A Comprehensive Review of Non-Newtonian Nanofluid Heat Transfer

Abstract

Nanofluids behave like non-Newtonian fluids in many cases and, therefore, studying their symmetrical behavior is of paramount importance in nanofluid heat transfer modeling. This article attempts to provide are flection on symmetry via thorough description of a variety of non-Newtonian models and further provides a comprehensive review of articles on non-Newtonian models that have applied symmetrical flow modeling and nanofluid heat transfer. This study reviews articles from recent years and provides a comprehensive analysis of them. Furthermore, a thorough statistical symmetrical analysis regarding the commonality of nanoparticles, base fluids and numerical solutions to equations is provided. This article also investigates the history of nanofluid use as a non-Newtonian fluid; that is, the base fluid is considered to be non-Newtonian fluid or the base fluid is Newtonian, such as water. However, the nanofluid in question is regarded as non-Newtonian in modeling. Results show that 25% of articles considered nanofluids with Newtonian base fluid as a non-Newtonian model. In this article, the following questions are answered for the first time: Which non-Newtonian model has been used to model nanofluids? What are the most common non-Newtonian base fluids? Which numerical method is most used to solve non-Newtonian equations?

1. Introduction

Most materials, ranging from industries to nature, are non-Newtonian and, for us, this is the main reason to investigate this critical subject. Fluids are divided into two types based on their behavior: Newtonian and non-Newtonian. To be more specific, it depends on the shear stress rate and the force or tension, as shown in Figure 1. When the slope of fluid versus the shear rate follows a linear rate, fluid is Newtonian but, on the other hand, if non-linear behavior occurs, it is called non-Newtonian fluid [1].

Non-Newtonian fluid is classified into two subdivided fluids: time-dependent and time-independent fluid [2]. For time-dependent fluid, there are two subdivisions: rheopectic and thixotropic. Figure 2 describes these types of fluids:

One promising application of nanotechnology in heat transfer is to increase the thermophysical properties of common fluids by adding nanoparticles. This suspension is called nanofluid. Some published papers about non-Newtonian fluids/nanofluids and their characteristics and applications can be seen in Refs [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. Nanofluids show higher conductivity than their base fluid, and the addition of nanoparticles increases the viscosity of the base fluid [4,12,27,29]. An increase in conductivity increases heat transfer and an increase in viscosity decreases it, especially in free convection [11,13]; the rheological behavior of nanofluid was first examined two decades ago [5]. To model the heat transfer of nanofluids with a Newtonian base fluid and with a low-volume fraction of nanoparticles, the nanofluid can be modeled as Newtonian, but in some cases, nanofluids with a Newtonian base fluid show the behavior of a non-Newtonian fluid [32,33,34,35,36,37,38,39,40,41]. It is known that heat transfer of nanofluids with non-Newtonian base fluids should be modeled in a non-Newtonian manner [42,43,44,45,46,47,48,49,50,51,52,53,54,55,56].

2. Non-Newtonian Nanofluid Heat Transfer

Cabaleiro et al. [57] researched rheological TiO2-ethylene glycol nanofluids at different temperatures ranging from 20 °C to 50 °C and they measured viscosity at different shear rates and showed that nanofluids have similar behavior with non-Newtonian Ostwald–de Waele models. The rheological behavior of polycarbonate containing 0.5 to 15 wt% was investigated by potschke et al. [58]. They reported that more than 2 wt% nanotubes show non-Newtonian behavior. Phuoc et al. [59] investigated the viscosity of multi-walled carbon nanotube nanofluids via experiments and found a 20% decrease in viscosity. Further, they reported, via increasing viscosity, that the fluid behaves as a non-Newtonian shear-thinning fluid. Aladag et al. [60] investigated temperature and shear rate on nanofluid nanotubes Al₂O₃-water and water-based CNT. They reported that CNT at a high shear rate is Newtonian, whereas Al₂O₃-water in the low-temperature range of the study shows non-Newtonian behavior. Tamjid and Guenther [61] investigated the rheology of silver nanofluids in diethylene glycol at 0.11–4.38 wt% and found that nanofluids behave as pseudoplastic. They compared the Herschel–Bulkley model and Bingham model by shear stress and shear rate curves. Moghadam et al. [62] investigated graphene–glycerol nanofluids at 0.0025–0.02 wt% and temperatures ranging from 20 to 60 °C. The results showed that nanofluid viscosity will decrease via increasing temperature, increasing by a mass fraction. They found that nanofluids act as non-Newtonian nanofluids. Some non-Newtonian models used for nanofluid behavior modeling can be seen in

Table 1. Models of non-Newtonian fluids with their applicability.

