Isotropic Viscosity Research Articles

Molecular theory for the Leslie viscosities of nematics

Abstract Simple analytic expressions for all six Leslie viscosities for nematics are obtained from the molecular theory for rigid prolate molecules. The theory relates the viscosities to the second and fourth moments, P 2 and P 4, of the molecular distribution function, the aspect ratio a of the molecules, and an isotropic viscosity that accounts for the effects of orientation-independent momentum transport. The theory is obtained by using expressions derived from a molecular theory by Kuzuu and Doi, as well as an approximate formula for the ‘tumbling parameter’ λ in terms of P 2 and P 4; this approximate formula agrees well with data for many small-molecule nematics. Using the Maier-Saupe potential to compute P 2 and P 4, we find that the temperature dependencies of all six Leslie viscosities are in surprisingly good agreement with measurements of these viscosities for MBBA.

Experimental study of three-dimensional separation on a large-scale model

Three-dimensional separation was studied by analyzing the flow past a large-scale model consisting of a half-prolate ellipsoid extended by a circular cylinder ending in a flat base at 45 deg with respect to the cylinder axis. The flow past the model, including boundary layer and vortical structures, was investigated in great detail using a three-component laser Doppler velocimetry system and three-hole pressure probes actuated by a displacement system installed inside the model. This last device allowed probing of the separating three-dimensional boundary layer very close to the surface. We observed how the boundary layer evolved as it gradually sheared into a vortex roll-up and then into an organized vortex. When skewing of the boundary layer increased, the difference in direction between the velocity gradient vector and the shear stress vector also increased. For this type of flow, turbulence models based on the assumption of isotropic turbulent viscosity are inadequate for numerical analysis.

Convection Instabilities in Nematic Liquid Crystals

Pattern formation in hydrodynamic instabilities has been studied intensely over the past few decades (Manneville 1990, Cross & Hohenberg 1993). Rayleigh-Benard convection (RBC) in simple fluids has been the prime example. Our goal in this review is to draw attention to the rich variety of scenarios found in nematic liquid crystals (LCs). Liquid crystals are materials made up of highly anisotropic organic molecules in a phase that reflects this anisotropy. The class of nematic LCs (nematics) is fully liquid without long-range translational, but with long-range uniaxial orienta tiona I ordering of the molecules. Thus in the well-established hydrodynamic description (Ericksen 1 976, Leslie 1 979, de Gennes 1974, Stephen & Straley 1974, Chandrasekhar 1977) the director n characterizing the preferred axis has to be included as an additional variable. One then has six shear viscosities lXI' • . • , 1X6 with 1X6 -1X5 = 1X3 + IXz (1X4/2 corresponds to the isotropic viscosity) in the momentum balance equation (generalized Navier-Stokes equation). In the director equation, which can be associated loosely with a balance of torque, there are two rotational viscosities, YI and Yz, which are expressible in terms of the shear viscosities. For the low-molecular-weight materials discussed here the vis­ cosities relevant in the following are of order 10-1 kg m -I S-I. One needs three orientational elastic modules, kl), k22' k33' to describe the three basic

Densification and creep in the final stage of sintering

Densification and creep by grain boundary diffusion is modelled for the late stage of sintering. The grain structure is described as a regular array of tetrakaidekahedra with pores at each grain boundary between next-nearest neighbors. The diffusion problem on the surface of the tetrakaidekahedron is solved numerically using the heat-conduction option of the finite element code ANSYS. From the calculated normal stresses on the boundary facets one assembles the macroscopic constitute behavior. Since the assumed grain shape is the Wigner-Seitz cell of the body-centered cubic lattice, the resulting viscosity tensor has cubic symmetry. Isotropic bulk and shear viscosities are obtained by applying the procedures developed for the elasticity theory of polycrystals. The resulting bulk viscosity is well approximated by a closed-form solution developed previously. Due to the pronounced cubic anisotropy of the model, the isotropic shear viscosity cannot be determined unambiguously. The model includes the effect of viscous grain boundary sliding. The influence of surface diffusion on the sintering rate is also explored.

