Nanofluids are engineered colloids made of a base fluid and nanoparticles (1-100 nm) Nanofluids have higher thermal conductivity' and single-phase heat transfer coefficients than their base fluids In particular the heat transfer coefficient increases appear to go beyond the mere thermal-conductivity effect, and cannot be predicted by traditional pure-fluid correlations such as Dittus-Boelter's In the nanofluid literature this behavior is generally attributed to thermal dispersion and intensified turbulence, brought about by nanoparticle motion To test the validity of this assumption, we have considered seven slip mechanisms that can produce a relative velocity between the nanoparticles and the base fluid These are inertia, Brownian diffusion, thermophoresis, diffusioplwresis, Magnus effect, fluid drainage, and gravity We concluded that, of these seven, only Brownian diffusion and thermophoresis are important slip mechanisms in nanofluids Based on this finding, we developed a two-component four-equation nonhomogeneous equilibrium model for mass, momentum, and heat transport in nanofluids A nondimensional analysis of the equations suggests that energy transfer by nanoparticle dispersion is negligible, and thus cannot explain the abnormal heat transfer coefficient increases Furthermore, a comparison of the nanoparticle and turbulent eddy time and length scales clearly indicates that the nanoparticles move homogeneously with the fluid in the presence of turbulent eddies so an effect on turbulence intensity is also doubtful Thus, we propose an alternative explanation for the abnormal heat transfer coefficient increases: the nanofluid properties may vary significantly within the boundary layer because of the effect of the temperature gradient and thermophoresis For a heated fluid, these effects can result in a significant decrease of viscosity within the boundary layer, thus leading to heat transfer enhancement A correlation structure that captures these effects is proposed

Convective Transport in Nanofluids

Turbulent friction and heat transfer behaviors of dispersed fluids (i.e., uttrafine metallic oxide particles suspended in water) in a circular pipe were investigated experimentally. Viscosity measurements were also conducted using a Brookfield rotating viscometer. Two different metallic oxide particles, γ-alumina (Al2O3) and titanium dioxide (TiO2), with mean diameters of 13 and 27 nm, respectively, were used as suspended particles. The Reynolds and Prandtl numbers varied in the ranges l04-I05 and 6.5-12.3, respectively. The viscosities of the dispersed fluids with γ-Al2O3 and TiO2 particles at a 10% volume concentration were approximately 200 and 3 times greater than that of water, respectively. These viscosity results were significantly larger than the predictions from the classical theory of suspension rheology. Darcy friction factors for the dispersed fluids of the volume concentration ranging from 1% to 3% coincided well with Kays' correlation for turbulent flow of a single-phase fluid. The Nusselt n...

Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles

We develop the dynamic renormalization group (RNG) method for hydrodynamic turbulence. This procedure, which uses dynamic scaling and invariance together with iterated perturbation methods, allows us to evaluate transport coefficients and transport equations for the large-scale (slow) modes. The RNG theory, which does not include any experimentally adjustable parameters, gives the following numerical values for important constants of turbulent flows: Kolmogorov constant for the inertial-range spectrumCK=1.617; turbulent Prandtl number for high-Reynolds-number heat transferPt=0.7179; Batchelor constantBa=1.161; and skewness factor¯S3=0.4878. A differentialK-\(\bar \varepsilon \) model is derived, which, in the high-Reynolds-number regions of the flow, gives the algebraic relationv=0.0837 K2/\(\bar \varepsilon \), decay of isotropic turbulence asK=O(t−1.3307), and the von Karman constantκ=0.372. A differential transport model, based on differential relations betweenK,\(\bar \varepsilon \), andν, is derived that is not divergent whenK→ 0 and\(\bar \varepsilon \) is finite. This latter model is particularly useful near walls.

