nonlinear spring, and conformation-dependent hydrodynamic properties. The latter include a conformation-dependent anisotropic bead friction coefficient, and a related conformation-dependent degree of inefficiency for rotation in straining flows. With these features, the dumbbell exhibits hydrodynamic behavior consistent with a particle of finite axis ratio over the complete set of possible polymer conformations, from random-coil to a fully extended thread-like configuration. The predicted rheological behavior in shear flow is improved, relative to data, by the inclusion of anisotropy and strain-inefficiency in the frictional properties of the model, while other desirable features such as the sudden onset of fully extended states at a critical value of the velocity gradient, the presence of a hysteresis-loop in end-to-end dimension as a function of the velocity gradient, and the correlation of end-to-end distance (or birefringence) with the eigenvalue of the velocity gradient tensor for a wide variety of two-dimensional flows, are maintained.     

Electrostatic probe for detecting charged particles in a gas-dynamic flow

The theoretical and practical aspects of detection of large (r 50 m) charged particles in a gas-dynamic flow by means of an electrostatic probe are considered. The basic equation that describes the interaction between such a particle and the probe is derived. A model problem that permits one to determine the electrostatic charge induced on the probe is formulated and solved. The interaction with the probe of charged conducting particles and dielectric particles is analyzed. It is shown that one can not only detect individual charged particles in a gas stream but also determine their charge and velocity by analyzing the time distributions of the measured signal. Actual constructions of an electrostatic probe and the corresponding measuring complexes are described. Methods are indicated for raising the sensitivity and resolution of the probe. The results are presented of diagnostics of charged gas-dynamic flows in a laboratory experiment, and also under test and full-scale conditions (measurement of charged particles in jet exhausts).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 72–81, January–February, 1980.We thank V. I. Grabovskii, D. G. Dubravskii, I. I. Il'yushenkovaya, and A. P. Strekalov for assistance during this work.     

A study of the interacting FENE dumbbell model for semi-dilute polymer solutions in extensional flows

The effect of polymer concentration on the conformation of semidilute polymer solutions in extensional flows is studied via the interacting elastic dumbbell model proposed by Hess (1984), here modified to include a nonlinear Warner spring (FENE dumbbell) instead of the linear Hookean spring of the original model. The length of flow-induced conformation changes for the polymer is predicted to be a decreasing function of concentration. In particular, increasing concentration tends to inhibit large extension of the polymer due to polymer-polymer interaction. The specific birefringence is thus proportional to c ^–1 for semi-dilute solutions, in contrast to dilute solutions where it is known to be independent of concentration. However, the correlation between birefringence and the principle eigenvalue of the velocity gradient tensor, also found originally for dilute solutions, is predicted to occur in the semi-dilute regime. All of these predictions agree qualitatively with experimental observations.Some recent exceptions to the neglect of segmental stretch can be found in Marrucci and Grizzuti (1988), Pearson et al. (1991), Mead et al. (1992).     

Time‐dependent turbulent cavitating flow computations with interfacial transport and filter‐based models

Turbulent cavitating flow computations need to address both cavitation and turbulence modelling issues. A recently developed interfacial dynamics‐based cavitation model (IDCM) incorporates the interfacial transport into the computational modelling of cavitation dynamics. For time‐dependent flows, it is known that the engineering turbulence closure such as the original k–ε model often over‐predicts the eddy viscosity values reducing the unsteadiness. A recently proposed filter‐based modification has shown that it can effectively modulate the eddy viscosity, rendering better simulation capabilities for time‐dependent flow computations in term of the unsteady characteristics. In the present study, the IDCM along with the filter‐based k–ε turbulence model is adopted to simulate 2‐D cavitating flows over the Clark‐Y airfoil. The chord Reynolds number is Re=7.0 × 10⁵. Two angles‐of‐attack of 5 and 8° associated with several cavitation numbers covering different flow regimes are conducted. The simulation results are assessed with the experimental data including lift, drag and velocity profiles. The interplay between cavitation and turbulence models reveals substantial differences in time‐dependent flow results even though the time‐averaged characteristics are similar. Copyright © 2005 John Wiley & Sons, Ltd.     

