## Abstract

The present work theoretically addresses the experimental observations of nanofluid flow exhibiting highly intensified laminar heat transfer rates at the leading edge of channels or tubes. The basis for this study is the continuum conservation equations for nanofluids. The Rayleigh–Stokes approximation is applied to the nonlinear advective effects and a perturbation scheme, in ascending powers of the nanoparticle volume fraction, is applied. The disparate thicknesses of momentum, heat and volume fraction is exploited to advantage in securing analytical, similar solutions. The volume fraction layer is ‘infinitely’ thin in that its effect on the momentum and thermal transport is essentially its bulk value far from the wall. The composite resulting zeroth- and first-order perturbations show that an increasing modification in the velocity and temperature profiles occur with increasing volume fraction and that this is caused, and quantitatively assessed, by inertial effects of advection and enhanced nanofluid transport properties. Some satisfactory explanations of experiments are made for aluminium oxide nanoparticles in water, in terms of the ratio of nanofluid to base fluid heat transfer coefficients, local heat transfer coefficient and the Nusselt number.

## 1. Introduction

Nanofluid refers to fluids containing dispersed nano-sized particles, particularly that of metallic particles. The purpose of this mixture is to enhance the thermal conductivity of the effective medium and provide heat transfer rates surpassing that of the base fluid. This has a wide range of engineering applications discussed, for instance, in the book by Das *et al*. (2008). The effective-thermal conductivity was first developed by Maxwell (1873) and Lord Rayleigh (1892) for dispersed spherical particles of much larger size. The formalism for calculating the effective-thermal conductivity, nevertheless, is adaptable to a nanofluid. The effective thermal conductivity is expressed as a function of the nanofluid volume fraction, the conductivity of the nanoparticle material and that of the base fluid. The formalism is developed further to incorporate effects of particle geometry and particle-fluid interfacial resistance by Nan *et al*. (1997). The theoretical results are found to be in good agreement with the recent experimental measurements at over 30 international laboratories, as reported in Buongiorno *et al*. (2009). In general, for small nanoparticle volume concentrations, the ratio of nanofluid conductivity to that of the base fluid is a linear function of the nanoparticle volume fraction.

The theoretical description of convective transport in nanofluids is given impetus by the recent paper of Buongiorno (2006). While the effective-medium description has certain features of a single phase fluid, the nanoparticles affect the convective heat transfer via the thermophysical properties. The latter are dependent on the nanoparticle volume fraction, which in turn is subject to convective diffusion. Even in the presence of small nanoparticle volume fraction, where the thermophysical properties are expressible as linear functions of the volume fraction, there is sufficient coupling to render the convective transport problem nonlinear. The monograph on nanofluids (Das *et al*. 2008) addressed advances to include most of the 2006 year, including controversies of the effective-thermal conductivity prior to Buongiorno *et al*. (2009).

Experimental measurements in micro and non-micro channels and tubes (e.g. Wen & Ding 2004; Jung *et al*. 2009) indicate that there is strong dependence of the heat transfer rates on nanoparticle concentration near the channel entrance region and that this is accentuated closer to the leading edge particularly for the higher Reynolds number ranges. A small increase in the volume fraction is accompanied by large increases in the surface heat transfer rates not explainable on basis of the thermal conductivity alone. In fact, Wen & Ding (2004) propose a ‘smart channel’ for enhanced heat transfer, consisting a series of entrance regions rather than letting the flow become fully developed in a single long channel. In the entrance region, the flow is mimicked by the leading edge of a laminar boundary layer. The behaviour of large heat transfer rates in this region is referred to as ‘anomalous heat transfer enhancement’ (e.g. Prabhat *et al*. 2010). Rea *et al.* (2009) presented experimental evidence that no abnormal enhancement occurs, including the entrance region. Ding *et al.* (2007) also stressed that in some cases no enhancement takes place. The present contribution hopes to supplement these observations by considering transport effects from the conservation equations. It also appears that a more rational ‘equation of state’ for nanofluid density/heat capacity, other than from ideal mixtures relation, is most likely to help complete the enhancement explanation puzzle as the present efforts show.