Rheology and flow-induced liquid crystal phase transitions in thermotropic polyethers

This work investigates the interdependence of the phase behaviour, viscosity, temperature, molecular weight and shear rates of thermotropic liquid crystalline polyethers. The viscosity of the isotropic and nematic phases are quantitatively compared; a positive variation in viscosity with respect to temperature is found, with the isotropic viscosity being about an order of magnitude higher than the liquid crystalline viscosity. The dependence of viscosity upon molecular weight of well defined fractions is investigated in both the liquid-crystal and isotropic phases. In the liquid crystalline state the viscosity scales with M3.5–5. Variations in the viscosity due to temperature changes affect the isotropic phase more than the liquid crystal phase. No evidence for a negative first normal stress difference is seen. Finally, it is shown how the phase diagram of the material can be altered by shearing the material in the isotropic phase. This is evident by the onset of a shear thinning region at temperatures slightly above Ti, which can be attributed to the formation of shear induced liquid crystallinity.

Pulsating strains

Pulsating strains are modelled analytically using a continuum model with a well defined heterogeneous rheology. The mechanical continuum comprises single circular cylindrical inclusions of isotropic homogeneous viscosity hosted in a less viscous matrix. The analysis concentrates on the behaviour of the competent inclusion and demonstrates that the progressive deformation of the inclusion is entirely controlled by its viscosity ratio to the host and the orientation of the far field stress. The rate of deformation is determined by the stress magnitude and the viscosity of the inclusion, but does not affect the deformation path followed. An extreme case is provided by pure shear deformations where, by definition, the shear rate vanishes from the deformation tensor. Vorticity will then be zero, and strains will develop coaxially without pulsation.

The prediction of riblet behaviour with a low-Reynolds number k-ε model

AbstractThe paper reports the first computational explorations of the flow in the vicinity of riblets with a two-equation turbulence model. For simplicity fully developed flow in a plane channel is considered though the results should be applicable to boundary-layer flow, as the riblets are confined within the (virtually) constantstress near-wall sublayer. The study of idealised L-shaped (or blade) riblets showed larger levels of drag reduction (up to 22%) than have hitherto been reported. When the usual triangular-profile riblets were considered, however, the maximum drag reduction was reduced to about 10%, in line with experiment.Although the maximum drag reduction is well predicted, there are serious differences from experiment for the values of h+ at which the optimum performance is achieved. Moreover, the computations suggest that at low Re for h+ < 10, drag increases result. These anomalies highlight two distinct and counteracting weaknesses in the model of turbulence: there is an insufficient sensitivity of the equation to three-dimensional straining in the jow-Reynolds-number region while use of an isotropic viscosity model enforces zero secondary circulation in the vicinity of the riblets.

Progressive deformation in anisotropic rocks

Field observations show that the majority of crustal rocks possess a penetrative foliation defined by either compositional layering, preferred orientations of crystals, or both. Ductile deformation involving planar anisotropy of viscosity can be characterized by an anisotropy factor δ = q N / r/ s , the ratio of the bulk viscosities in pure and simple shear, respectively. This ratio of the normal and shear viscosities may be determined analytically if the anisotropy is resolvable in a multilayer sequence with individual laminae of isotropic viscosity. In that case, the resistance to normal compression will be largely controlled by the competent layers, whereas the resistance to shear is controlled by the soft layers. More specifically, the viscosities of the individual laminae add up like resistances in series and in parallel in the expressions for the normal and shear viscosities, respectively. The reorientation of the bulk stress within an anisotropic multilayer is systematically investigated for a range of anisotropy factors. The mode of progressive deformation is controlled only by the anisotropy factor and the orientation of the principal deviatoric stress. The rate of deformation can be scaled if the bulk normal viscosity and the magnitude of the principal deviatoric stress are also known. The results are illustrated in a series of nomograms showing the spectrum of strain and rotation histories possible in rock volumes deforming with planar anisotropy. It appears that with increasing anisotropy factor the deformation spectrum will narrow on simple shear, irrespective of the orientation of the background or bulk deviatoric stress axes. Plane strain and isochoric conditions are assumed throughout the analysis.

The weighted particle method for convection-diffusion equations. I. The case of an isotropic viscosity

The aim of this paper is to present and study a particle method for convection-diffusion equations based on the approximation of diffusion operators by integral operators and the use of a particle method to solve integro-differential equations described previously by the second author. The first part of the paper is concerned with isotropic diffusion operators, whereas the second part will consider the general case of a nonconstant matrix of diffusion. In the former case, the approximation of the diffusion operator is much simpler than in the general case. Furthermore, we get two possibilities of approximations, depending on whether or not the integral operator is positive.

