Bingham Plastic Fluid Research Articles

Analysis of biological mechanism of blood flow containing nanoparticles through an arterial stenosis

ABSTRACT This article aims to discuss the flow of nanoparticles in the blood through a stenosed artery in the presence of a magnetic field and periodic body acceleration. The overlapping shape of stenosis is chosen to show the impact of blood flow. The Bingham plastic fluid model is utilized to capture the non-Newtonian behavior of blood in the stenosed artery under diseased conditions. The resultant equations are solved analytically using the regular perturbation method. The impact of various emerging parameters on velocity, flow rate, effective viscosity, and wall shear stress are presented through graphs and discussed in detail. It is noticed that liquid velocity and flow rate increase whereas effective viscosity decreases with an increase in slip velocity and Womersley’s frequency parameter. It is also noted that liquid velocity and flow rate are decreased by enhancing the Lorentz force. The increase in body acceleration enhances fluid velocity.

CFD-based investigation on the flow of Bingham plastic fluids through 90^circ bends

In this paper, CFD-based results are presented on the estimation of losses in bends with validated numerical models for single-phase, incompressible Bingham fluids. Five fittings were investigated: 90^circ bends of different R/D ratios between 1 and 10. Bingham plastic fluids were studied with Hedström numbers in the wide range from 1 to 10^9. Loss coefficients are given as the function of the generalized Reynolds number ({textrm{Re}_textrm{gen}}= 0.1–17 800) and the generalized Dean number ({textrm{De}_textrm{gen}}= 0.1–12 600). The paper presents friction factors as well that were consistent with the literature. It was shown that at low generalized Dean numbers of {textrm{De}_textrm{gen}}< 40, the losses of the bends were caused purely by the wall friction and can be estimated from the {textrm{De}_textrm{gen}}. In a broad range of higher {textrm{Re}_textrm{gen}} and {textrm{De}_textrm{gen}} numbers, the loss coefficients were separated at the basis of the Hedström numbers. This phenomenon was mainly observable in the case of high curvature and was attributed to the flow patterns. Our study investigated this range with the bend of R/D=1 in detail: flow patterns in the bend and at the upstream side of the fitting were examined. An approximating equation and the fitting parameters were also given for loss coefficients.

CFD-based Estimation of Friction Factor in Rough Pipes with Herschel-Bulkley Fluids

The appropriate estimation of frictional losses in a pipeline system is essential. So far, little attention has been paid to determining the friction factors with non-Newtonian fluids, especially in rough pipes. This study aims at calculating the friction factor using validated three-dimensional Computational Fluid Dynamics models in Ansys CFX. Steady-state computations are performed with three different incompressible Herschel-Bulkley fluids in rough pipes with relative roughness of the inner pipe surface ε = 0.0005 – 0.01. A power-law type bath gel as a test fluid is used for experiments to validate our numerical model. The numerical results are compared with the measured values and also with numerous existing friction factor estimation models with the help of generalization of the Reynolds number in the relevant engineering range of Regen = 100 – 40,000. This paper shows that the existing approximations can not accurately describe the friction factor with pseudoplastic fluids in rough pipes. On the contrary, in the case of Bingham plastic fluid, a new, explicit calculation relation is found in a unified form accepted by the literature.

Study of multilayer flow of a bi-viscous Bingham fluid sandwiched between hybrid nanofluid in a vertical slab with nonlinear Boussinesq approximation

Bi-viscosity Bingham plastic fluids are used to understand the rheological characteristics of pigment–oil suspensions, polymeric gels, emulsions, heavy oil, etc. In many industrial and engineering problems involving high-temperature situation, a linear density-temperature variation is inadequate to describe the convective heat transport. Therefore, the characteristics of the nonlinear convective flow of a bi-viscous Bingham fluid (BVBF) through three layers in a vertical slab are studied. The two outer layers of the oil-based hybrid nanofluid and the intermediate layer of BVBF are considered. The thermal buoyancy force is governed by the nonlinear Boussinesq approximation. Continuity of heat flux, velocity, shear stress, and temperature are imposed on the interfaces. The governing equations are derived from the Navier–Stokes equation, conservation of energy, and conservation of mass for three layers. The nonlinear multi-point (four-point) boundary value problem is solved using the differential transform method (DTM). Converging DTM solutions are obtained, and they are validated. The entropy equation and Bejan number were also derived and analyzed. It is established that the nonlinear density–temperature variation leads to a significant improvement in the magnitude of the velocity and temperature profiles due to the increased buoyancy force, and as a result, the drag force on the walls gets reduced. The drag force on the slab gets reduced by decreasing the volume fraction of nanoparticles. Furthermore, nonlinear convection and mixed convection give rise to an advanced rate of heat transport on the walls and thereby to an enhanced heat transport situation.