## 2. Basic equations of the dynamics and thermodynamics of nanofluid flow

With Bird *et al*. (2001) as a guide (see also, Probtein 1994), Buongiorno (2006) generalized the conservation equations for the continuum description of the dynamics and thermodynamics of nanofluid flow. The composite fluid consists of a mixture of base fluid and dispersed embedded nanoparticles; the latter phase is in macroscopic momentum and thermodynamic equilibrium with the base fluid. The microscopic deviation of nanoparticle velocity from the mass-averaged plane manifests itself in the mass conservation for the nanoparticles and is modelled, in absence of imposed external forces, by Brownian and thermal diffusion (Buongiorno 2006). Such a continuum formulation is also discussed in Tzou (2008) and Pfautsch (2008). The resulting conservation equations, in the case of steady nanofluid flow in the leading edge region of a channel, are essentially the boundary layer equations (Pfautsch 2008). The global continuity is
2.1the streamwise momentum equation for plane flow is
2.2where the shear stress is
the normal momentum equation in the boundary layer approximation is
2.3where *u*, *v* are the streamwise and normal velocity components in the *x*, *y*- directions, respectively; the nanofluid properties are: *ρ* is the density, *p* the pressure and *μ* the viscosity. The leading edge problem, far from the developing region, is considered here so that the pressure at the edge of the boundary layer *p*_{e} is set equal to constant subsequently. The thermodynamic energy equation in boundary layer form and for (very) low Mach number flows is
2.4where *c* is the nanofluid heat capacity, *T* is the temperature; the normal component of heat flux *q*_{y} is accomplished by thermal conduction via the nanofluid thermal conductivity *k* and the transport of nanoparticle-specific enthalpy *h*_{p} by the normal diffusion flux *j*_{p,y},
It is very similar to the heat flux in a binary reacting gas including the diffusion-transport of thermal energy elucidated by Lees (1956). The diffusion flux is composed of Brownian diffusion flux and thermal diffusion (Buongiorno 2006)
The nanophase continuity is written in terms of the nanophase volume fraction *ϕ* and its diffusion flux:
2.5subscript p denotes properties of the nanoparticle phase (subscript f denotes that of the base fluid phase). For low Mach number flow, the rate of viscous dissipation and the work done by the pressure gradients are neglected from (2.4).

The boundary conditions are
and
where the subscript W denotes condition at the wall and that in the bulk fluid far away from the wall. The zero-flux boundary condition *j*_{p,y=0}=0 is first suggested by Buongiorno (2006), as the natural one for non-porous, inert-solid wall.

The thermophysical properties of the composite fluid are expressed, following Buongiorno (2006), in terms of the constituent components. On the basis of mixtures, the nanofluid density is expressed as *ρ*=*ϕρ*_{p}+(1−*ϕ*)*ρ*_{f}, the heat capacity as *c*=*ϕ*(*ρ*_{p}/*ρ*)*c*_{p}+(1−*ϕ*)(*ρ*_{f}/*ρ*)*c*_{f}. On the basis of correlations, the nanofluid viscosity is expressed as *μ*=*μ*_{f}(1+*a*_{1}*ϕ*+*a*_{2}*ϕ*^{2}), and thermal conductivity *k*=*k*_{f}(1+*b*_{1}*ϕ*+*b*_{2}*ϕ*^{2}), where *a*_{1},…,*b*_{1},… are nanoparticle-dependent correlation constants. The diffusion coefficients are: for Brownian diffusion, Stokes–Einstein relation: *D*_{B}=*k*_{B}*T*/3*πμd*_{p}, where *k*_{B} is the Boltzmann constant, *d*_{p} the particle diameter. The thermal diffusion coefficient is *D*_{T}=*μβϕ*/*ρ*, where *β*=*β*(*k*,*k*_{p})=*c*_{k}*k*/(2*k*+*k*_{p}), *c*_{k}≅0.26, *k*_{p} is the thermal conductivity for the nanoparticle material, is a tentative correlation for nanofluids as it is an extension of the correlation for micrometre-size particles. Nanofluid viscosity, measured by Pak & Cho (1998) showed no non-Newtonian effects. This is plausible as the nanoparticles essentially follow the motion of the base fluid (water), which is here Newtonian.

