Rate Of Viscous Energy Dissipation Research Articles

Analysis of dissipative mechanisms of cluster heterogeneous structures in gas–solid riser

In present study, a model of the criterion for cluster heterogeneous flow is presented in terms of the rates of hydrodynamic and collisional energy dissipations of the occurrence of particle clusters. The dynamic threshold distinguishes homogeneous flow from heterogeneous flow according to the collisional and hydrodynamic energy dissipations. Computation simulations indicate that the gas viscous energy dissipation rate is low about one to two order of magnitudes in comparison to hydrodynamic and collision energy dissipation rates. The energy dissipation of interface interaction is large in comparison to the energy dissipations of the dilute and dense phases. The dilute and dense collisional energy dissipations play different roles at the low and high solids volume fractions. As the relative velocity of gas and particles decreases, particle clusters are formed because the hydrodynamic energy dissipation becomes dominant. With the decrease of granular temperature, the formation of particle clusters is induced by collisional energy dissipation.

Non-dimensional energy dissipation rate near the turbulent/non-turbulent interfacial layer in free shear flows and shear free turbulence

The non-dimensional dissipation rate $C_{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D700}L/u^{\prime 3}$, where $\unicode[STIX]{x1D700}$, $L$ and $u^{\prime }$ are the viscous energy dissipation rate, integral length scale of turbulence and root-mean-square of the velocity fluctuations, respectively, is computed and analysed within the turbulent/non-turbulent interfacial (TNTI) layer using direct numerical simulations of a planar jet, mixing layer and shear free turbulence. The TNTI layer that separates the turbulent and non-turbulent regions exists at the edge of free shear turbulent flows and turbulent boundary layers, and comprises both the viscous superlayer and turbulent sublayer regions. The computation of $C_{\unicode[STIX]{x1D700}}$ is made possible by the introduction of an original procedure, based on local volume averages within spheres of radius $r$, combined with conditional sampling as a function of the location with respect to the TNTI layer. The new procedure allows for a detailed investigation of the scale dependence of several turbulent quantities near the TNTI layer. An important achievement of this procedure consists in permitting the computation of the turbulent integral scale within the TNTI layer, which is shown to be approximately constant. Both the non-dimensional dissipation rate and turbulent Reynolds number $Re_{\unicode[STIX]{x1D706}}$ vary in space within the TNTI layer, where two relations are observed: $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-1}$ and $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-2}$. Specifically, whereas the viscous superlayer and part of the turbulent sublayer display $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-2}$, the remaining of the turbulent sublayer exhibits $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-1}$, which is consistent with non-equilibrium turbulence (Vassilicos, Annu. Rev. Fluid Mech. vol. 47, 2015, pp. 95–114).

Turbulent Rayleigh-Bénard convection with polymers: Understanding how heat flux is modified

We study how polymers affect the heat flux in turbulent Rayleigh-Bénard convection at moderate Rayleigh numbers using direct numerical simulations with polymers of different relaxation times. We find that heat flux is enhanced by polymers and the amount of heat enhancement first increases and then decreases with the Weissenberg number, which is the ratio of the polymer relaxation time to the typical time scale of the flow. We show that this nonmonotonic behavior of the heat flux enhancement is the combined effect of the decrease in the viscous energy dissipation rate due to the viscosity of the Newtonian fluid and the increase in the energy dissipation rate due to polymers when Weissenberg number is increased. We explain why the viscous energy dissipation rate decreases with the Weissenberg number. Then by carrying out a generalized boundary layer analysis supplemented by a space-dependent effective viscosity from the numerical simulations, we provide a theoretical understanding of the change of the heat flux when the viscous energy dissipation rate is held constant. Our analysis thus provides a physical way to understand the numerical results.