Effect of Engineered Roughness on the Performance of Journal Bearings Lubricated by Bingham Plastic Fluid Using Computational Fluid Dynamics (CFD)

A journal bearing is a machine element that is used to keep the shaft rotating about its axis. The increasing demand for journal bearing applications in high-speed machines that are efficient and economical has resulted in the need for improvements to the acoustic and tribological performance of journal bearings. In order to improve the tribological and acoustic performance, this study aims to propose a novel journal bearing design by introducing a roughness condition in a specific zone of the stationary bearing surface. In addition, the impact of the application of engineered roughness on the performance of Bingham-plastic-lubricated bearings is investigated in more detail. Considering the effect of cavitation, the analysis was conducted using a 3D computational fluid dynamics (CFD) model of a journal bearing. In comparison with the Reynolds equation—which is inertialess—for lubrication analysis, the use of a 3D CFD model based on Navier–Stokes equations reflects more detailed flow characteristics. Moreover, in this work, variations in the area of surface roughness were employed, resulting in various roughness patterns on the surface of the journal bearing, so that the acoustic and tribological performances of the journal bearing were anticipated to be enhanced. The findings of this study show that under non-Newtonian lubrication of the bearing, the engineered roughness has a strong effect in altering the tribological performance. Furthermore, the well-chosen roughened surface was proven to be more pronounced in enhancing the load support and reducing the friction force. The simulation results also show that using an engineered surface has little effect on the noise of the bearing.

Rheology of debris flow materials is controlled by the distance from jamming

Debris flows are dense and fast-moving complex suspensions of soil and water that threaten lives and infrastructure. Assessing the hazard potential of debris flows requires predicting yield and flow behavior. Reported measurements of rheology for debris flow slurries are highly variable and sometimes contradictory due to heterogeneity in particle composition and volume fraction ([Formula: see text]) and also inconsistent measurement methods. Here we examine the composition and flow behavior of source materials that formed the postwildfire debris flows in Montecito, CA, in 2018, for a wide range of [Formula: see text] that encapsulates debris flow formation by overland flow. We find that shear viscosity and yield stress are controlled by the distance from jamming, [Formula: see text], where the jamming fraction [Formula: see text] is a material parameter that depends on grain size polydispersity and friction. By rescaling shear and viscous stresses to account for these effects, the data collapse onto a simple nondimensional flow curve indicative of a Bingham plastic (viscoplastic) fluid. Given the highly nonlinear dependence of rheology on [Formula: see text], our findings suggest that determining the jamming fraction for natural materials will significantly improve flow models for geophysical suspensions such as hyperconcentrated flows and debris flows.

Wall Effects for Spheroidal Particle in Confined Bingham Plastic Fluids

The wall effects on the sedimentation motion of a single spheroidal particle in cylindrical tubes filled with Bingham plastic fluid are investigated with the fixed computational domain using the Computational Fluid Dynamic (CFD) model in steady-state mode. The CFD model is validated with literature in both bounded and unbounded mediums. The rheological model of the Bingham plastic fluid is regularized with a smoothly varying viscosity. The retardation effects of the tube wall are presented in functions of Reynolds number Re, radius ratio λ (the radius of the tube to the semiaxis of the particle normal to the flow λ = R/r), aspect ratio E (the ratio of the semiaxis of the particle along the flow to r, E = b/r), and Bingham number Bn. The simulation results demonstrate that the drag coefficient C D declines with the rise in Reynolds number. The relative contribution to drag coefficient from the pressure force increases with larger Bingham number comparing with that from the friction force. The formation and size of the recirculation wake is suppressed by the yield stress. While Bn is approaching infinity, the limiting behavior is observed in the location of yield surface and the value of yield-gravity parameter. The values of critical yield-gravity parameter are explicitly given at different values of E, showing independence with Re and λ. For the flow with Bn ≥ 100, the influence of wall can be even ignored while λ is larger than 5.