## 3. Approximate description of the leading edge of channel entrance region

The entrance region length *x*_{E} of a channel (or tube, when the viscous layer thickness is small relative to the tube diameter) can be estimated from that of the base fluid in absence of nanoparticles (Schlichting 1955) *x*_{E}/*D*≈0.04 *Re*_{f} for the velocity entrance length, where *D* is the channel width (or hydraulic diameter), *Re*_{f}=*UD*/*ν*_{f} is the base fluid Reynolds number, *U* is the entrance velocity and *ν*_{f}=*μ*_{f}/*ρ*_{f} is the kinematic viscosity of the base fluid. For experiments conducted in channels and tubes, the nanofluid properties are those of the bulk fluid. Since the viscous spreading rate involves the nanofluid kinematic viscosity, it is anticipated that the entrance region length of nanofluid to that of the base fluid is simply the ratio of kinematic viscosities . For instance, for aluminium oxide nanoparticles at about 1 per cent volume concentration, the nanofluid entrance length is shortened by about 25 per cent.

In Jung *et al*.'s (2009) experiments (e.g. their fig. 6), their *Re*=286, based on an averaged velocity, the momentum entrance length is very nearly *x*_{E}/*D*≈11. The thermal entrance length is estimated as *x*_{E,th}/*D*≈0.04 *RePr*, which is lengthier by a Prandtl number factor. The reduction of thermal entrance length in a nanofluid is approximated by . But still, for order of magnitude estimates, *x*_{E,th}/*x*_{E}≈*Pr* which ranges 5 to 10 for liquids with small concentration of nanoparticles as in the experiments. Within this leading edge region, the heat transfer rates are spectacular and increases many fold from that of the base fluid for a small increase in nanoparticle volume fraction. But for their *Re*=60 case, the measured heat transfer rates are already in the fully developed region and are thus very nearly flat. We therefore direct our studies towards the leading edge region, for which simplifying approximations are made and discussed in the following.

In the leading edge region, the left sides of (2.2, 2.4, 2.5) and the advective part of the continuity equation (2.1), the nonlinear advective operators are modelled by the Rayleigh–Stokes approximation,
3.1where the nonlinear advection velocities are replaced by entrance velocity *U*, but with the respective right sides remaining the same; this approximation is also known as the ‘plug flow’ model. In this case, the time-like effect of advection becomes obvious (advection time ≈*x*/*U*). In the leading edge of the entrance region, the boundary layers on the wall are not yet interacting with the bulk fluid, thus ∂*p*_{e}/∂*x*=0 in equation (2.2).

The base fluid, with properties denoted by the subscript f, is taken as incompressible, so are the nanoparticle properties with subscript p, but not implying that the nanofluid density *ρ* is necessarily ‘incompressible’ because of its involvement with the volume fraction and its diffusion equation. However, a common assumption is that the composite nanofluid is taken as *ρ*≈*ρ*_{f} incompressible (Buongiorno 2006; Tzou 2008), because of the smallness of the volume fraction in practice. However, this argument can be equally put forth for other thermophysical properties, and in so doing, the nanofluid would then be altogether assumed away. In a stratified flow, the incompressible assumption implies that the nanofluid density remains constant along each streamline as it is advected by the fluid. In the present approximation, the global continuity equation becomes, with the Rayleigh–Stokes approximation
3.2retaining the compressible effect that *ρ*≠*ρ*_{f}, consistent with other nanofluid thermophysical properties (*ρ*,*c*,*μ*,*k*,…) remaining ‘compressible’ because of the presence of the nanoparticle volume fraction *ϕ*, which causes the nonlinearity. Properties of the nanoparticle phase (such as *ρ*_{p},*c*_{p},…) and of the base fluid (*ρ*_{f},*c*_{f},…) are individually ‘incompressible’.

The continuity equation, within our considerations, would be used, if desired, to calculate the normal velocity *v* from *ρ*, which is expressed as a function of *ϕ*, the latter, *u* and *T* are calculated from the closed set of volume concentration, energy and *x*-momentum equations,

The following entrance region characteristic quantities are used to render the simplified conservation equations dimensionless: *U*, *D*, *ρ*_{f}, *μ*_{f}, *c*_{f}, *k*_{f}, *D*_{B,ref}, *D*_{T,ref}, , , *T*_{W}–. The dimensionless quantities
and
The thermophysical properties in dimensionless form become
with