Energy balance in viscous liquid containing a bubble: Rise due to buoyancy

The total energy balance of a system consisting of a bubble (radius 0.74 mm) freely rising in viscous liquid under the action of buoyancy was considered. Values of bubble acceleration, local and terminal velocities, and shape deformations calculated by numerically solving the Navier‐Stokes equation are compared with experimentally determined values. Additionally, the energies of the system associated with bubble motion (kinetic, potential, and rate of viscous energy dissipation), obtained from simulations, are given. On the basis of energy balances obtained for the system, two independent relations describing the motion of a bubble (Lagrange and Joseph equations) were used to show that, for steady‐state conditions, the constant (terminal) velocity of the bubble is a result of balance between buoyancy force and the drag originating from energy dissipation due to fluid viscosity. According to the calculations performed, the discrepancy between the expected and actual results of the Lagrange and Joseph equations for the system considered was ∼1 %. The estimated added mass coefficient was in close agreement with expected theoretical value for deformed spheres, and increased with decreasing tube radius.

A high‐speed tomographic PIV system for measuring zooplanktonic flow

Quantification of the hydrodynamic disturbance of free‐swimming zooplankton provides insight into propulsion as well as organism sensory interactions. Current flow measurement techniques, such as planar particle image velocimetry (PIV), are limited by their two‐dimensional nature. These techniques also are challenged by the small spatial scale of zooplankton, by the high speeds achieved by many zooplankton species, and by zooplankton photosensitivity to a broad range of wavelengths. We present a high‐speed tomographic PIV system using near‐infrared laser illumination that is capable of measuring three‐dimensional velocity vectors in a volume surrounding a plankter. This technique is assessed by recording and analyzing the time‐resolved flow field created by a high‐speed escape of a copepod (Calanus finmarchicus). Persistent body and wake vortices are created by the impulsive momentum transfer to the fluid surrounding the animal. It is shown that some aspects of this flow can be analytically modeled as an impulsive stresslet. Azimuthal asymmetry of the strength and position of the wake vortex is analyzed and attributed to the strong ventral flows created by the metachronally beating swimming legs. In addition, the energy required by a copepod escape jump is estimated by calculating the viscous energy dissipation rate using the spatial gradients of the measured three‐dimensional velocity field. Finally, the challenges and benefits of the tomographic PIV technique are discussed.

Minimax principle on energy dissipation of incompressible shear flow

The energy dissipation rate is an important concept in the theory of turbulence. Doering-Constantin’s variational principle characterizes the upper bounds (maximum) of the time-averaged rate of viscous energy dissipation. In the present study, an optimization theoretical point of view was adopted to recast Doering-Constantin’s formulation into a minimax principle for the energy dissipation of an incompressible shear flow. Then, the Kakutani minimax theorem in the game theory is applied to obtain a set of conditions, under which the maximization and the minimization in the minimax principle are commutative. The results explain the spectral constraint of Doering-Constantin, and confirm the equivalence between Doering-Constantin’s variational principle and Howard-Busse’s statistical turbulence theory.

Energy dissipation in a shear layer with suction

The rate of viscous energy dissipation in a shear layer of incompressible Newtonian fluid with injection and suction is studied by means of exact solutions, nonlinear and linearized stability theory, and rigorous upper bounds. The injection and suction rates are maintained constant and equal and this leads to solutions with constant throughput. For strong enough suction, expressed in terms of the entry angle between the injection velocity and the boundaries, a steady laminar flow is nonlinearly stable for all Reynolds numbers. For a narrow range of small but nonzero angles, the laminar flow is linearly unstable at high Reynolds numbers. The upper bound on the energy dissipation rate—valid even for turbulent solutions of the Navier–Stokes equations—scales with viscosity in the same way as the laminar dissipation in the vanishing viscosity limit. For both the laminar and turbulent flows, the energy dissipation rate becomes independent of the viscosity for high Reynolds numbers. Hence the laminar energy dissipation rate and the largest possible turbulent energy dissipation rate for flows in this geometry differ by only a prefactor that depends only on the angle of entry.

