Scaling analysis for the investigation of slip mechanisms in nanofluids
© Savithiri et al; licensee Springer. 2011
Received: 30 November 2010
Accepted: 26 July 2011
Published: 26 July 2011
The primary objective of this study is to investigate the effect of slip mechanisms in nanofluids through scaling analysis. The role of nanoparticle slip mechanisms in both water- and ethylene glycol-based nanofluids is analyzed by considering shape, size, concentration, and temperature of the nanoparticles. From the scaling analysis, it is found that all of the slip mechanisms are dominant in particles of cylindrical shape as compared to that of spherical and sheet particles. The magnitudes of slip mechanisms are found to be higher for particles of size between 10 and 80 nm. The Brownian force is found to dominate in smaller particles below 10 nm and also at smaller volume fraction. However, the drag force is found to dominate in smaller particles below 10 nm and at higher volume fraction. The effect of thermophoresis and Magnus forces is found to increase with the particle size and concentration. In terms of time scales, the Brownian and gravity forces act considerably over a longer duration than the other forces. For copper-water-based nanofluid, the effective contribution of slip mechanisms leads to a heat transfer augmentation which is approximately 36% over that of the base fluid. The drag and gravity forces tend to reduce the Nusselt number of the nanofluid while the other forces tend to enhance it.
Nanofluid was first proposed by Choi and Eastman  about a decade ago, to indicate engineered colloids composed of nanoparticles dispersed in a base fluid. Contrary to the milli- and micro-sized particle slumped explored in the past, nanoparticles are relatively close in size to the molecules of the base fluid and thus can realize very stable suspensions with little gravitational settling over long periods of time. It has long been recognized that suspensions of solid particles in liquid have great potential as improved heat management fluids. The enhancement of thermal transport properties of nanofluids was even greater than that of suspensions of coarse-grained materials. In the recent years, many studies show that there is an abnormal increase in single phase convective heat transfer coefficient relative to the base fluid . Such an increase mainly depends on factors such as the form and size of the particles and their concentration, the thermal properties of the base fluid as well as those of the particles, kinetics of particle in flowing suspension, and nanoparticle slip mechanisms. The enhancement mechanism of heat transfer in nanofluid can be explained based on the following two aspects: (1) The suspended nanoparticles increase the thermal conductivity of the two-phase mixture and (2) the chaotic movement of the ultrafine particles due to the slip between the particles and the base fluid resulting in thermal dispersion plays an important role in heat transfer enhancement. Slip mechanisms of the particles increase the energy exchange rates in the nanofluid. Thermal dispersion will flatten the temperature distribution inside the nanofluid and make the temperature gradient between the fluid and wall steeper, which augments heat transfer rate between the fluid and the wall . Understanding the effect of different forces that bring about the slip mechanism is therefore essential in the study of convective transport of nanofluids.
An overall understanding of the effect of nanoparticle slip mechanisms for the augmentation of heat transport in nanofluids is in its infancy. In the past, several authors have attempted scaling analysis for convective transport of nanofluids to show the effect of slip mechanisms. Scaling analysis [4–6] is an effective tool to apply and develop mathematical models for describing transport processes. Through scaling analysis, the solution for any quantity that can be obtained from the governing equations can be reduced to a function of the dimensionless independent variables and the dimensionless groups. Ahuja  examined the augmentation in heat transport of flowing suspensions due to the contribution of rotational and translational motions by an order of magnitude analysis and concluded that the translational motion is expected to be negligibly small compared to that of the rotational motion of the particles. Savino and Paterna  performed order of magnitude analysis for Soret effect in water/alumina nanofluid and concluded that the thermofluid-dynamic behavior may be influenced by gravity and the relative orientation between the residual gravity vector and the imposed temperature gradient. Khandekar et al.  used scaling analysis for different nanofluids to show that entrapment of nanoparticles in the grooves of surface roughness leads to deterioration of the thermal performance of nanofluid in closed two-phase thermosyphon. Hwang et al. , in his study for water/alumina nanofluid, showed that both thermophoresis and Brownian diffusion have major effect on the particle migration and that the effect of viscosity gradient and non-uniform shear rate can be negligible. Buongiorno  estimated the relative importance of different nanoparticle transport mechanisms through scaling analysis for water/alumina nanofluid and concluded that Brownian diffusion and thermophoresis are the two most important slip mechanisms. Also, he ascertained that these results hold good for any nanoparticle size and nanofluid combination.