A few of the resulting characteristic parameters that appear in the dimensionless conservation equations differ slightly from that of Buongiorno (2006) because of the anticipated perturbation in terms of the smallness of , which is extracted out of these parameters; but similar symbols and similar dimensionless parameters are defined: The Reynolds number is *Re*=*UD*/*ν*_{f}, defined with the base fluid kinematic viscosity; *Pr*_{f}=*ν*_{f}/*α*_{f} is the base fluid Prandtl number, where *α*_{f}=*k*_{f}/*ρ*_{f}*c*_{f} is the base fluid thermal diffusivity; the Lewis number is defined similarly as Buongiorno (2006), *Le*_{f}=*k*_{f}/*ρ*_{p}*c*_{p}*D*_{B,ref}, but without the perturbation parameter in the denominator, as is the Brownian to thermal diffusion coefficient ratio in which is absent from the numerator; the Schmidt number is defined as *Sc*_{f}=*ν*_{f}/*D*_{B,ref}.

## 4. Perturbation for small nanofluid volume fraction

The intermediate forms of dimensionless conservation equations are similar in to the dimensional ones and thus will not be stated here. Even to first order, as the thermophysical properties are expressed as linear functions of the volume fraction, the flow structure is nonlinear involving products of the volume fraction and flow quantities. The Rayleigh–Stokes approximation simplifies the nonlinear effects of advection, but not the nonlinearities owing to ‘compressibility effects’ through the volume fraction.

In order to systematically study the effect of small nanoparticle volume fraction on the flow structure, a perturbation procedure is devised by expressing flow quantities and thermophysical properties in ascending powers of the volume fraction. This is a linearization procedure about the flow of the base fluid for small volume fraction. In practice , so that any flow quantity or theromophysical property is systematically represented as a truncated perturbation series
4.1the zeroth-order quantity is that of the base fluid, the effect of nanoparticle volume fraction appears as perturbations in ascending series in powers of , in which only the *n*=0,1 terms are retained. As indicated by the right side of (4.1), all dimensionless asterisks are dropped in the perturbed quantities. For :
4.2The thermophysical properties are already in convenient perturbation form:
4.3The diffusion coefficients are expanded as
4.4where in more compact notation, , *T*^{(1)}=−(*T**_{W}−1)*θ*^{(1)}, and *ν*^{(0)}=1, *β*^{(0)}=1.

Substituting the expansions (4.1)–(4.4) into the conservation equations (3.7–3.9) and equating terms of like order in the perturbation parameter and omitting the asterisk on the dimensionless independent variables, a series of problems are obtained and individually described in the following.

### (a) The zeroth-order problem

The zeroth-order problem in the Rayleigh–Stokes approximation reduces to the familiar ‘heat equation’ 4.5with boundary conditions and The similarity variables are defined in terms of the respective estimates of momentum and thermal diffusion layer thicknesses so that and . The zeroth-order Rayleigh–Stokes problems for the velocity and temperature in similarity form are then 4.6which has solutions in terms of the error function 4.7The volume fraction enters only into the first-order perturbation, where and its similarity variable, in comparison, is defined as .

### (b) The scale of diffusion layers and the first-order problems

The nanofluids studied by Buongiorno (2006), Pfautsch (2008) and Tzou (2008) theoretically and experimentally by Jung *et al*. (2009), Wen & Ding (2004), is typified by Al_{2}O_{3} of about 10 nm diameter in water at standard conditions. The latter gives *Pr*_{f}≈5.9, *Sc*_{f}≈2×10^{4}, *Le*_{f}≈3.7×10^{3}, thus , Also, from similar estimates. In the following, the disparate relative thicknesses with respect to the volume fraction diffusion thickness are significant and are used to advantage in simplifying the system of perturbation conservation equations.

Within finite *η*_{u}, *η*_{T} respective regions of interest to the momentum and heat transfer problems, because *δ*_{u}, *δ*_{T}≫*δ*_{ϕ}, the perturbation volume fraction *Φ*^{(1)}(*η*_{ϕ}) and its derivatives take on their free-stream or bulk values so that *Φ*^{(1)}(*η*_{ϕ})→1, d*Φ*^{(1)}(*η*_{ϕ})/d*η*_{ϕ}→0. In this simplification, the momentum and thermodynamic problems become uncoupled from the concentration problem and are thus uncoupled from one another. On the other hand, within finite *η*_{ϕ} region of interest to the volume fraction diffusion layer, the temperature problem takes on its wall values , *θ*^{(0)}→0, .