Radiation hydrodynamic calculation of super-Eddington accretion disks

The results of nonrelativistic radiation-hydrodynamic calculations of axisymmetric supercritical accretion disks around Newtonian quasi-black holes are reported. Anisotropic and isotropic constant kinematic viscosity models are used, with radiation transport described by a gray Thomson scattering opacity and flux-limited diffusion. The resulting solutions have four distinct zones: (1) centered on the disk midplane is a thick, dense region of turbulent convection; (2) above this is an accretion zone in which low angular momentum matter rapidly flows onto the black hole; (3) the accretion zone is bounded by a photocone, in which the matter becomes optically thin; (4) inside the photocone, surrounding the angular momentum axis, is a broad subrelativistic jet of expelled matter. Applications of these results to SS 433 and to extragalactic jets are discussed. 44 references.

On the Calculation of Flow and Heat Transfer Characteristics for CANDU-Type 19-Rod Fuel Bundles

A numerical study is reported of flow and heat transfer in a CANDU-type 19 rod fuel bundle. The flow domain of interest includes combinations of triangular, square, and peripheral subchannels. The basic equations of momentum and energy are solved with the standard k–ε model of turbulence. Isotropic turbulent viscosity is assumed and no secondary flow is considered for this steady-state, fully developed flow. Detailed velocity and temperature distributions with wall shear stress and Nusselt number distributions are obtained for turbulent flow of Re = 4.35 × 104, 105, 2 × 105, and for laminar flow of Re ∼ 2400. Friction factor and heat transfer coefficients of various subchannels inside the full bundle are compared with those of infinite rod arrays of triangular or square arrangements. The calculated velocity contours of peripheral subchannel agreed reasonably with measured data.

Turbulent air flow over a moving curved surface

A numerical model of turbulent air flow over a curved surface is described. The model is based on two-dimensional nonlinear Reynolds equations and continuity equations written in a coordinate system moving with the profile of the curved surface. The Reynolds stresses are represented in the form of the product of the isotropic turbulent viscosity coefficient, which increases linearly with height, and the deformation tensor of the mean velocity field. Flow over a stationary sinusoidal surface and a sinusoidal gravity wave on water is simulated. The structure of the velocity and pressure wave fields is obtained. The differences in flow over stationary and moving surfaces are analyzed.

A constitutive law for sea ice and some applications

A constitutive law for sea ice is constructed taking into account empirical facts of ice motion. The Reiner–Rivlin theory is used as a frame for the formulation, in which the isotropic stress and shear viscosity depends on the ice concentration and thickness. The latter depends also on the strain rate invariants. It is shown that winds normal to the shore can enhance the along-shore velocity component, which has a direction to the right of the wind due to the Coriolis force. This enhancement is mainly a result of the balance between the stresses, which is a nongeostrophic balance. Only after adding a small along-shore wind blowing to the left, can the geostrophic leftward along-shore jet be recovered. It is concluded that the wind direction is crucial for the direction of the along-shore ice velocity near the shore in this viscous sea-ice formulation.

The prediction of turbulent swirling jet flow

A space marching integration procedure is used to solve the Reynolds equations governing the axisymmetric incompressible turbulent swirling jet flow. Turbulence is modelled by the k− ε model with an isotropic turbulent viscosity. Besides mean velocity field, turbulent properties—such as Reynolds stresses, turbulent kinetic energy and dissipation rate—are obtained and the results are compared with experimental data. Agreement is quite encouraging and shows that the assumption of isotropic turbulent viscosity seems plausible.

Super‐rotation and diffusion of axial angular momentum: I. ‘Speed limits’ for axisymmetric flow in a rotating cylindrical fluid annulus