Renormalization group analysis of turbulence I. Basic theory

Recent advances in theory and experimen- tation motivate a thorough reassessment of the physics of debris flows. Analyses of flows of dry, granular solids and solid-fluid mixtures provide a foundation for a com- prehensive debris flow theory, and experiments provide data that reveal the strengths and limitations of theoret- ical models. Both debris flow materials and dry granular materials can sustain shear stresses while remaining stat- ic; both can deform in a slow, tranquil mode character- ized by enduring, frictional grain contacts; and both can flow in a more rapid, agitated mode characterized by brief, inelastic grain collisions. In debris flows, however, pore fluid that is highly viscous and nearly incompress- ible, composed of water with suspended silt and clay, can strongly mediate intergranular friction and collisions. Grain friction, grain collisions, and viscous fluid flow may transfer significant momentum simultaneously. Both the vibrational kinetic energy of solid grains (mea- sured by a quantity termed the granular temperature) and the pressure of the intervening pore fluid facilitate motion of grains past one another, thereby enhancing debris flow mobility. Granular temperature arises from conversion of flow translational energy to grain vibra- tional energy, a process that depends on shear rates, grain properties, boundary conditions, and the ambient fluid viscosity and pressure. Pore fluid pressures that exceed static equilibrium pressures result from local or global debris contraction. Like larger, natural debris flows, experimental debris flows of ;10 m 3 of poorly sorted, water-saturated sediment invariably move as an unsteady surge or series of surges. Measurements at the base of experimental flows show that coarse-grained surge fronts have little or no pore fluid pressure. In contrast, finer-grained, thoroughly saturated debris be- hind surge fronts is nearly liquefied by high pore pres- sure, which persists owing to the great compressibility and moderate permeability of the debris. Realistic mod- els of debris flows therefore require equations that sim- ulate inertial motion of surges in which high-resistance fronts dominated by solid forces impede the motion of low-resistance tails more strongly influenced by fluid forces. Furthermore, because debris flows characteristi- cally originate as nearly rigid sediment masses, trans- form at least partly to liquefied flows, and then trans- form again to nearly rigid deposits, acceptable models must simulate an evolution of material behavior without invoking preternatural changes in material properties. A simple model that satisfies most of these criteria uses depth-averaged equations of motion patterned after those of the Savage-Hutter theory for gravity-driven flow of dry granular masses but generalized to include the effects of viscous pore fluid with varying pressure. These equations can describe a spectrum of debris flow behav- iors intermediate between those of wet rock avalanches and sediment-laden water floods. With appropriate pore pressure distributions the equations yield numerical so- lutions that successfully predict unsteady, nonuniform motion of experimental debris flows.

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The physics of debris flows

Dynamic strength of molecular adhesion bonds.

The effect of Brownian motion of particles in a statistically homogeneous suspension is to tend to make uniform the joint probability density functions for the relative positions of particles, in opposition to the tendency of a deforming motion of the suspension to make some particle configurations more common. This smoothing process of Brownian motion can be represented by the action of coupled or interactive steady ‘thermodynamic’ forces on the particles, which have two effects relevant to the bulk stress in the suspension. Firstly, the system of thermodynamic forces on particles makes a direct contribution to the bulk stress; and, secondly, thermodynamic forces change the statistical properties of the relative positions of particles and so affect the bulk stress indirectly. These two effects are analysed for a suspension of rigid spherical particles. In the case of a dilute suspension both the direct and indirect contributions to the bulk stress due to Brownian motion are of order o2, where o([Lt ] 1) is the volume fraction of the particles, and an explicit expression for this leading approximation is constructed in terms of hydrodynamic interactions between pairs of particles. The differential equation representing the effects of the bulk deforming motion and the Brownian motion on the probability density of the separation vector of particle pairs in a dilute suspension is also investigated, and is solved numerically for the case of relatively strong Brownian motion. The suspension has approximately isotropic structure in this case, regardless of the nature of the bulk flow, and the effective viscosity representing the stress system to order ϕ2 is found to be
 \[
 \mu^{*} = \mu(1+2.5\phi + 6.2\phi^2).
 \]
 The value of the coefficient of o2 for steady pure straining motion in the case of weak Brownian motion is known to be 7[sdot ]6, which indicates a small degree of ‘strain thickening’ in the o2-term.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/72B16FA20D2A14415CDED6A4F042809F/S0022112077001062a.pdf/div-class-title-the-effect-of-brownian-motion-on-the-bulk-stress-in-a-suspension-of-spherical-particles-div.pdf