However, the different slip mechanisms between nanoparticle and the base fluid are dependent on several factors such as the shape, size, and volume fraction of the particle. Also, the thermophysical properties of the nanofluid used in the scaling analysis affect the magnitude of the slip forces in nanofluids which were not taken into consideration in the previous studies discussed above. Therefore, the objective of the present work is to carry out a detailed scaling analysis to understand the effect of seven different slip mechanisms in both water- and ethylene glycol-based nanofluids. A comprehensive parametric study has been carried out by varying the shape, size, concentration, and temperature of the nanoparticle in the fluid in order to understand the relative effect of these parameters on the magnitude of slip forces. The study is extended across different nanoparticles such as gold, copper, alumina, titania, silica, carbon nanotube (CNT), and graphene, suspended in the base fluid. The effect of slip mechanism on heat transfer augmentation in these nanofluids due to the slip mechanisms is also studied.
where R is the universal gas constant, T is the temperature, dm is the molecular diameters of base fluid, NA is the Avagodra's constant, and P is the pressure.
For water and ethylene glycol, the values of molecular mean free path are 0.278 and 0.26 nm, respectively. Therefore, for the nanoparticles in range of interest (1-100 nm), the Knudsen number is relatively small (Kn < 0.3); thus, the assumption of continuum is reasonable.
Continuous fluid phase
The governing equations for the continuous phase include the continuity equation (mass balance), equation of motion (momentum balance), and energy equation (energy balance). They are given, respectively, in the following:
Abbreviations for different forces
The coupling between the continuous fluid phase and discrete phase is realized through the Newton's third law of motion. The inter-particle forces such as the Van der Waals and electrostatic forces are neglected in the analysis due to their relatively negligible contributions in nanofluids.
The above forces are computed separately as shown below.
and A is the actual surface area of the non-spherical particle.
where Vp is the volume of the particle, ρp is the density of the nanoparticle, ρbf is the density of the base fluid, and g is acceleration due to gravity.
where KB is the Boltzmann constant, T is the temperature, h is the length of the non-spherical particle, dp is the particle diameter, μnf is the dynamic viscosity of the nanofluid. and vB is the Brownian velocity and is a function of temperature and diameter of the particle.
knf is the thermal conductivity of nanofluid and kp is the particle thermal conductivity.
Saffman's lift force
Particle rotational force
Thermophysical properties of nanofluids
Thermophysical properties of nanofluids
ρnf = ϕρp + (1 - ϕ)ρbf
(ρCnf) = ϕ(ρCp)p + (1 - ϕ) (ρCp)bf
μnf = μbf (T) (1 + 2.5ϕ)
Water/Cu, gold, CNT, grapheme
μnf = μbf (T) (1 + 39.11ϕ + 533.9ϕ2)
μnf = μbf (T) (1 + 5.45ϕ + 108.2ϕ2)
μnf = μbf (T) (1 + 56.5ϕ)
μnf = μbf (T) (1 + 11ϕ)
EG/Cu, gold, CNT, grapheme
μnf = μbf (T) (1 - 0.19ϕ + 3.6ϕ2)
μnf = μbf (T) (1 + 10.6ϕ + 112.36ϕ2)
knf - kbf = kbf (0.764ϕ + 0.0186T - 0.46215)
knf = kbf (1 + 2.92ϕ - 11.99ϕ2)
Scaling analysis methodology
In this section, the forces contributing to the slip between the particle and base fluid are analyzed through scaling analysis. A Reynolds number is introduced for each force depending on the velocity of the particle and the base fluid velocity. The time scale is defined as the time that a nanoparticle takes to diffuse a length scale that is equal to its diameter under the effect of that mechanism. In the present study, scaling analysis is to understand the order of magnitude for the forces involved in the slip mechanism of nanofluids.
Saffman's lift force
Thus, from scaling analysis, we arrive at an expression for the particle Reynolds number which is dependent on the Reynolds number of each of the slip mechanisms, the relaxation time of the particle, and the time scale of each mechanism.