Instability of a Charged Droplet in an Inhomogeneous Electrostatic Field of a Rod of Finite Thickness

The instability of a charged droplet of an ideal liquid in an inhomogeneous electrostatic field of a rod of finite thickness maintained at a constant electrostatic potential is investigated within the framework of analytic asymptotic calculations. It is shown that the mode amplitudes and the drop oscillation frequencies increase with the rod thickness. The critical conditions of instability of the droplet reduce by several times as compared with the critical conditions of implementation of its instability in the electrostatic field of an infinitely thin filament maintained at a constant electrostatic potential. An analytic dependence between the charge and field parameters, critical for implementation of the instability of a charged droplet in an inhomogeneous electrostatic field and dependent on the rod thickness, is found.     

含氮杂环类润滑油添加剂CoMFA-QSTR及CoMSIA-QSTR抗磨损性能模型的构建

依据摩擦学定量构效关系理论(QSTR),采用比较分子力场分析(CoMFA)和比较分子相似性指数分析(CoMSIA)这两种方法研究了含氮杂环类润滑油添加剂的抗磨损性能的摩擦学三维定量构效关系(3D-QSTR),并建立了相应的3D-QSTR模型.结果表明:仅利用静电场构建CoMFA或CoMSIA模型时,模型预测能力最好,r~2,q~2均大于0.5.根据CoMFA或CoMSIA模型等高线图分析得出:分子静电场对含氮杂环类润滑油添加剂的抗磨损性能影响最大,在特定区域的引入带负电荷或带正电荷的基团将有助于抗磨性能的提高.     

Effective Cohesive Behavior of Layers of Interatomic Planes

A simple model of cleavage in brittle crystals consists of a layer of material containing N atomic planes separating in accordance with an interplanar potential under the action of an opening displacement δ prescribed on the boundary of the layer. The problem addressed in this work concerns the characterization of the constrained minima of the energy E_N of the layer as a function of δ as N becomes large. These minima determine the effective or macroscopic cohesive law of the crystal. The main results presented in this communication are: (i) the computation of the Γ limit E₀ of E_N as N → ∞; (ii) the characterization of the minimum values of E₀ as a function of the macroscopic opening displacement; (iii) a proof of uniform convergence of the minima of E_N for the case of nearest-neighbor interactions; and (iv) a proof of uniform convergence of the derivatives of E_N, or tractions, in the same case. The scaling on which the present Γ-convergence analysis is based has the effect of separating the bulk and surface contributions to the energy. It differs crucially from other scalings employed in the past in that it renders both contributions of the same order.     

Thermal diffusion of dilute polymer solutions

In this paper diffusion of a dilute solution of elastic dumbbell model macromolecules under non-isothermal conditions is studied. Using the center of mass definition for the local polymer concentration, the diffusive flux contains a thermal diffusion dyadic d _T .  To get some idea of thermal diffusion d _T is evaluated for steady state isothermal conditions. Explicit results are presented for some homogeneous flows. It is shown that if the polymeric number density is defined via the beads (of the dumbbell) – termed n _b – then the diffusive flux j contains , where τ ^c is the intramolecular contribution to the bulk stress. Though the form of the diffusion equation for n _b thus differs from the corresponding one for n, it is shown that for essentially unbounded systems differences between n and n _b are small. Since the results involve the translational diffusion coefficient they can readily be taken over for Rouse coils. Received: 23 September 1997 Accepted: 5 June 1998     

Progress in subgrid-scale transport modelling for continuous hybrid non-zonal RANS/LES simulations

The partially integrated transport modelling (PITM) method can be viewed as a continuous approach for hybrid RANS/LES modelling allowing seamless coupling between the RANS and the LES regions. The subgrid turbulence quantities are thus calculated from spectral equations depending on the varying spectral cutoff location [Schiestel, R., Dejoan, A., 2005. Towards a new partially integrated transport model for coarse grid and unsteady turbulent flow simulations. Theoretical and Computational Fluid Dynamics 18, 443–468; Chaouat, B., Schiestel, R., 2005. A new partially integrated transport model for subgrid-scale stresses and dissipation rate for turbulent developing flows. Physics of Fluids, 17 (6)] The PITM method can be applied to almost all statistical models to derive its hybrid LES counterpart. In the present work, the PITM version based on the transport equations for the turbulent Reynolds stresses together with the dissipation transport rate equation is now developed in a general formulation based on a new accurate energy spectrum function E(κ) valid in both large and small eddy ranges that allows to calibrate more precisely the c_sgs2 function involved in the subgrid dissipation rate _sgs transport equation. The model is also proposed here in an extended form which remains valid in low Reynolds number turbulent flows. This is achieved by considering a characteristic turbulence length-scale based on the total turbulent energy and the total dissipation rate taking into account the subgrid and resolved parts of the dissipation rate. These improvements allow to consider a large range of flows including various free flows as well as bounded flows. The present model is first tested on the decay of homogeneous isotropic turbulence by referring to the well known experiment of Comte-Bellot and Corrsin. Then, initial perturbed spectra E(κ) with a peak or a defect of energy are considered for analysing the model capabilities in strong non-equilibrium flow situations. The second test case is the classical fully turbulent channel flow that allows to assess the performance of the model in non-homogeneous flows characterised by important anisotropy effects. Different simulations are performed on coarse and refined meshes for checking the grid independence of solutions as well as the consistency of the subgrid-scale model when the filter width is changed. A special attention is devoted to the sharing out of the energy between the subgrid-scales and the resolved scales. Both the mean velocity and the turbulent stress computations are compared with data from direct numerical simulations.     