Thermal instability and evaporation of accretion disc atmospheres

We investigate the vertical structure of the outer layers of accretion discs in which the local viscous energy dissipation rate scales with the pressure as for standard Shakura‐Sunyaev discs. It has been pointed out by several authors that a thermal instability occurs in the outer layers of such discs when the gas pressure drops below a certain value. When the density becomes too low thermal equilibrium can no longer be maintained and the gas heats up, forming a hot corona or possibly a wind. To assess the importance of this effect we estimate the pressure and temperature at which this instability will occur, where the instability point lies with respect to the total vertical disc structure, and whether the instability is likely to be important for the disc as a whole. The main difference between our work and earlier estimates lies in a more detailed treatment of the heating and cooling processes and the inclusion of the effects of an external radiation field. By solving for the accretion disc vertical structure using the grey two-stream approximation instead of the usual diffusion approximation for the radiative transfer, we first show that the thermal structure of the optically thin outer layers is in first approximation independent of radiative transfer effects, and follows the thermal equilibrium curve for optically thin plasmas in the pressure‐temperature plane. We then calculate the thermal structure using the detailed photoionization code mappings, which includes much more accurate heating and cooling physics than the mean opacity used in the vertical structure calculations. This approach also allows a straightforward inclusion of the effects of an external radiation field from the centre of the accretion flow. We apply our method to cataclysmic variable (CV) and stellar mass black hole discs, and show that evaporation due to the thermal instability can be important under a variety of conditions. In the case of CVs, radiative heating by photons emanating from the boundary layer can increase the evaporation rate significantly over the non-irradiated case, but for steady-state CV discs the evaporation by the mechanism considered here is still not sufficient to evaporate the entire disc. It may become important however in non-steady discs in dwarf novae if the accretion-heated white dwarf plays a role in irradiating the disc after an outburst. In the case of black hole soft X-ray transients, the evaporation can have a significant effect on the structure of the outer regions of the disc, resulting in mass loss rates comparable to the local mass accretion rate through the disc for _

Reading the copepod personal ads: increasing encounter probability with hydromechanical signals

Females of the calanoid copepodTemora longicornisreact to chemical exudates of male conspecifics with little hops, quite distinct from their normal smooth uniform swimming motion. These hops possibly serve to create a hydrodynamical signal in the surrounding water, to increase encounter probability with potential mates. Laser sheet particle image velocimetry was used to investigate the flow fields associated with these hops and to compare them to the flow of the feeding current of an adult female. During, and immediately after a hop, the flow field around the copepod showed a marked difference from that of a foraging animal. During foraging, the highest velocity gradients were located around the feeding appendages of the copepod. During a hop, high velocity gradients are located behind the animal. About 0.5 seconds after the start of swimming leg movement, effects of the hop had virtually dissipated and the flow field resembled that around a foraging animal. The estimated volume of influence (i.e. the volume around the copepod where the animal has a significant influence on the water) increased about 12–fold during the hop compared with the situation around a foraging animal. Furthermore, the rate of viscous energy dissipation within the copepods' volume of influence increased nearly 80–fold. Hops may serve to increase encounter probability, but due to the short duration of the effect and the high energetic costs they would only be adaptive when other cues have indicated that suitable sexual partners are in the vicinity.

SPH simulations of tidally unstable accretion discs in cataclysmic variables

We numerically study the precessing disk model for superhump in the SU~UMa subclass of cataclysmic variables, using a two dimensional SPH code specifically designed for thin disk problems. Two disk simulations for a binary with mass ratio $q=\frac{3}{17}$ (similar to OY~Car) are performed, in order to investigate the Lubow (1991 a,b) tidal resonance instability mechanism. In the first calculation, a disk evolves under steady mass transfer from $L_1$. In the second simulation, mass is added in Keplerian orbit to the inner disk. The two disks follow similar evolutionary paths. However the $L_1$ stream-disk interaction is found to slow the disk's radial expansion and to circularise gas orbits. The initial eccentricity growth in our simulations is exponential at a rate slightly less than predicted by Lubow (1991a). We do not observe a clearing of material from the resonance region via the disk's tidal response to the $m=2$ component of the binary potential as was described in Lubow (1992). Instead the $m=2$ response weakens as the disk eccentricty increases. Both disks reach an eccentric equilibrium state, in which they undergo prograde precession. The rate of viscous energy dissipation in the disks has a periodic excess with a period matching the disk's rotation. The source is identified as a large region in the outer disk, and the mechanism by which it is produced is identified. The time taken for the periodic excess to develop is consistent with the first appearance of superhumps in a superoutburst.

