Nanofluid was first proposed by Choi and Eastman [1
] 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 [2
]. 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 [3
]. 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
] 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 [7
] 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 [8
] 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. [9
] 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. [10
], 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 [11
] 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.
The fluid surrounding the nanoparticles will be assumed to be continuum. Knudsen number is defined as the ratio of the molecule mean free path of base fluid molecules to the nanoparticles diameter [11
is the particle diameter and λ
is the molecule mean free path of base fluid molecules and is given by:
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:
in Equation 2 is the stress tensor defined as:
is the shear viscosity of the base fluid phase and I
is the unit vector. Sp
in Equation 4 is the source term representing the momentum transfer between the fluid and particle phases and is obtained by computing the momentum variation due to several forces of slip, ∑ F
, experienced by the control volume as:
In the Lagrangian frame of reference, the equation of motion of a nanoparticle is given by:
The equation of motion of nanoparticles contains the drag force FD
, gravity FG
, Brownian motion force FB
, thermophoresis force FT
, Saffman's lift force FL
, rotational force FR
, and Magnus effect FM
and is given in the following equation:
The abbreviations for the different forces are listed in Table .
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.
Drag is the force generated in opposition to the direction of motion of a particle in a fluid. Drag force is proportional to the relative velocity between the base fluid and nanoparticle and is expressed by [12
is the velocity of the base fluid, vp
is the particle velocity, mp
is the mass of the particle, β
is the interphase momentum exchange coefficient:
is the Reynolds number due to drag:
is the drag coefficient for spherical particles and is given by:
For non-spherical particles [13
Here, Ψ is the shape factor:
is the surface area of the sphere of the same volume as the non-spherical particle:
and A is the actual surface area of the non-spherical particle.
Gravity force is proportional to the volume of the particle, and the relative density of nanoparticle and base fluid is expressed as:
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.
The random motion of nanoparticles within the base fluid is called Brownian motion and results from continuous collisions between the nanoparticles and the molecules of the base fluid. Brownian force is a function of concentration gradient, surface area of the particle, and the Brownian diffusion coefficient [11
is the Brownian diffusion coefficient (DB
) for spherical particles [11
For non-spherical particles:
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.
For non-spherical particles:
Small particles suspended in a fluid that has a temperature gradient experience a force in the direction opposite to that of the gradient. This phenomenon is known as the thermophoresis. Thermophoresis is a function of thermophoretic velocity, temperature gradient, Knudsen number, thermal conductivity, dynamic viscosity, and density of the nanofluid [11
is the volume fraction of the particle, vT
is a thermophoretic velocity:
is the density of nanofluid, μnf
is the dynamic viscosity of the nanofluid, T
is the temperature gradient, and Kn is the Knudsen number:
knf is the thermal conductivity of nanofluid and kp is the particle thermal conductivity.
Saffman's lift force
A free-rotating particle moving in a shear flow gives rise to a lift force. Lift due to shear, FL
has been derived by Saffman [14
] and it can be expressed as [15
is the radius of the particle, vbf
is the velocity of the base fluid, vnf
is the kinematic viscosity of the fluid,
is the shear rate
= 81.2, and D
is the diameter of the tube.
Particle rotational force
The force experienced by the particle due to rotational motion around a fixed axis is given as [7
Under the effect of the shear stress, a particle rotates about an axis perpendicular to the main flow direction. If a relative axial velocity exists between the particle and the fluid, a force perpendicular to the main flow direction will arise. This is known as the Magnus effect. It is a function of the difference between the axial velocity and radial velocity of the particle [11
The empirical relation proposed by Segre and Silberberg [16
] for the velocity of radial motion of the particles (vM
) is used:
is the mass flow rate of the particle, vm
is the mean velocity,
Thermophysical properties of nanofluids
The correlations used to compute the physical and thermal properties of the nanofluids are listed in Table . In this table, the subscripts p, bf, and nf refer to the particles, the base fluid, and the nanofluid, respectively. The density and specific heat of nanofluid are assumed to be a linear function of volume fraction due to lack of experimental data on their temperature dependence. Widely accepted correlations for determining the dynamic viscosity and thermal conductivity as a function of volume fraction for different nanofluids as shown in Table are used in the analysis. For Al2
-water nanofluids, the equation for effective thermal conductivity as suggested by Li and Peterson [17
] is used.
Thermophysical properties of nanofluids
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.
The drag force acting on the particle, FD
is given by:
The velocity of nanoparticle due to gravitational settling, vG
, can be calculated from a balance of buoyancy and viscous forces[11
The corresponding Reynolds number and the force due to gravity can be expressed as:
and the time scale for gravity:
The corresponding Reynolds number and the force due to Brownian motion can be expressed as:
and the time scale for Brownian:
The corresponding Reynolds number and the force due to thermophoresis can be expressed as:
and the time scale for thermophoresis:
Saffman's lift force
The corresponding Reynolds number and the force due to lift can be expressed as:
and the time scale for lift:
The corresponding Reynolds number and the force due to particle rotation can be expressed as:
and the time scale for rotational:
The corresponding Reynolds number and the force due to Magnus effect can be expressed as:
and the time scale for Magnus effect:
Applying the scaling analysis, to Equation 8, the acceleration term is normalized by the ratio of particle velocity to the relaxation time of the particle:
The particle relaxation time can be expressed as [11
and the particle Reynolds number as:
By substituting all the forces in Equation 9, we get the Reynolds number of the particle as a function of the Reynolds numbers of all the seven slip mechanisms:
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.