Abstract‘Super‐rotation’ can be defined with respect to the limitations on the magnitude of specific angular momentum, m, in an inviscid fluid. In a viscous fluid, a steady, super‐rotating axisymmetric flow is shown to require diffusion of m against its gradient ∇m in the meridional plane. The conditions under which m can be diffused in an incompressible fluid by molecular viscosity against ∇m in a cylindrical system are shown to be consistent with the normal properties of isotropic Newtonian viscosity (i.e. appropriate for the laminar flow of a viscous liquid in the laboratory) when cylindrical curvature is fully represented. A series of numerical simulations of thermally‐driven axisymmetric circulations in a cylindrical fluid annulus, subject to various mechanical boundary conditions, is then presented. The role of diffusion in the angular momentum budget of the simulations is examined by diagnosing m, its diffusive flux F (due to molecular viscosity) and divergence ∇·F from the final equilibrium (steady) flow. Up‐gradient diffusion (with respect to ∇m) is found to be particularly important for flows in a system with stress‐free top and side boundaries and a non‐slip base. Similar up‐gradient diffusion can also result in angular momentum expulsion effects in a system confined entirely by stress‐free boundaries. The characteristic dynamics of super‐rotation in a viscous fluid, and the associated limitations upon its magnitude and dependence upon the external conditions and fluid properties, are then explored in a scale analysis for the cylindrical annulus with a non‐slip base (based on a scheme due to Hignett, Ibbetson and Killworth). The most rapid super‐rotation (with zonal Rossby number > 1) is found to be favoured at moderate rotation rates, with strong cylindrical curvature, and a large meridional aspect ratio and Prandtl number. The results of the scale analysis are verified by means of further numerical experiments.

Two-phase flow equations for a dilute dispersion of gas bubbles in liquid

Equations of motion correct to the first order of the gas concentration by volume are derived for a dispersion of gas bubbles in liquid through systematic averaging of the equations on the microlevel. First, by ensemble averaging, an expression for the average stress tensor is obtained, which is non-isotropic although the local stress tensors in the constituent phases are isotropic (viscosity is neglected). Next, by applying the same technique, the momentum-flux tensor of the entire mixture is obtained. An equation expressing the fact that the average force on a massless bubble is zero leads to a third relation. Complemented with mass-conservation equations for liquid and gas, these equations appear to constitute a completely hyperbolic system, unlike the systems with complex characteristics found previously. The characteristic speeds are calculated and shown to be related to the propagation speeds of acoustic waves and concentration waves.

A minimum dissipation principle for the Rayleigh-Taylor problem in viscous magnetohydrodynamics

A variational principle for magnetohydrodynamic gravitational modes, including isotropic viscosity and the shear of magnetic field lines, is given. It is shown to imply a minimum dissipation requirement for the modes. A sufficient condition for stability given in the literature for the case of ordinary fluids is corrected here.

Precessing twisted accretion disks, with an application to Hercules X-1

An eigenvalue equation is derived for the steady-state precession frequency of a twisted, thin accretion disk in a binary system. We describe an algorithm which solves this equation by time-evolution of the twisted disk equations. Applying the parameters of the X-ray binary Her X-1, and assuming that the 35 day cycle of that system is due to periodic obscuration by a precessing disk, we find that the observations place severe restrictions on the possible values of the viscosity in the disk. For a (twisted) ''standard'' disk model with isotropic viscosity, we find that ..cap alpha.., the dimensionless ratio of viscous-to-pressure forces in the disk, must lie in the range 1/5< or approx. =..cap alpha..< or approx. =2, with a most likely value of about 1.

Low-shear limit of the secondary electroviscous effect

A theory has been developed for the isotropic zero-shear viscosity of electrostatically stabilized suspensions of spheres at concentrations dilute enough that pair interactions dominate. Brownian motion of and viscous forces on individual spheres are included, but the weak flow coupled with strong electrical repulsions allows hydrodynamic pair interactions and double-layer distortion to be neglected. The viscosity, expressed in terms of the volume fraction φ as μ μ 0 −1=A θ+ B (ακ) 5 θ 2+…, includes the secondary electroviscous effect in B∾(ln α lnα ) 4.5 where α = electrostatic forces/Brownian forces ⪢ 1. The results compare favorably with available data but suggest the importance of multiparticle electrostatic interactions as φ increases.

Nonstationary Rayleigh problem for magnetized plasma

A solution is obtained for the problem of onset of oscillatory convection in a horizontal layer of fully ionized plasma located in constant vertical gravitational and magnetic fields. In case of significant anisotropy in the thermal conductivity with isotropic viscosity (only the electrons are magnetized) the onset of convection at large Hartmann numbers is accompanied by oscillations if the ''magnetic Prandtl number'' is much larger than unity. The results obtained here are a natural generalization of the well-known Chandrashekar's criteria and can be used in the theory of MHD wave generation in a strongly magnetized plasma.