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.     

Smoothing the discontinuities of an electrical charge in electrodynamics as a result of diffusion processes

Electrohydrodynamic flows in which there are zones of abrupt changes in the electric charge (while remaining bounded, by assumption) are investigated. In a diffusionless approximation such flows are characterized by a discontinuity in the electric charge q. Examples of such motions are nonstationary flows with moving electrical charge fronts [1], stationary flows in which the electrical charge is lumped in just part of the hydrodynamic stream [2, 3], flows with discontinuity in q [4–7], boundary layers near an electrode grid mounted perpendicularly to the electrohydrodynamic stream. Diffusion effects of charged particles should cause smoothing of the electrical charge discontinuities. The diffusion structure of such discontinuities is studied for high electrical Peclet numbers. The distribution of q in gasdynamic jumps is analyzed taking account of the viscous and diffusion structure of the discontinuities in the small parameter approximation of the electrogasdynamic interaction. Three problems about flows with charged particle diffusion are examined: the problem of scattering of a finite electric charge in a medium at rest, initially concentrated at a point on a line of unit length; the boundary layer on an electrode grid perpendicular to the direction of the charged fluid stream; electrogasdynamic flows with an abrupt change in velocity not accompanied by the appearance of a surface charge.     

The application of the method of characteristics for the simulation of nearly horizontal flow in two and three spatial dimensions

Certain free surface flows exhibit in nature negligible vertical accelerations and as a result the pressure within the fluid remains hydrostatic. The method of characteristics is developed as a solution technique for the integration of the partial differential equations describing this kind of flow. The equations are integrated over the depth to provide a two-dimensional model which is then tested and validated by comparing its results with tide-induced flows occurring in a number of cases where either analytical or observational data are available for comparison. On the basis of the results of the 2D model, a finite difference 3D model is developed which provides the values of the unknown velocities u, v and w along the three axes x, y and z. This combined 2D–3D model is verified by applying it in cases of wind-induced flow inside closed or open basins for which the classical Ekman solution may be used as a testing means.     

Green's function,continual weighted composition operators along trajectories,and hyperbolicity of linear extensions for dynamical systems

The hyperbolicity of linear skew-product flows with infinite-dimensional fibersE over a dynamical system on a compact metric spaceX is described in terms of the existence and uniqueness of Green's function and in terms of the spectra for family of the semigroups of weighted composition operators acting inL ₂(R;E) and parametrized by the points ofX.     

Electrokinetic Mixing and Displacement of Charged Droplets in Hydrogels

Mixing in droplets is an essential task in a variety of microfluidic systems. Inspired by electrokinetic mixing, electric field-induced hydrodynamic flow inside a charged droplet embedded in an unbounded polyelectrolyte hydrogel is investigated theoretically. In this study, the polyelectrolyte hydrogel is modeled as a soft, and electrically charged porous solid saturated with a salted Newtonian fluid, and the droplet is considered an incompressible Newtonian fluid. The droplet-hydrogel interface is modeled as a surface, which is located at the plane of shear, with the electrostatic potential \(\zeta \) . The fluid inside the droplet attains a finite velocity owing to hydrodynamic coupling with the electroosmotic flow arising from the droplet and polymer charge. The fluid velocity inside the droplet is linearly proportional to the electroosmotic flow velocity in the charged gel and the electroosmotic flow velocity beyond the electrical double layer of a charged interface. It is found that the polymer boundary condition at the droplet surface and the viscosities of the fluids inside and outside the droplet significantly modulate the interior fluid flow. The ionic strength and the permeability of the polymer network impact the flow differently depending on whether the flow arises from the droplet or polymer charge. Finally, the displacement of a charged droplet embedded in a gel under the influence of an external electric field is undertaken. This work is motivated by experimental attempts, which can register sub-nanometer-scale inclusion displacements in hydrogels, to advance electrical microrheology as a diagnostic tool for probing inclusion-hydrogel interfaces. In the absence of polymer charge, a close connection is found between the electrical response of a charged droplet when it is immobilized in an uncharged incompressible gel and when it is dispersed in a Newtonian electrolyte.     