Influence of Body Acceleration and Slip Velocity on Fluid Flow in a Multi-Stenotic Artery

Blockage of arteries by the formation of plaques causing stenosis is a major health problem among human beings.In this study, a model is developed for the flow through a multi-stenosed elastic artery segment under the effect of body acceleration and slip velocity. Fluid is regarded as Non-Newtonian Bingham Plastic Fluid. The equations representing the model are solved analytically using Perturbation Method. Result is discussed for velocity, wall shear stress and volumetric flow rate by varying several parameters like pulse frequency, slip parameter and Womersley Frequency parameter graphically.

Blood Flow Modeling in Constricted Arteries Under Body Acceleration and Wall Slip Using Two-Layered Bingham Plastic Fluid

Analysis is done on a two-layered Bingham Plastic model of restricted, thin arteries with periodic body acceleration. The model essentially consists of a centre layer with a core of suspended red blood cells and an outer layer with a peripheral plasma layer. It has been assumed that the rheology of blood in the core region has been classified as a Newtonian fluid with the PPL and a non-Newtonian fluid obeying the law of Bingham plastic model. This model has been used to investigate how blood flow in stenotic arteries is affected by body acceleration, the non-Newtonian character of blood, and a velocity slip at the wall. Analytical equations for axial velocity, flow rate, wall shear stress, and apparent viscosity are produced by using the perturbation method, and their variations with respect to various parameters are shown in the figures and explained in this article. Due to a wall slip, it is seen that while velocity and flow rate rise, effective viscosity falls. The impact of body acceleration significantly increases flow rates and speed. The physiological effects of this theoretical modelling for blood flow conditions are also briefly looked at.

Modeling lost-circulation in natural fractures using semi-analytical solutions and type-curves

Drilling is a requisite operation for many industries to reach a targeted subsurface zone. During operations, various issues and challenges are encountered, particularly drilling fluid loss. Loss of circulation is a common problem that often causes interruptions to the drilling process and a reduction in efficiency. Such incidents usually occur when the drilled wellbore encounters a high permeable formation such as faults or fractures, leading to total or partial leakage of the drilling fluids.In this work, a semi-analytical solution and mud type-curves (MTC) are proposed to offer a quick and accurate diagnostic model to assess the lost-circulation of Herschel-Bulkley fluids in fractured media. Based on the observed transient pressure and mud-loss trends, the model can estimate the effective fracture conductivity, the time-dependent cumulative mud-loss volume, and the leakage period. The behavior of lost-circulation into fractured formation can be quickly evaluated, at the drilling site, to perform useful diagnostics, such as the rate of fluid leakage, and the associated effective fracture hydraulic properties. Further, derivative-based mud-type-curves (DMTC) are developed to quantify the leakage of drilling fluid flow into fractures. The developed model is applied for non-Newtonian fluids exhibiting yield-power-law (YPL), including shear thickening and thinning, and Bingham plastic fluids. Proposing rigorous dimensionless groups generates the dual type-curves, MTC and DMTC, which offer superior predictivity compared to traditional methods. Both type-curve sets are used in a dual trend matching, which significantly reduces the non-uniqueness issue that is typically encountered in type-curves.Usage of numerical simulations is implemented based on finite elements to verify the accuracy of the proposed solution. Data for lost circulation from several field cases are presented to demonstrate the applicability of the proposed method. The semi-analytical solver, combined with Monte Carlo simulations, is then applied to assess the sensitivity and uncertainty of various fluid and subsurface parameters, including the hydraulic property of the fracture and the probabilistic prediction of the rate of mud leakage into the formation. The proposed approach is based on a robust semi-analytical solution and type-curves to model the flow behavior of Herschel-Bulkley fluids into fractured reservoirs, which can serve as a quick diagnostic tool to evaluate lost-circulation in drilling operations.