### (c) The first-order temperature distribution

The first-order conservations are systemically obtained in physical coordinates and transformed into the respective similarity variables and incorporating the discussions of scale effects. The effect of order (*Le*_{f}*N*_{BT,ref})^{−1}≪1 is neglected in energy equation so that heat transport between the free stream and the wall is due to thermal conduction alone (Buongiorno 2006). Substituting the zeroth-order functions , *n*=1,2 into the first-order energy equation, the resulting simplified energy equation in similarity form is an inhomogeneous, source-like second-order heat equation with the same differential operator as the zero-order equation,
4.8The boundary conditions are homogeneous, , since the physical boundary conditions are already satisfied by the zeroth-order problem. The right side of (4.8) is a source- or sink-like effect if the sign of the bracketed terms is positive or negative, respectively. The enhanced nanofluid conductivity has the effect of smoothing the temperature profile, thus working against the enhanced conductivity in the calculation of heat conduction at the wall.

Integrating equation (4.8) once as a first-order inhomogeneous differential equation for d*θ*^{(1)}/d*η*_{T} yields the general solution (Murphy 1960, p. 14),
4.9where *C*_{1} is an integration constant and is the contribution to the enhanced surface heat transfer rate from the conservation equations, d*θ*^{(1)}(0)/d*η*_{T}=*C*_{1}. A straightforward integration of equation (4.9) then gives the first-order perturbation temperature profile
4.10The homogeneous condition at the wall gives *C*_{2}=0. The condition at infinity gives
4.11which provides a positive (enhanced) contribution to the surface heat transfer rate in nanofluids, if the parameters in the bracket remains positive. The error function terms in the profile (4.10) cancel and the first-order perturbation temperature profile becomes
4.12The profile function in the solution (4.12) is weighted by the coefficient , produces a modification in the overall temperature profile. This modification depends linearly on the volume fraction according to the perturbation expansion. The profile modification could conceivably be experimentally detected against variations of nanofluid properties and volume fraction (private communication, Buongiorno 2012).

### (d) The first-order velocity distribution

The (*Le*_{f}*N*_{BT,ref})^{−1}≪1 approximation brings similarity between the energy and momentum equations, the first-order velocity distribution can thus be inferred from the steps taken in solving the temperature problem (4.12). Thus, from (4.9)
4.13where the integration constant, inferred from (4.11) is
4.14The first-order perturbation velocity distribution is
4.15In (4.15), the factor (*ρ**_{p}−1−*a*_{1}), when multiplied by , contributes to a modification of the velocity distribution in much the same way as the temperature distribution. Similarly, the velocity profile modifications could be measured experimentally against variations of nanofluid properties.

### (e) Solution for the volume fraction

The volume fraction diffusion layer is deeply embedded inside the thermal and momentum boundary layers according to the scale-effects already discussed. The volume fraction conservation equation (2.5) is recast into the form:
4.16with the zero-wall flux boundary condition (Buongiorno 2006) and that in the free stream
where, for compactness, defined are
4.17The general solution is obtained systematically, similar as that for the energy and momentum equations,
4.18where *C*_{1,ϕ}, *C*_{2,ϕ} are integration constants. *Re*-define the independent variable as
which further simplifies the solution representation. The general solution is then, from Gröbner & Hofreiter (1949, p. 109) for the integral in (4.18),
4.19where the integration constants are identified with the volume fraction *Φ*^{(1)}(0)=*C*_{2,ϕ} and its slope d*Φ*^{(1)}(0)/d*η*_{ϕ}=*C*_{1,ϕ} at the wall. The boundary conditions for (4.16) give
It can be deduced from the representation for *a*, *b* in (4.17) that and that , so that for , the solution for zero-wall flux condition in presence of a uniform free-stream volume fraction is very nearly approximated by the ‘insulated wall’ condition which renders, at the wall, a slightly negative but nearly zero slope of the volume fraction and a wall volume fraction very nearly that of the free stream. These are anticipated if the approximation were directly applied to the volume fraction differential equation and wall boundary condition.

