# Mathematical Modeling and Computer Simulations of Nanofluid Flow with Applications to Cooling and Lubrication

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Use of Nanofluids in Cooling and Lubrication

#### 2.1. Experimental Evidence

#### 2.1.1. Cooling with Nanofluids

_{2}O

_{3}-water nanofluid flow in rectangular microchannels in the laminar flow regime with NP-volume fractions up to 1.8%. They found that the Nusselt number of the nanofluid flow can be correlated by $\text{Nu}=0.014{\mathsf{\phi}}^{0.095}{\mathrm{Re}}^{0.4}{\mathrm{Pr}}^{0.6}$. The measured increase in convection heat transfer coefficient at 1.8% NP volume fraction was 32% over that of the distilled water, while the measured friction losses were similar for both the nanofluid and the base fluid. Ho et al. [31] experimentally assessed the thermal and hydraulic performance of Al

_{2}O

_{3}-water nanofluid flow in a copper MCHS with rectangular microchannels in the laminar flow regime. It was found that the friction factor in the microchannel heat sink cooled by the nanofluid, containing up to 2 vol % NPs, slightly increased relative to the pure water. On the other hand, a 70% increase in average heat transfer coefficient and 25% decrease in MCHS thermal resistance were found by adding 1 vol % NPs to the base fluid at Re = 1676. Using nanofluids was found to be more beneficial at higher flow rates. In contrast, the experimental study of Anoop et al. [32] showed that, when compared to water, nanofluid heat transfer enhancements occurred at lower flow rates in a microchannel, and heat transfer degradation occurred at higher flow rates due to NP deposition on the microchannel wall. Byrne et al. [33] measured the fluid flow and heat transfer characteristics of CuO-water nanofluids in microchannels with NP volume fractions from 0.005% to 0.1%. While a modest improvement in heat transfer was found, the use of a surfactant was essential in maintaining a proper suspension of NPs in the liquid. Rimbault et al. [34] evaluated transitional and turbulent CuO-water flow in a MCHS with NP-diameters of 29 nm and volume fractions ranging from 0.24% to 4.5%. Little heat transfer enhancement was obtained by adding NPs at low concentrations, while a clear decrease was found at high concentrations. On the other hand, a large pressure drop increase (up to 70% at 4.5 vol %) with respect to water was found for all NP-loading ranges. This is the only study presenting results of transitional and turbulent nanofluid flow in microsystems, which is opposite to the beneficial findings for thermal nanofluid flow in the laminar regime. Indeed, the observed deterioration in heat transfer was confirmed by the authors of [35], who analyzed Al

_{2}O

_{3}-water flow in a mini-channel. Rimbault et al. [34] suggested that particle migration towards the wall and enhanced local particle deposition may be the reasons. Manay and Sahin [36] experimentally determined the effect of the aspect ratio of microchannels on nanofluid heat transfer. While nanofluids provided better heat transfer performance without causing excessive pressure drop, reduced heat transfer and increased pressure drop were generated by nanofluid flow in microchannels with high aspect ratios. Azizi et al. [37] studied the thermal performance and friction factor of a cylindrical MCHS cooled by Cu-water nanofluids. The results showed that the presence of NPs enhanced the entrance Nusselt number up to 43%; however, the friction factor also increased up to 45.5% when compared to the base fluid.

#### 2.1.2. Nanoparticles in Lubrication

_{2}, Al

_{2}O

_{3}, CuO, ZnO, ZrO

_{2}, and ZnAl

_{2}O

_{4}) and their composites, sulfides (e.g., WS

_{2}, MoS

_{2}, CuS, and ZnS), nonmetals (e.g., diamond, SiO

_{2}, carbon nanotubes, and fullerenes), inorganic oxythiomolybdates (e.g., Cs

_{2}MoO

_{2}S

_{2}, ZnMoO

_{2}S

_{2}), and baron-based nanoparticles (e.g., calcium borate, zinc borate, boron nitride, and alkali borates). Most of the evidence indicates that adding NPs to the lubricants can significantly improve lubrication performance by reducing both friction and wear, thereby allowing for an increase in load-carrying capacity [21,40,41,42,43,44,45,46,47,48,49,50,51]. A number of mechanisms have been proposed to explain the superior performance of lubricants with nanoparticle additives, including surface adsorption, penetration into asperities, and tribo-chemical reaction to reduce wear, as well as size effect, colloidal effect, exfoliation, protective film, and third-body effect to reduce friction (see Figure 5 and Figure 6).