Shear and extensional rheology of carbon nanofiber suspensions

The morphology and rheology of carbon nanofiber (CNF) suspensions were studied. The CNFs, produced by decomposing organic vapors at elevated temperature in the presence of metal catalysts, have characteristic diameter and length of 100 nm and 20–100 m, respectively. The CNFs, as delivered, have a strong tendency to clump into mm-sized agglomerates. The efficacy of CNF/glycerol-water suspensions was studied vs. two processing parameters: mechanical sonication and chemical treatment. Experimental measurements revealed that sonication alone reduces the size of CNF clumps from millimeter to micrometer scale, but cannot achieve uniform dispersion. The chemically untreated sonicated suspensions contain clumps of nanofibers with a characteristic size of 20×50 m, together with smaller aggregations of partially dispersed nanofibers. In response to this unsatisfactory dispersion, the effect of acid treatment before dispersion was investigated. This acid treatment, which makes the surface of the CNFs more hydrophilic, greatly improves dispersion in the aqueous solution: treatment followed by sonication results in a uniform dispersion of individual nanofibers. At the same time, however, we observed that surface treatment and subsequent sonication greatly shorten the nanofibers.The rheology of CNF/glycerol-water suspensions is highly non-Newtonian both in shear and extensional flows, with strong dependence on the dispersion, particle length, and concentration of the CNFs. As the solvent is Newtonian, all of the elastic and strain-rate dependent behavior in the CNF/aqueous suspensions derives from the addition of nanofibers. The steady shear viscosity of the untreated-sonicated (poorly dispersed, with longer fibers) suspensions is highly shear thinning with a viscosity that increases three orders of magnitude as concentration varies from 0.5 wt% to 5 wt%. Beyond 5 wt% the suspensions are too viscous to be effectively mixed by sonication. When the CNFs are chemically treated and then sonicated (resulting in much better dispersion but shorter fibers), the viscosity exhibits little shear thinning, and only varies by a factor of two from pure solvent to 5 wt%. In small amplitude oscillatory shear measurements, we found strong indications of elastic behavior in both the treated and untreated suspensions, with elastic modulus G greater than loss modulus G. In particular, for both systems G exhibits a low-frequency plateau when nanofiber concentration is 3 wt% or above, a characteristic of elastic solidlike response. Again, there is a strong dependence on CNF dispersion and fiber length: At low frequencies, the elastic modulus of the 5 wt% untreated suspension (with agglomerates and longer fibers) is four orders of magnitude larger than that of the 5 wt% treated suspension (with uniformly dispersed, shorter fibers). In addition, G of untreated suspensions is a much stronger function of concentration than that of treated suspensions, indicative of network formation.The rheology and morphology of nanofiber suspensions were related by identifying morphology of the suspensions with the assumptions of the kinetic theory-based elastic and rigid dumbbell constitutive models; the approach is to specify the parameters in the kinetic theory models in terms of microscale morphological features measured in the SEM. Of those investigated, the elastic dumbbell model with anisotropic hydrodynamic drag is the most successful, effectively modeling the small amplitude oscillatory shear and steady shear behavior of the treated sonicated suspensions. As for the treated unsonicated and untreated sonicated suspensions, which contain mesoscale agglomerates not present in the underlying assumptions of the dumbbell models, it is discovered that the elastic dumbbell with parameters assigned from morphological measurements predicts the correct trends in the steady shear experiments, but fails to accurately predict the small amplitude oscillatory shear experiments.     