The effect of Brownian motion on the bulk stress in a suspension of spherical particles

When some external agency imposes on a fluid large-scale variations of some dynamically passive, conserved, scalar quantity θ like temperature or concentration of solute, turbulent motion of the fluid generates small-scale variations of θ. This paper describes a theoretical investigation of the form of the spectrum of θ at large wave-numbers, taking into account the two effects of convection with the fluid and molecular diffusion with diffusivity k. Hypotheses of the kind made by Kolmogoroff for the small-scale variations of velocity in a turbulent motion at high Reynolds number are assumed to apply also to small-scale variations of θ.Previous contributions to the problem are reviewed. These have established that the spectrum of θ varies as , the result being given by (4.8). The same result is obtained, using essentially the same approximation about the velocity field, from a different kind of analysis in terms of velocity and θ correlations. Finally, the relation between this work and Townsend's model of the small-scale variations of vorticity in a turbulent fluid is discussed.

Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivity

The purpose of the paper is to consider in general terms the properties of the bulk stress in a suspension of non-spherical particles, on which a couple (but no force) may be imposed by external means, immersed in a Newtonian fluid. The stress is sought in terms of the instantaneous particle orientations, and the problem of determining these orientations from the history of the motion is not considered. The bulk stress and bulk velocity gradient in the suspension are defined as averages over an ensemble of realizations, these averages being equal to integrals over a suitably chosen volume of ambient fluid and particles together when the suspension is statistically homogeneous. Without restriction on the type of particle or the concentration or the Reynolds number of the motion, the contribution to the bulk stress due to the presence of the particles is expressed in terms of integrals involving the stress and velocity over the surfaces of particles together with volume integrals not involving the stress. The antisymmetric part of this bulk stress is equal to half the total couple imposed on the particles per unit volume of the suspension. When the Reynolds number of the relative motion near one particle is small, a suspension of couple-free particles of constant shape is quasi-Newtonian; i.e. the dependence of the bulk stress on bulk velocity gradient is linear. Two significant features of a suspension of non-spherical particles are (1) that this linear relation is not of the Newtonian form and (2) that the effect of exerting a couple on the particles is not confined to the generation of an antisymmetrical part of the bulk stress tensor. The role of surface tension at the particle boundaries is described.In the case of a dilute suspension the contributions to the bulk stress from the various particles are independent, and the contributions arising from the bulk rate of strain and from the imposed couple are independent for each particle. Each particle acts effectively as a force doublet (i.e. equal and opposite adjoining ‘Stokeslets’) whose tensor strength determines the disturbance flow far from the particle and whose symmetrical and antisymmetrical parts are designated as a stresslet and a couplet. The couplet strength is determined wholly by the externally imposed couple on the particle; but the stresslet strength depends both on the bulk rate of strain and, for a non-spherical particle, on the rate of rotation of the particle relative to the fluid resulting from the imposed couple. The general properties of the stress system in a dilute suspension are illustrated by the specific and complete results which may be obtained for rigid ellipsoidal particles by use of the work by Jeffery (1922).