Other possible wall boundary conditions include an equivalent constant wall condition by setting *Φ*^{(1)}(0)=0. The constant volume fraction wall boundary condition is used by Tzou (2008) in his work on natural convection in nanofluids and by Pfautsch (2008) in the direct numerical integration of the forced convection boundary layer problem with free stream at zero volume fraction. In both cases, the constant volume fraction wall condition is interpreted physically as that controlled by mass exchange through a porous wall matrix (Buongiorno 2010, personal communication). The solution for *Φ*^{(1)}(0)=0, is found from (4.19) to be
which, again for , reduces to *Φ*^{(1)}(*η*_{ϕ})≈*erf*(*η*_{ϕ}/*T**_{W}) as also anticipated directly from the differential equation and its boundary conditions. In this case, in the overall region of interest to convective heat and momentum transfer, where *δ*_{ϕ}≪*δ*_{u}, *δ*_{T}, by taking *Φ*^{(1)}(*η*_{ϕ})→1 in finite regions of *η*_{u}, *η*_{T} earlier is shown to be a good approximation for either of the volume fraction wall boundary conditions.

## 5. Heat transfer

The surface heat transfer rate for boundary layer-type nanofluid flows is,
which includes the mechanisms of heat conduction and diffusion transport of nanoparticle enthalpy to the wall. For a non-porous wall, the wall boundary condition advanced by Buongiorno (2006) is that the diffusion flux at the wall is zero, (*j*_{p,y})_{0}=0. In this case, heat transfer at the wall is accomplished by conduction alone,
In terms of the dimensionless quantities and similarity variable already defined, directly from the outcome of the perturbation scheme, to first order
5.1where , and from the volume fraction solution with Buongiorno's wall condition, . In this case, the local heat transfer coefficient formed from (5.1) is
5.2for the same wall and free-stream temperatures for the base and nanofluids. The base fluid heat transfer coefficient from the Rayleigh–Stokes approximation is
5.3The present perturbation scheme brings out the explicit mechanisms contributing to nanofluid heat transfer. The nanofluid contribution to the heat transfer coefficient (5.2) is accounted for through the thermal conductivity in the definition of heat conduction at the wall which give rise to +*b*_{1}; −*b*_{1}/2 comes from the effect of increased heat conductivity in the energy equation. The inertia effect of the nanofluid density/heat capacity contributes to the factor . If only the nanofluid thermal conductivity in the definition of heat transfer at the wall is taken into account, but nanofluid effects on the fluid temperature profile according to the conservation equations are not, then the ratio *h*_{x}/*h*_{x,f} reduces to the ratio of thermal conductivities at the wall, , which is essentially the same as the free-stream nanofluid thermal conductivity ratio .

In the developing region, the definition of the local Nusselt number consistent with the local heat transfer coefficient is *Nu*_{x}=*h*_{x}*x*/*k*_{0}, where the streamwise development variable *x* is appropriately used instead of *D* and that the thermal conductivity used is appropriately the same as that in the definition of heat conduction relation at the wall, but that as shown. The local nanofluid Nusselt number is
5.4As the base fluid is assumed incompressible in the very beginning, the conductivity *k*_{f} is taken as a constant. The nanofluid thermal conductivity enters as a normalizing factor in the definition of the Nusselt number thus only the temperature profile modification coefficient of (4.12) appears in (5.4).

Of importance in convective heat transfer is the Stanton number, in which the surface heat transfer rate is normalized by the free-stream nanofluid convection. In this case, it is consistent to define the local Stanton number as 5.5resulting in 5.6The free-stream convective/inertia factor in the definition of the Stanton number incurs a correction to order .

The present perturbation considerations brought out three aforementioned mechanisms contributing to heat transfer modification from that of the base fluid: (i) transport property effect of nanofluid thermal conductivity, as coefficient of the temperature gradient, in evaluating the heat conduction at the wall, (ii) nanofluid thermal conductivity effect on the temperature profile through the energy equation, and (iii) inertial effects of convective transport in terms of the density/heat capacity on the temperature profile via the energy equation. The heat transfer coefficient exhibits all three effects explicitly, the Nusselt number involves only effects (ii) and (iii), whereas the Stanton number involves thermal conductivity effects (i) and (ii), modified by free-stream nanofluid convection.