_{2}-suspensions using a four-ball tester. The author found that finer particles only improved wear performance at high loads with smooth surfaces. He suggested that temperature, material, and surface roughness must be taken into account in order to transfer these results to conditions different from the four-ball tester. Cusano and Sliney [54,55] added sub-micron graphite and MoS

_{2}particles to a mineral oil and found deleterious effects under boundary lubrication conditions of a Hertzian contact.

_{2}nanoparticles with organic agents, activating the inorganic nanoparticles. They proposed that the nanoparticles could bond to specific active agents and act as nano-reservoirs to store these additives. Once the NPs steer through the asperities of the contacting surfaces, they can deliver the organic agents and thus achieve specific functions. This idea is in the same vein as the methodology of targeted nanodrug delivery [5]. Recent experimental studies, focusing on reduced friction, anti-wear, and enhanced load-carrying capacity, have explored different NP materials and various surfactants to stabilize the mixture [52,70,83,84,85,86,87,88,89,90].

#### 2.2. Theoretical Background

#### 2.2.1. Forces Acting on Nanoparticles

**Random force**. The random force $\overrightarrow{{F}_{B}(t)}$ that derives from the collisions between particles and the randomly moving molecules of the base fluid gives rise to the Brownian motion of the nanoparticles. Based on the Langevin equation and the equi-partition theorem, the stochastic process causes an average instantaneous nanoparticle velocity of ${\overrightarrow{v}}_{p}$, which can be expressed as

**Thermophoretic force**. Particles can diffuse under the effect of a temperature gradient, where particles migrate from hotter to colder regions and cause a higher concentration of nanoparticles in the colder regions. This phenomenon is called thermophoresis and is equivalent to the Soret effect for gaseous or liquid mixtures. The thermophoresis effect can be expressed in terms of a thermophoretic velocity:

**Body force**. Nanoparticles may experience different field forces depending on the types of applications. Typically, the gravitational force applies due to the density difference between the nanoparticles and the base fluids:

**Stokes force**. When a nanoparticle moves relative to the base fluid, it experiences a drag force. At particle Reynolds numbers of $\mathrm{Re}\le 1$, the drag force can be expressed according to Stokes’ law as

**Inter-particle forces**. When two particles approach each other due to Brownian motion, the attractive van der Waals forces and the repulsive electrostatic force become important.

**Relative importance of these forces**. An order-of-magnitude analysis of the Stokes force and the inter-particle forces in nanofluids has been provided in [96]. In dilute nanofluids, the surface forces, i.e., the van der Waals forces and the electrostatic force, are of the same order as the Stokes force if multi-particle interactions are considered. Meanwhile, Buongiorno [97] has shown that body forces are negligible compared to the random force that causes Brownian diffusion. However, the surface forces are only important when nanoparticles are well dispersed in the base fluid. When nanoparticle aggregation occurs, the surface forces become less important, while wake interactions due to Brownian motion dominate.

#### 2.2.2. Nanoparticle Aggregation

#### 2.2.3. Thermo-Physical Properties of Nanofluids

**Thermal conductivity of nanofluids**. Certain nanofluids have shown significantly enhanced thermal conductivities over their base fluids. It has been shown that the effective thermal conductivity of the mixture ${k}_{nf}$ increases with the nanoparticle volume fraction and temperature, and with the decrease of particle size (see Figure 5). In addition, particle shape, pH, and aggregation also have significant effects on ${k}_{nf}$.

_{2}O

_{3}-water nanofluids. The characteristic time interval ${\mathsf{\tau}}_{a}$ is expressed as

**Viscosity of nanofluids**. Measurements of the rheological properties of nanofluids indicate that most nanofluids behave as Newtonian-fluid mixtures at low particle volume concentrations [103,104]. Particle concentration, size [105], shape [106], temperature, shear rate [107], surfactants, and pH have direct impacts on the viscosity of nanofluids. For example, viscosity increases with higher particle volume fractions, while it decreases with elevated temperatures. However, due to the aggregation effect of nanoparticles, it is difficult to determine how the particle size affects the effective viscosity. Surfactants and pH affect ${\mathsf{\mu}}_{nf}$ mainly by enhancing the dispersion of nanoparticles.