Influence of the Induced Magnetic and Rotation on Mixed Convection Heat Transfer for the Peristaltic Transport of Bingham plastic Fluid in an Asymmetric Channel

In this paper, the peristaltic flow under the impact of heat transfer, rotation and induced magnetic field of a two dimensional for the Bingham plastic fluid is discussed. The coupling among of momentum with rotational, energy and the induced magnetic field equations are achieved by the perturbation approximation method and the mathematica software to solve equations that are nonlinear partial differential equations. The fluid moves in an asymmetric channel, and assumption the long wavelength and low Reynolds number, approximation are used for deriving a solution of the flow. Expression of the axial velocity, temperature, pressure gradient, induced magnetic field, magnetic force, current density are developed the effect of rotation, a stress on the lower and upper channel and stream function. The quantities flow have been tested for variant parameters. The impact of the Bingham, Brinkman, Hartman and Grashof numbers are also tested for different values to indicate the effect on the move of flow fluid. The applications can be seen through many graphics.

Heat transfer enhancement compared to entropy generation by imposing magnetic field and hybrid nanoparticles in mixed convection of a Bingham plastic fluid in a ventilated enclosure

PurposeThe purpose of this study is to analyze the nonhomogeneous model on the mixed convection of Al2O3–Fe3O4 Bingham plastic hybrid nanofluid in a ventilated enclosure subject to an externally imposed uniform magnetic field. Entropy generation and the pressure drop are determined to analyze the performance of the heat transfer. The significance of Joule heating arising due to the applied magnetic field on the heat transfer of the yield stress fluid is described.Design/methodology/approachThe ventilation in the enclosure of heated walls is created by an opening on one vertical wall through which cold fluid is injected and another opening on the opposite vertical wall through which fluid can flow out.FindingsThis study finds that the inclusion of Fe3O4 nanoparticles with the Al2O3-viscoplastic nanofluid augments the heat transfer. This rate of enhancement in heat transfer is higher than the rate by which the entropy generation is increased as well as the enhancement in the pressure drop. The yield stress has an adverse effect on the heat transfer; however, it favors thermal mixing. The magnetic field, which is acting opposite to the direction of the inlet jet, manifests heat transfer of the viscoplastic hybrid nanofluid. The horizontal jet of cold fluid produces the optimal heat transfer.Originality/valueThe objective of this study is to analyze the impact of the inclined cold jet of viscoplastic electrically conducting hybrid nanofluid on heat transfer from the enclosure in the presence of a uniform magnetic field. The combined effect of hybrid nanoparticles and a magnetic field to enhance heat transfer of a viscoplastic fluid in a ventilated enclosure has not been addressed before.

Non-Newtonian Flow in Bifurcated Piping and Arterial Networks

Flows in multi-branch piping systems are modeled for linear Newtonian and nonlinear Power Law, Bingham Plastic and Herschel-Bulkley fluids. The nonlinear models are often used to describe multiphase fluids consisting of liquid and solid phases, including cement slurries and drilling muds in petroleum engineering. Rheological and heterogeneous multiphase effects are important to blood flows in single and bifurcated arteries, capillaries and veins – biological applications are also emphasized to highlight their increasing importance in medical research. Unfortunately, conventional engineering approaches, many oversimplified, employ idealized assumptions for total energy conservation. Others are unrealistic and based on empirical “head loss” coefficients related to fittings, roughness, bend and expansion effects. In our approach, mass conservation is assumed, leading to momentum and energy reductions and a credible predictive algorithm is devised. Bifurcation results are given for several fluid rheologies, either in closed analytical form or using numerical algorithms based on closed form solutions. When the entry flow rate and all outlet pressures are given, along with needed geometric details, solutions give pressure drop versus flow rate relations useful in pumping and power calculations. Also calculated are pressure levels that ensure safe, burst-free operations and wall viscous shear stress values that support cleaning and remediation operations. Analytical formulas are derived for circular flow cross-sections while numerical extensions are summarized for general clogged flows (applications in piping conduits of arbitrary cross-sectional geometry). Example calculations are given demonstrating the usefulness and versatility of the new methods.

Onset of Flow Reversal in a Vertical T‐channel: Bingham Fluids

AbstractThe commencement of the flow reversal phenomenon is investigated which occurs in the daughter branch of a vertical T‐channel for the flow of Bingham plastic fluids. The combined influences of the imposed forced flow and buoyancy‐driven flow are analyzed at low Reynolds numbers. The mixed convection flow of a Bingham model fluid is explored for a wide range of conditions in terms of Prandtl number, Richardson number, and Bingham number. The critical Richardson number is strongly influenced by the Prandtl number and the yield stress of the Bingham fluids. The dimensionless pressure is seen to decrease with the critical Richardson number. Also, the fractional flow passing through the main branch exhibits a strong direct dependence on the Richardson number while an opposite relationship with Bingham and Prandtl number is observed.