## 6. Skin friction and the nanofluid Reynolds analogy

The skin friction is defined with the nanofluid viscosity coefficient evaluated at the wall *τ*_{0}=(*μ*∂*u*/∂*y*)_{0}. The velocity profile is computed via the momentum problem in terms of it similar solution; the nanofluid viscosity at the wall is , in which *Φ*^{(1)}(0)≈1 as shown earlier. The local skin friction in terms of the corresponding base fluid skin friction is , where *τ*_{0,f}=(*μ*_{f}*U*/*D*)(*Re*/*πx*/*D*)^{1/2}. The skin friction coefficient is obtained by consistently normalizing the skin friction by the free-stream dynamic pressure, where , then the ratio of local nanofluid skin friction coefficient is , where *c*_{fx.f}=2(*Reπx*/*D*)^{−1/2}.

The Reynolds analogy between heat transfer and skin friction, which follows from that of the base fluid, *c*_{fx,f}/2=*st*_{x,f}(*Pr*_{f})^{1/2}, to first order in is
Another form is obtained in terms of ratio to the base fluid for the same volume fraction
The largeness of the (measured) constant *a*_{1} limits the validity of the perturbation range of and the usefulness of the analogy is thus limited, but it illustrates and assesses the skin friction drag penalty accompanying nanofluid heat transfer enhancement.

## 7. Discussion of the results

The experimental results on heat transfer enhancement in nanofluids and its possible anomaly is recently extensively analysed and discussed in Prabhat *et al*. (2010, 2011). These authors compared experimental measurements with standard laminar heat transfer correlations but using as properties those of nanofluids, without appealing to the nanofluid transport effect via the conservations equations. This comparison is valuable and has shown that, especially in the leading edge of channels, large enhanced *h*_{x} were measured for small increases in . The enhancement experiments are typified by , Jung *et al.* (2009) and Lai *et al.* (2008) in the entrance region at higher Reynolds numbers.

On the other hand, Rea *et al.* (2009) presented convective heat transfer data for alumina and zirconia nanofluids in laminar flow, showing no abnormal enhancement with respect to the traditional Shah's correlation, including the entrance region. Ding *et al.* (2007) also noticed that in some cases the convective heat transfer coefficient is significantly larger than the thermal conductivity enhancement and not so in other cases. The present calculations, from the conservation equations, as approximate as it is, could provide possible explanations of some of the apparent incongruities and not in some other cases. The heat transfer coefficient derived in (5.2) is focused upon.

In the spirit of Rea *et al.* (2009), the heat transfer coefficient of the base fluid, (5.3), is extended empirically to hold for nanofluids provided that nanofluid properties are used as in the perturbation form from (4.3). Systematic substitution and expansion for small volume fractions recover the same relation (5.2). This shows that empirically one can obtain the same heat transfer relation, without addressing the mechanisms, as obtained from the differential equation derivation which accounts for the temperature profile modifications. Thus, the derived nanofluid heat transfer relation is embedded in the otherwise empirically used base fluid ‘correlation’ in the simplified present example.

In order to explicitly obtain the extent of enhancement, the enhancement relation from (5.2), (5.3) should be examined. For enhancement to exceed merely the thermal conductivity effect, then . However, for examples of nanofluids under discussion, . It is recalled that presently the ideal mixture relations is used for both the nanofluid density and heat capacity per unit volume. Most recently, Puliti *et al.* (2011), on the basis of molecular dynamics computations, found that for gold-water nanolayer mixtures, the molecular dynamics-obtained heat capacity is nearly twice that from ideal mixture results, (*ρc*)_{MD}≈1.85(*ρc*)_{ideal/mix}. The implication to explaining heat transfer enhancement is enormous in that inertia effects, in addition to thermal conductivity, give rise to enhanced heat transfer.

The temperature and velocity profile modifications, (4.12) and (4.15), respectively, could conceivably be measured in the laboratory. It can be seen that awaited is also improvements to the ideal mixture relations (Puliti *et al.* 2011), in the form of a similar equation of state, that can be incorporated into the explanation of heat transfer enhancement in nanofluids within the continuum transport description.

## Acknowledgements

I am indebted to E. Pfautsch and J. Buongiorno for my introduction to nanofluids (Buongiorno 2006; Pfautsch 2008). The preliminary versions of this work were presented at the *Int. Conf*. *Nanofluids: Fundamentals and Applications II*, 15–19 August 2010 Montréal, 63*rd* DFD/APS Meeting, 19–21 November 2010, San Diego and at the 64*th* DFD/APS Meeting, 20–22 November 2011, Baltimore.

- Received November 9, 2011.
- Accepted February 6, 2012.

- This journal is © 2012 The Royal Society