_{bf}is the equivalent diameter of a base fluid molecule, given by

_{0}= 293 K.

**Convection heat transfer coefficient**. Convection heat transfer performance is the most important measure of thermal nanofluid flow. Heat transfer performance for Fourier-type processes is characterized by the convective heat transfer coefficient or the Nusselt number, i.e.,

_{w}is the local wall heat flux, and Nu

_{x}is the Nusselt number at location x.

#### 2.3. Transport Equations and Modeling Approaches

#### 2.3.1. Single-Phase Approach

_{p}< 100 nm, that they closely follow the fluid streamlines, causing the nanofluid to behave like a homogeneous mixture [103]. Thus, the single-phase Navier-Stokes equations can be used with the effective thermo-physical properties of nanofluids, as follows:

- (Laminar flow)$$\frac{\partial ({\mathsf{\rho}}_{nf}{u}_{i})}{\partial t}+\frac{\partial}{\partial {x}_{j}}({\mathsf{\rho}}_{nf}{u}_{i}{u}_{j})=-\frac{\partial p}{\partial {x}_{i}}+\frac{\partial}{\partial {x}_{j}}\left[{\mathsf{\mu}}_{nf}\left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}\right)\right],\mathrm{and}$$
- (Turbulent flow)$$\frac{\partial ({\mathsf{\rho}}_{nf}{u}_{i})}{\partial t}+\frac{\partial}{\partial {x}_{j}}({\mathsf{\rho}}_{nf}{u}_{i}{u}_{j})=-\frac{\partial p}{\partial {x}_{i}}+\frac{\partial}{\partial {x}_{j}}\left[{\mathsf{\mu}}_{nf}\left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}\right)-{\mathsf{\rho}}_{nf}\overline{{u}_{i}{u}_{j}}\right];$$

- (Laminar flow)$$\frac{\partial \left({\mathsf{\rho}}_{nf}h\right)}{\partial t}+\frac{\partial}{\partial {x}_{i}}\left[{\left(\mathsf{\rho}{c}_{p}\right)}_{nf}{u}_{i}T\right]=\frac{\partial}{\partial {x}_{i}}\left({k}_{nf}\frac{\partial T}{\partial {x}_{i}}\right)+{\mathsf{\mu}}_{nf}\mathrm{\Phi},\mathrm{and}$$
- (Turbulent flow)$$\frac{\partial \left({\mathsf{\rho}}_{nf}h\right)}{\partial t}+\frac{\partial}{\partial {x}_{i}}\left({\mathsf{\rho}}_{nf}{u}_{i}{h}_{tot}\right)=\frac{\partial}{\partial {x}_{i}}\left({k}_{nf}\frac{\partial T}{\partial {x}_{i}}-{\mathsf{\rho}}_{nf}\overline{{u}_{i}h}\right)+{\mathsf{\mu}}_{nf}\mathrm{\Phi}-\frac{\partial {u}_{i}}{\partial {x}_{j}}\left({\mathsf{\rho}}_{nf}\overline{{u}_{i}{u}_{j}}\right),$$

#### 2.3.2. Two-Phase Approach

#### 2.3.3. Nanofluid Transport in Porous Media

_{0}is a reference temperature, and $\mathsf{\beta}$ is the volume expansion coefficient. The superficial velocity ${u}_{i}$ is used in the equations.

#### 2.3.4. Magnetic Nanofluids

#### 2.3.5. Nanofluid Transport in Thin Films

## 3. Nanofluid Flow Applications

#### 3.1. Entropy Generation

#### 3.2. Nanodroplet-Vapor-Air Mixture Dynamics

^{−3}·s

^{−1}), i.e.,

#### 3.3. Nanofluids for Microsystem Cooling

_{nf}model, whose predictions agree well with available measured data in the literature. They suggested that nanofluids measurably enhance the thermal performance of microchannel mixture flow with a small penalty in pumping power. Lelea [141] and Lelea and Nisulescu [142] numerically investigated nanofluid flow in microchannels. They found that the enhancement of heat transfer due to the presence of NPs rises along the channels. The heat transfer enhancement for cooling and heating are different for the same NP concentration. Additionally, neglecting viscous heating effects would give incorrect predictions. Ting et al. [143] investigated the effects of streamwise conduction on the thermal performance of nanofluid flow in MCHS’s under exponentially decaying wall heat flux. The Peclet number was found to strongly affect the temperature distribution when the streamwise conduction is incorporated. The effect of streamwise conduction became important when Pe < 10.