Mixed convection from semi-circular cylinder in Bingham plastic fluids: Adding buoyancy

Mixed convection heat transfer from a heated semi-circular cylinder in Bingham plastic fluids with adding buoyancy configuration has been studied numerically. Extensive results of fluid flow and heat transfer characteristics are presented in terms of the velocity and temperature profiles, shape and size of yield/unyielded zone. In the steady flow regime, Reynolds number, Prandtl number, and Richardson number are expected to be positively dependent on the average Nusselt number (which decreases the convection boundary layer). In contrast, the average Nusselt number decreases as the value of the Bingham number increases from its maximum value in Newtonian media to its conduction limit when most of the fluid is frozen.. Finally, using a simple expression, the numerical results of average Nusselt number have been predicted in terms of the j-factor for new applications.

Experimental and Theoretical Studies on Taylor Bubbles Rising in Stagnant Non-Newtonian Fluids in Inclined Non-Concentric Annuli

Experiments have been conducted measuring the drift velocity (Ud) of single Taylor bubbles rising in stagnant Bingham plastic fluids in an inclined (4° ≤θ≤ 45° from vertical) and fully eccentric 0.1524 × 0.1016-m (6 × 4-in) annulus. The results presented contribute to addressing the knowledge gap of how Taylor bubbles migrate through water-based fluids used in the process of drilling oil, gas and geothermal wells. The experimental results indicate that for all fluids tested,(1) Ud is generally maximized at an inclination angle (θ) of ∼30o,(2) Ud decreases with increasing plastic viscosity (μp) and yield point (τy),(3) Ud decreases when rotating the inner pipe of the annulus at a commonly used drilling operation speed of 100 rpm and(4) Ud is unaffected by the length of the Taylor bubble (LTB).It was also observed that LTB decreases as μp and τyincrease, and that pipe rotation has no effect on LTB.The characteristic dimension (D*), used in determining both the Froude number (Frθ) and apparent viscosity (μapp), has been selected as (Do+Di). These, inclination angle (θ), and experimental results from this and other studies form the basis for developing a drift velocity correlation. Additionally, a theoretical model applying a slot-flow approach is developed for predicting bubble rise velocities for both concentric and fully eccentric annuli. The drift velocity predicted from both the theoretical model and the correlation exhibit good agreement with the experimental results of the present work and those reported in literature.

Effect of real and whole blood rheology on flow through an axisymmetric stenosed artery

This study presents an extensive numerical investigation on the flow phenomena of a blood analogous fluid through an axisymmetric stenosed artery under both steady and pulsatile flow conditions. Most of the previous investigations, carried out for this benchmark problem, used either a simple Newtonian fluid model or a generalized Newtonian (GNF) fluid model like the power-law or Bingham plastic fluid model to represent the rheological behaviour of blood. However, many prior rheological studies showed that the real and whole blood exhibits both the shear-thinning and viscoelastic properties. In this study, for the first time, a multi-mode sPTT (simplified Phan-Thein-Tanner) model is used to carry out the numerical computation, which accounts for both the shear-thinning and viscoelastic rheological properties of blood. Additionally, a realistic set of viscoelastic model parameters, obtained by fitting the response of real and whole blood in standard viscometric flows, is used in this study. An excellent agreement is seen between the experimentally determined apparent viscosity (in simple shear flows) of blood with that predicted by the present viscoelastic model and other prior models developed for blood. Therefore, we believe that this study presents more accurate and realistic numerical results for the flow of blood through a stenosed artery than that presented by the previous studies in the literature. A simple Newtonian fluid model is also used in the present analysis to compare and show how the flow behavior of blood can be influenced by its complex rheological properties under otherwise identical conditions. We find a significant difference in the flow characteristics (in terms of the streamline profiles, velocity magnitude, pressure drop, etc.) obtained with the Newtonian and viscoelastic fluid models. For instance, the velocity gradient is seen to be large near the artery wall for the viscoelastic fluid model than that seen for the Newtonian model; whereas, a reverse trend is seen for the pressure drop across the stenosis. Furthermore, we find that the flow dynamics is strongly modulated by the stenosis geometry, Reynolds number and flow types, i.e., whether it is steady or pulsatile.