Calculation of turbulent combustion of propane in furnaces

An evaluation of some numerical methods for turbulent reacting flows in furnace-like geometries is carried out. The Reynolds averaged Navier–Stokes equations and the two-equation k–? model together with either finite-rate or infinite-rate reaction models are solved numerically. Either single- or multiple-step reactions together with the ‘eddy dissipation concept’ (EDC) are used to model reacting flows with finite reaction rates. The numerical scheme is finite difference based, together with a multi-grid method and a local grid refinement technique. These methods have been used to calculate the combustion of propane in a single- and multiple-burner configurations. In the former case, the sensitivity of the solution to variations in some model parameters (determining the reaction rate) and numerical parameters (mesh spacing) has been studied. It is noted that different dependent variables exhibit different levels of sensitivity to the variation in model parameters. Thus, calibration and validation of models for reacting flows require that one compares the most sensitive variables. For engineering purposes, on the other hand, one may calibrate and validate models with respect to the most relevant variables. Our conclusion is that since sensitivity of the temperature distribution is relatively mild, one can still use EDC-like methods in engineering applications where details of the temperature field are of minor importance.     

Theory of regular electrostatic beams of charged particles

In some papers concerned with the exact solution of the equations of a nonrelativistic single-energy beam of charged particles (e.g., [1, 2]), the opinion has been expressed that, while the method of separation of the variables has possibilities, serious difficulties can arise in obtaining the actual systems with separated variables. In particular, it has become popular, when investigating regular electrostatic flows, to transform to a coordinate system connected with the trajectory. In this system the velocity vector only has one component, say v={i_x t, 0}, so that flow only occurs in the x¹ direction (x¹ flow). We shall also refer to a single-component flow, as in [3], This method is thought (e.g., [4]) to be effective for a wide class of flows. The question of the coordinate systems that allow flows in the¹ direction is more specialized than the general problem of separation of variables.The concept of an x¹ flow is discussed in §1 from the point of view of its utility for obtaining solutions of the regular electrostatic beam equations. Transformation to a coordinate system connected with the trajectory is found to be only justifiable for four orthogonal systems: cartesian, cylindrical, helical-cylindrical, and spherical. It is shown that, in the case of two-dimensional systems on a plane with conformal metric, the condition required for an x¹ flow and the conditions for the space to be euclidean can be used effectively to establish the existence in the given class of coordinate systems of an x¹ flow starting from a fictitious emitter (§2). The usual tensor notation is employed.     

Stability of some shear flows for concentrated suspensions

In this paper we investigate the stability of some viscometric flows for a concentrated suspension model which allows for the effects of shear-induced migration, including plane and circular Couette and Poiseulle flows, and unbounded and bounded torsional flows. In the bounded torsional flow, where its radial outer boundary is assumed frictionless, an exact closeform solution is given. With the exception of torsional flows, we find that a limit point for all the steady-state solutions can exist for certain range in the parameter values. In all cases, disturbances can persist for a long time, O (H ²/a ²), where H is a dimension of the flow field, and a is the particles' radius.     

Multiscale simulation of viscoelastic free surface flows

A micro–macro approach based on combining the Brownian configuration fields (BCF) method [M.A. Hulsen, A.P.G. van Heel, B.H.A.A. van den Brule, Simulation of viscoelastic flow using Brownian configuration fields, J. Non-Newtonian Fluid Mech. 70 (1997) 79–101] with an Arbitrary Lagrangian–Eulerian (ALE) Galerkin finite element method, using elliptic mesh generation equations coupled with time-dependent conservation equations, is applied to study slot coating flows of polymer solutions. The polymer molecules are represented by dumbbells with both linear and non-linear springs; hydrodynamic interactions between beads are incorporated. Calculations with infinitely extensible (Hookean) and pre-averaged finitely extensible (FENE-P) dumbbell models are performed and compared with equivalent closed-form macroscopic models in a conformation tensor based formulation [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on the conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409]. The BCF equation for linear dumbbell models is solved using a fully implicit time integration scheme which is found to be more stable than the explicit Euler scheme used previously to compute complex flows. We find excellent agreement between the results of the BCF based formulation and the macroscopic conformation tensor based formulation. The computations using the BCF approach are stable at much higher Weissenberg numbers, (where λ is the characteristic relaxation time of polymer, and is the characteristic rate of strain) compared to the purely macroscopic conformation tensor based approach, which fail beyond a maximum Wi. A novel computational algorithm is introduced to compute complex flows with non-linear microscopic constitutive models (i.e. non-linear FENE dumbbells and dumbbells with hydrodynamic interactions) for which no closed-form constitutive equations exist. This algorithm is fast and computationally efficient when compared to both an explicit scheme and a fully implicit scheme involving the solution of the non-linear equations with Newton’s method for each configuration field.