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112070000745

The stress system in a suspension of force-free particles

The dispersion considered consists of a large number of identical small rigid spheres with random positions which are falling through Newtonian fluid under gravity. The volume fraction of the spheres (c) is small compared with unity. The dispersion is statistically homogeneous, and the axes of reference are chosen so that the mean volume flux across any stationary surface is zero. The problem is to determine the mean value of the velocity of a sphere (U). In §3 there is described a systematic and rigorous procedure which overcomes the familiar difficulty presented by the occurrence of divergent integrals, essentially by the choice of a quantity V whose mean value can be found exactly and which has the same long-range dependence on the position of a second sphere as U so that the mean of U – V can be expressed in terms of an absolutely convergent integral. The result is that, correct to order c, the mean value of U is U0(1 – 6.55 c), where U0, is the velocity of a single sphere in unbounded fluid. The only assumption made in the calculation is that the centres of spheres in the dispersion take with equal probability all positions such that no two spheres overlap; arguments are given in support of this assumption, which is expected to be valid only when the spheres are identical. Calculations which assume a simple regular arrangement of the spheres or which adopt a cell model of the hydrodynamic interactions give the quite different result that the change in the mean speed of fall is proportional to , for reasons which are made clear.The general procedure described here is expected to be applicable to other problems concerned with the effect of particle interactions on the average properties of dispersions with small volume fraction of the particles.

/pdf/sedimentation-in-a-dilute-dispersion-of-spheres-17ampt33yn.pdf

Sedimentation in a dilute dispersion of spheres

An exact formula is obtained for the term of order c2 in the expression for the bulk stress in a suspension of force-free spherical particles in Newtonian ambient fluid, where c is the volume fraction of the spheres and c [Lt ] 1. The particles may be of different sizes, and composed of either solid or fluid of arbitrary viscosity. The method of derivation circumvents the familiar obstacle, of non-absolutely convergent integrals representing the effect of all pair interactions in which one specified particle takes part, by the judicious use of a certain quantity which is affected by the presence of distant particles in a similar way and whose mean value is known exactly. The bulk stress is in general of non-Newtonian form and depends on the statistical properties of the suspension which in turn are dependent on the type of bulk flow.The formula contains two functions which are parameters of the flow field due to two spherical particles immersed in fluid in which the velocity gradient is uniform at infinity. One of them, p(r, t), represents the probability density for the vector r separating the centres of the two particles. The variation of p(r, t) for a moving material point in r-space due to hydrodynamic action is found in terms of a function q(r), and this gives p(r, t) explicitly over the whole of the region of r-space occupied by trajectories of one particle centre relative to another which come from infinity. In a region of closed trajectories, steady-state hydrodynamic action alone does not determine the relation between the values of p (r, t) for different material points. The function q(r) is singular when the spheres touch, and the contribution of nearly-touching spheres to the bulk stress is evidently important. Approximate numerical values of all the relevant functions are presented for the case of rigid spherical particles of uniform size.In the case of steady pure straining motion of the suspension, all trajectories in r-space come from infinity, the suspension has isotropic structure and the stress behaviour can be represented (to order c2) in terms of an effective viscosity . It is estimated from the available numerical data that for a suspension of identical rigid spherical particles
 \[
 {\mathop\mu\limits^{*}}/\mu = 1 + 2.5c + 7.6c^2,
 \]
 the error bounds on the coefficient of c2 being about ∓ 0.8. In the important case of steady simple shearing motion, there is a region of closed trajectories of one sphere centre relative to another, of infinite volume. The stress system is here not of Newtonian form, and numerical results are not obtainable until the probability, density p(r, t) can be made determinate in the region of closed trajectories by the introduction of some additional physical process, such as three-sphere encounters or Brownian motion, or by the assumption of some particular initial state.In the analogous problem for an incompressible solid suspension it may be appropriate to assume that for many methods of manufacture p(r, t) is uniform over the accessible part of r-space, in which event the solid suspension has ‘Newtonian’ elastic behaviour and the ratio of the effective shear modulus to that of the matrix is estimated to be 1 + 2·5c + 5·2c2 for a suspension of identical rigid spheres.

G. K. Batchelor

Papers

The effect of Brownian motion on the bulk stress in a suspension of spherical particles

Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivity

The stress system in a suspension of force-free particles

Sedimentation in a dilute dispersion of spheres

The determination of the bulk stress in a suspension of spherical particles to order c 2