Results and discussion
Parameters and their ranges used for scaling analysis
Particle size (nm)
Particle concentration (%)
Particle temperature (°C)
Effect of particle shape
Effect of particle size
Effect of particle concentration
Effect of particle temperature
Slip mechanisms in different nanofluids
The role of slip mechanisms in different nanofluids is also understood from the scaling analysis in Figures 1, 2, 3, 4, 5, 6, 7, and 8, and it is observed that for gold and copper nanoparticles suspended in water, the values of gravity, Saffman's, Brownian, Magnus, and rotational forces are higher than that of other nanoparticles. The value of drag force is found to be higher in silica- and alumina-based nanofluids. The value of thermophoresis force is also found to be very high in alumina-based nanofluids. The trend observed for water-based nanofluid is found to hold good for ethylene glycol-based nanofluids as well. Comparisons of forces in both the nanofluids are discussed below. The Brownian force is found to be 100 orders of magnitude higher in water-based nanofluids as compared to ethylene glycol-based nanofluids due to higher viscosity of water-based nanofluids. Thermophoresis force is 10 orders of magnitude higher in ethylene glycol-based nanofluids compared to water-based nanofluids. The value of rotational and drag forces are an order of magnitude higher in ethylene glycol-based nanofluids. However, the values of gravity and Magnus forces are found to be an order of magnitude higher in water-based nanofluids.
Role of slip mechanisms on heat transfer augmentation
The objective of this work is to understand the effect of nanoparticle slip mechanisms in different nanofluids through scaling analysis. The role of each mechanism is studied by considering the parameters of nanoparticle such as its shape, size, concentration, and temperature. From the scaling analysis, it is found that all of the slip mechanisms are dominant in particles of cylindrical shape as compared to the spherical and sheet particles. The parametric study on the effect of nanoparticle size is conducted by considering the particle size in the range of 1-80 nm. It is found that the magnitudes of slip mechanisms are higher for particles of large size between 10 and 80 nm. However, the Brownian and drag forces are found to dominate in smaller particles below 10 nm. The volume fraction of the nanoparticles in the suspension plays an important role in the thermophysical properties of the nanofluids. The random motion of the particles is enhanced in dilute suspensions; hence, Brownian force dominates at lower volumetric loading of nanoparticles. The Brownian and Magnus forces are found to be dominating at higher temperatures while the other forces are negligible.
With respect to forces, it is found that the Brownian force is more active for nanoparticles of cylindrical shape and in smaller-sized particles with a volumetric concentration of 1%. This force is predominant in water-based nanofluids due to higher viscosity as compared to ethylene glycol-based nanofluids. Thermophoresis force is found to be dominant for cylindrical nanoparticles at lower temperature of nanofluid. The drag force is found to increase with the increasing volume fraction and temperature of the nanoparticles and decrease with the increasing particle size. In terms of time scales, the Brownian and gravity forces act considerably over a longer duration than the other forces. This result holds good for both water- and ethylene glycol-based nanofluids. For copper-water-based nanofluids, the effective contributions of slip mechanisms lead to a heat transfer augmentation which is approximately 36% over that of the base fluid. The drag and gravity forces tend to reduce the Nusselt number of the nanofluid while the other forces tend to enhance it.
A, actual surface area of the non-spherical particle; A, area (m2); Asp, surface area of the sphere of the same volume as the non-spherical particles; C, specific heat (J/kg K); Cm, coefficient in Equation 18; Cs, coefficient in Equation 18; Ct, coefficient in Equation 18; dp, particle diameter (nm); D, diameter of the tube (m); dm, molecular diameter; F, force acting on the particle (N); g, gravity acceleration (m/s2); h, length of the cylindrical particle; I, unit vector; k, thermal conductivity (W/mK); K, thermal conductivity ratio (knf/kp); KB, Boltzmann constant = 1.3806504 × 10-23 (J/K); Kn, Knudsen number; KL, coefficient in Equation 19; mp, mass of the particle (kg); , mass flow rate of the particle (kg/s); NA, Avagadro's number = 6.0221367 × 1023/mol; Nu, Nusselt number; P, pressure (Pa); Pe, Peclet number; R, radius of the tube (m); R, universal gas constant = 8.3145 J/mol.K; Re, Reynolds number; Sp, source term; t, time (s); T, temperature (K); Tbf, stress tensor of nanofluids; v, velocity (m/s); V, relative velocity of nanofluid (m/s); Vp, volume of the particle (m3); z, location; z, location inside the tube (m)
β, inter-phase momentum exchange coefficient; , shear rate (s-1); λ, mean free path of the fluid (m); μ, dynamic viscosity (kg/ms); Ψ, shape factor; ρ, fluid density (kg/m3); τ, time (s); τp, relaxation time of the particle; ϕ, volume fraction of the nanoparticle; ν, kinematic viscosity (m2/s)
B, Brownian; bf, base fluids; D, drag; G, gravity; L, lift; m, mean; M, Magnus; nf, nanofluids; p, particle; R, rotational; T, thermophoresis
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