#### 3.4. Nanoparticles for Enhanced Lubrication

_{2}nanoparticles in engine oil can reduce end-leakage as well as friction, thus improving the load capacity of the bearing featuring negative radial and tilt adjustments. Binu et al. [179] used the modified Krieger-Dougherty viscosity model to incorporate the nanoparticle effect on the fluid rheology. They introduced coupled stresses to the Reynolds equation to account for the impact of nanoparticles. The model predicted a significant increase in load-carrying capacity of the journal bearing by using a TiO

_{2}-based lubricant. Nicoletti [180] found that adding nanoparticles to the base oil modifies the volumetric heat capacity of the lubricant, resulting in lower temperature and thus higher viscosity for an improved load-carrying capacity.

## 4. Conclusions and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

MCHS | microchannel heat sink |

MD | molecular dynamics |

NP | nanoparticle |

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**Figure 1.**Applications of nanofluids for (

**a**) enhanced microsystem cooling; (

**b**) improved lubrication; and (

**c**) drug delivery [26].

**Figure 2.**Experimental evidence of thermal performance of nanofluid flow in microsystems. The thermal performance ratio in this figure represents ${h}_{nf}/{h}_{bf}$ or $\left(1-{R}_{nf}/{R}_{bf}\right)+1$ depending on how the data were originally presented in the references, where R is the heat sink thermal resistance.

**Figure 3.**Experimental evidence of hydraulic characteristics of nanofluid flow in microsystems. The pressure drop ratio in this figure represents $\Delta {p}_{nf}/\Delta {p}_{bf}$ or ${f}_{nf}/{f}_{bf}$ depending on how the data were originally presented in the references, where $f$ is the friction factor.

**Figure 4.**Thermal (

**a**) and hydraulic (

**b**) performance of nanofluid flow in a MCHS in laminar, transitional, and turbulent regime. The dashed line is used as a guide [34].

**Figure 6.**Schematic presentation of the exfoliation process: individual nanotube exfoliation. Exfoliation can also happen on nano-sheets from aggregates, from smearing the nanotubes, and by breaking large NPs into smaller pieces (see [52]). The arrows indicate the sliding direction.

**Figure 7.**Performance of nanoparticle-enhanced lubricants in friction reduction as a function of NP concentration. The friction reduction is calculated as $1-{f}_{nf}/{f}_{bf}$, where $f$ is the friction coefficient.

**Figure 8.**Performance of nanoparticle-enhanced lubricants in wear reduction as a function of NP concentration. The wear reduction is calculated as $1-{\mathsf{\epsilon}}_{nf}/{\mathsf{\epsilon}}_{bf}$, where $\mathsf{\epsilon}$ represents wear scar diameter, wear volume, or wear rate, depending on the form of the data provided in the original studies.

**Figure 9.**Performance of nanoparticle-enhanced lubricants in improved load-carrying capacity as a function of loading. The wear reduction is calculated as $1-{\mathsf{\epsilon}}_{nf}/{\mathsf{\epsilon}}_{bf}$, where $\mathsf{\epsilon}$ represents wear scar diameter, wear volume, or wear rate, depending on the form of the data provided in the original studies.

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Kleinstreuer, C.; Xu, Z.
Mathematical Modeling and Computer Simulations of Nanofluid Flow with Applications to Cooling and Lubrication. *Fluids* **2016**, *1*, 16.
https://doi.org/10.3390/fluids1020016

**AMA Style**

Kleinstreuer C, Xu Z.
Mathematical Modeling and Computer Simulations of Nanofluid Flow with Applications to Cooling and Lubrication. *Fluids*. 2016; 1(2):16.
https://doi.org/10.3390/fluids1020016

**Chicago/Turabian Style**

Kleinstreuer, Clement, and Zelin Xu.
2016. "Mathematical Modeling and Computer Simulations of Nanofluid Flow with Applications to Cooling and Lubrication" *Fluids* 1, no. 2: 16.
https://doi.org/10.3390/fluids1020016