Variational bounds on energy dissipation in incompressible flows: Shear flow.

A variational principle for upper bounds on the time averaged rate of viscous energy dissipation for Newtonian fluid flows is derived from the incompressible Navier-Stokes equations. When supplied with appropriate test ``background'' flow fields, the variational formulation produces explicit estimates for the energy dissipation rate. This dissipation rate is related to the drag of the fluid on the boundaries, and so these estimates translate into bounds on the drag. We analyze the problem of boundary-driven shear flow in detail, comparing the rigorous estimates obtained from the variational method with both recent experimental results and predictions of a conventional closure approximation from statistical turbulence theory.

Dynamic simulations of flows of bubbly liquids at large Reynolds numbers

Results of dynamic simulations of bubbles rising through a liquid are presented. The Reynolds number of the flow based on the radius and the terminal speed of bubbles is large compared to unity, and the Weber number, which is the ratio of inertial to surface tension forces, is small. It is assumed that the bubbles do not coalesce when they approach each other but rather bounce instantaneously, conserving the momentum and the kinetic energy of the system. The flow of the liquid is assumed to be irrotational and is determined by solving the many-bubble interaction problem exactly. The viscous force on the bubbles is estimated from the rate of viscous energy dissipation. It is shown that the random state of bubbly liquids under these conditions is unstable and that the bubbles form aggregates in planes transverse to gravity. These aggregates form even when the size distribution of the bubbles is non-uniform. While the instability results primarily from the nature of inertial interaction among pairs of bubbles, which causes them to be attracted toward each other when they are aligned in the plane perpendicular to gravity, it is shown that the presence of viscous forces facilitates the process.

A fracture model for a weak interface in a viscoelastic material (small scale yielding analysis)

The problem of adhesion at the interface between two polymers is investigated using a fracture mechanics approach. The effect of viscoelastic deformation of the bulk polymers and the micromechanical behavior of the interface on the fracture toughness of the interface was investigated by analyzing the problem of a steadily growing interface crack under the condition of small scale yielding. This condition assumes that the region of displacement discontinuity across the interface (the adhesive zone) due to loss of adhesion is small compared with typical specimen dimensions. It is found that the amount of inelastic or viscous energy dissipation is strongly coupled to the interface behavior and that the measured interface fracture toughness can be much higher than the intrinsic interface fracture toughness. For sufficiently high crack growth rates, a simple expression is obtained which relates the rate of viscous energy dissipation and the intrinsic interface fracture toughness to the viscoelastic properties of the bulk polymer. The dependence of the stress field inside the adhesive zone and the length of the adhesive zone on the viscoelastic properties of the polymer, the applied loading, the interface micromechanical model and the crack growth rate ȧ is obtained by numerically solving an integral equation. The size of the region of the viscoelastic energy dissipation is found to scale with ȧτr , where τr is the retardation time of the bulk polymer.

Study by laser Doppler anemometry of the turbulent flow induced by a Rushton turbine in a stirred tank: Influence of the size of the units—II. Spectral analysis and scales of turbulence

The effects of the size of the vessels and of the rotational speed of the turbine on the spectral analysis of anemometric signals were studied in an agitated tank; the energy spectrum and correlation coefficient, time and spectral integral scales, the turbulent diffusion coefficient, the viscous energy dissipation rate and the turbulence microscale were determined and discussed.