Effect of rheology and interfacial tension on spreading of emulsion drops impacting a solid surface

This paper presents an experimental and theoretical study of Newtonian and non-Newtonian (Bingham plastic) emulsion drop impact on a solid non-heated surface. The utilization of different emulsifiers at a constant concentration of continuous and dispersed phases in emulsions allows the considerable variation of the surface tension at the liquid–liquid interface. Our data for the maximum spreading diameter of water, n-decane, and emulsion drops impacting on a surface are compared with that predicted from the existing models for single-phase liquid drops. All selected models underpredict the experimental data. As a result, the importance of considering the capillary effects at the internal interfaces of the emulsion drops and the careful examination of all rheological properties in the case of Bingham plastic fluids is confirmed experimentally and is taken into account theoretically. The models of Pasandideh-Fard et al. [“Capillary effects during droplet impact on a solid surface,” Phys. Fluids 8, 650 (1996)] and Ukiwe and Kwok [“On the maximum spreading diameter of impacting droplets on well-prepared solid surfaces,” Langmuir 21, 666–673 (2005)] are modified and adapted to the emulsion drop by means of including the additional surface energy term at the liquid–liquid interface of the emulsion drop in the energy conservation equation and the non-Newtonian Reynolds number. The predictions of the maximum spreading diameter give good agreement with the measured one. Several constraints and future lines of research that relate to a specific behavior of the compound liquid drops at the impact on a solid surface are highlighted.

Stability criteria and convective mass transfer from the falling spherical drops, part I: Bingham plastic fluids

AbstractThis work deals with the flow and mass transfer around a falling spherical drop in Bingham plastic fluids. Bubbles, drops, and particles are often dispersed in a suspending fluid of non‐Newtonian nature. For example, suspensions of particles in a polymer solution are used in hydraulic fracturing slurries and swarms of bubbles are used in airlift bioreactors containing various biomacromolecules, as well as in foam production, degassing, and de‐volatilization of polymer melts. Governing equations for momentum and mass transfer and the Bingham constitutive equation are solved over a wide range of dimensionless groups as Reynolds number, ; Schmidt number, ; Bingham number, ; and viscosity ratio (0.1 and 10). The local underline transport processes are expressed using streamlines, concentration contours, and sheared and un‐sheared regions, whereas the average transport quantities are reported in terms of drag coefficient, critical yield‐stress parameter, and Sherwood number. A co‐existence of sheared and un‐sheared regions occurs within the flow domain due to the fluid‐yield stress. The sheared regions progressively diminish with the increasing value of the Bingham number. On the other hand, inertial effects (increasing ) promotes the expansion of the sheared region in the flow domain. The new modified dimensionless groups are defined by re‐scaling the governing equations based on the effective viscosity of the fluid. Finally, the present numerical values of the drag and average Sherwood number are correlated using the new modified dimensionless numbers.

Effect of Sinusoidally Varying Flow of Yield Stress Fluid on Heat Transfer From a Cylinder

Abstract The effect of pulsating laminar flow of a Bingham plastic fluid on heat transfer from a constant temperature cylinder is studied numerically over wide ranges of conditions as Reynolds number (0.1 ≤ Re ≤ 40) and Bingham number (0.01 ≤ Bn ≤ 50) based on the mean velocity, Prandtl number (10 ≤ Pr ≤ 100), pulsation frequency (0 ≤ ω* ≤ π), and amplitude (0 ≤ A ≤ 0.8). Results are visualized in terms of instantaneous streamlines, isotherms, and apparent yield surfaces at different instants of time during a pulsation cycle. The overall behavior is discussed in terms of the instantaneous and time-averaged values of the drag coefficient and Nusselt number. The size of the yielded zone is nearly in phase with the pulsating velocity, whereas the phase shift has been observed in both drag coefficient and Nusselt number. The maximum augmentation (∼30%) in Nusselt number occurs at Bn = 1, Re = 40, Pr = 100, ω* = π, and A = 0.8 with respect to that for uniform flow. However, the increasing yield stress tends to suppress the potential for heat transfer enhancement. Conversely, this technique of process intensification is best suited for Newtonian fluids in the limit of Bn → 0. Finally, a simple expression consolidates the numerical values of the time-averaged Nusselt number as a function of the pertinent dimensionless parameters, which is consistent with the widely accepted scaling of the Nusselt number with ∼Pe1/3 under these conditions.