High shear viscometry of concentrated solutions of poly (alkylmethacrylate) in a petroleum lubricating oil

High shear capillary viscometry at 37.8°C (100°F) of concentrated solutions of a series of polyalkylmethacrylate viscosity index improvers in a petroleum lubricating oil is reported. Viscosity average molecular weights of the four polymers varied from 355 000–1650 000 and solution concentrations varied from 2–20 wt.-%. An approximating function based on the error function was computerfit to the complete flow curves by correlating the distribution of apparent viscosity with the product ( $$\dot \gamma $$ τ), the rate of viscous energy dissipation. This gave an estimate of the secondNewtonian viscosity (η ∞ ) and two parameters of the approximating function. The fourth quantity required to completely define the flow curve is the low shear or firstNewtonian viscosity (η 0 ). Representation of the original data was within 2%, by this technique. The parameters of the flow function — the energy level at the inflection point and the slope of the transformed flow function — were found to vary in a regular manner with both molecular weight of the polymer and polymer concentration, expressed as relative viscosity (η rel). The limiting asymptotes of the approximating function —η 0 andη ∞ — could not be treated according to the conventionalHuggins equation, but they were fit adequately by theMartin equation: log(η sp/C)=log[η]+K[η]C. The intrinsic viscosities thus determined for both low shear ([η M ]0) and high shear ([η M ]0) demonstrate aMark-Houwink relationship, i.e., [η M ]0=5.668×10−5 M 0.660 and [η M ]∞=2.574×10−5 M 0.0669 so that ([η M ]∞≈[η M ]0/2) over the range studied. The relationship of these results to other reports of high shear viscometry of polymer solutions is discussed.

Controlling Emissions from Gas Turbines—the Importance of Chemical Kinetics and Turbulent Mixing

Abstract An ideal combustor composed of a fuel-air mixing zone followed by a perfectly stirred reactor (PSR) and a series of plug-flow reactors is studied to determine emission levels of NOx CO, unburned hydrocarbons and soot. It is concluded that these emissions can be reduced to negligible levels provided the stirred reactor is operated at high mass loading rates (Q = [mdot]/P2V) and mixing is perfectly controlled. Fuel and air must be premixed before entering the PSR and local φ must be less than 1.2. Perfect stirring in the PSR is necessary. A controlled amount of dilution air is instantaneously added at the PSR exit to quench NO formation but not CO and hydrocarbon oxidation. The turbulent mixing process required to obtain this ideal is discussed. Mixing times are related to the time scale for large scale eddy breakup (τ1, ∼ l/u) which in turn can be related to the rate of viscous energy dissipation (∊∼u3/l). It is shown that to obtain PSR conditions, l must approach LC , the characteristic chamber d...

Extremum principles for slow viscous flows with applications to suspensions

Helmholtz stated and Korteweg proved that of all divergenceless velocity fields in a domain, with prescribed values on the boundary, the solution of the Stokes equation minimizes the rate of viscous energy dissipation. Hill & Power and also Kearsley proved the corresponding reciprocal maximum principle involving the stress tensor. We prove generalizations of both these principles to the flow of a liquid containing one or more solid bodies and drops of another liquid. The essential point in doing this is to take account of the motion of the solids or drops, which must be determined along with the flow. We illustrate the use of these principles by deducing several consequences from them. In particular we obtain upper and lower bounds on the effective coefficient of viscosity and a lower bound on the sedimentation velocity of suspensions of any concentration. The results involve the statistical properties of the distribution of suspended particles or drops. Graphs of the bounds are shown for special cases. For very low concentrations of spheres, both bounds on the effective viscosity coefficient are the same, and agree with the results of Einstein and Taylor.

Flow properties of a linear–gel polymer system

Flow curves (shear stress versus shear rate) were obtained for poly-(ethyl acrylate) containing up to 40% of gel particles over a range of shear rates from 3 × 10−3 to 7 × 103 sec.−1 with a specially designed biconical rheometer and a standard Instron capillary viscometer. Crosslink density of the gels was varied by emulsion-polymerizing ethyl acrylate with 3, 10, and 30% triethyleneglycol dimethacrylate. Particle sizes were varied by adjusting the concentration of emulsifying agent and by seeding techniques. Samples for study were prepared by blending and co-precipitating the linear and gel latices. The addition of gel always increases the shear stress at a given shear rate. However, this effect is greatest at the lowest shear rates. The flow properties of the gelcontaining materials are related quantitatively to those of the linear material through a new parameter, the rate of viscous energy dissipation, and by two material constants. These are a yield stress and a pseudo zero-shear viscosity, both of these being functions of the amount, crosslink density, and particle size of the gel phase.