	Model Name	Applicability
1	PowerLaw [63] (Ostwald de Waele)	The most popular model
2	Carreau [3]	At extra high and extra low shear rates
3	Ellis [64]	for low shear rates, it would be better to use the Ellis model
4	Cross [65]	at low and high shear rates
5	Bingham plastic [66]	Shear-Thickening or Dilatant Viscoelastic fluid
6	Herschel–Bulkley [67]	Derivation from Bingham for nonlinear fluid in
7	Casson [68]	Best model for foodstuffs and biological materials: blood and other fluids in food industrial yoghurt, chocolate and….
8	Maxwell Model [47]	viscoelastic
9	Jeffreys Model [46]	viscoelastic
10	Upper Convected Maxwell (UCM) Model [63]	viscoelastic
11	Oldroyd-B Model [67]	second simplest nonlinear viscoelastic model very popular model
12	Bautista–Manero Model [69]	viscoelasticity and the Fredrickson’s kinetic equation
13	Godfrey Model [67]	time dependence of thixotropic fluids

.

In the following, the flow and heat transfer of nanofluids modeled with non-Newtonian models mentioned in Table 1 are given:

Maleki et al. [70] considered CMC–water as a base flow and Al₂O₃,T_iO₂,CuO, and Cu as nanoparticles and by modeling non-Newtonian with power law and comparing the Newtonian model and non-Newtonian models by changing the power-law index which represents different types of non-Newtonian fluids, showing that by adding nanoparticles to the Newtonian model, there is no significant rise in heat transfer. Still, in non-Newtonian models, this is completely different. As shown in Figure 3, the Nusselt number increases via increasing the nanofluid volume fraction.

Ibrahim and Negera [71] investigated and compared upper-convected Maxwell and Williamson Nanofluid via the KBM method and found that melting and magnetic field parameter have more effect on the velocity boundary layer of Williamson nanofluid than upper-convected Maxwell nanofluid. The temperature distribution increases with an increase in the Eckert number. Upper-convected Maxwell fluids are less affected by the melting parameter when compared with Williamson fluids. Moreover, the velocity, temperature, and concentration distributions decrease for both fluids when the permeability parameter increases. This manner can be seen for Pr number (Figure 4). Furthermore, the temperature distribution increases with an increase in the Eckert number.

Abbasian Arani et al. [72] presented a cross model for Al₂O₃-Water through a lid-driven enclosure. They stated that with nanoparticles, the average Nusselt number is increasing in non-Newtonian nanofluid, but for Newtonian nanofluid, it is related to natural or forced convection flow and may decrease or increase via increasing the Re. The heat transfer rate increases for both Newtonian and non-Newtonian approaches (Figure 5).

Naseem et al. [73] focused on the gyrotactic microorganisms in the MHD biconvective flow of Powell Eyring nanofluid over a stretched surface. They compared Newtonian and Powell–Eyring fluid and found that the magnitude of velocity is greater in the case of Powell–Eyring fluid, and a comparison of temperature and velocity for Newtonian and non-Newtonian is shown in the Figure 6 as shown below:

Kamran et al. [74] investigated the magneto hydrodynamics flow in Casson nanofluid combined with Joule heating and the KBM method was considered. They found that the Nusselt number declines and the Sherwood number rises by increasing the Eckert number. By increasing the Eckert number, the Nusselt number decreases and the Sherwood number increases. Hartman number has a direct effect on declining both Sherwood and Nu numbers. As shown in Figure 7, an increase in the Casson parameter decreases the fluid velocity, the opposite to temperature.

Sulochana and Ashwinkumar [75] investigated Carreau fluid for film flow in magnetic nanofluids and found that CoFe2O4–water nanofluid has a higher heat transfer rate compared to Fe3O4–water nanofluid, and when the Weissenberg number is increasing, the heat transfer will increase as well (Figure 8).

Eldabe and Abou-zeid [76] investigated MHD heat transfer intheBingham model through a non-Darcy porous medium using HPM. They found that velocity decreases by increasing thermophoresis, whereas it decreases (increases) the temperature; the velocity decreases when B increases. They also reported that the temperature distribution experiences an enhancement via increasing the Ecker number, while it decreases by increasing the Darcy number (Figure 9).

Hayat et al. [77], using the Jeffrey model, found that the Hartman number on the temperature and concentration distributions is similar in quantity. Temperature and concentration distributions are increased by increasing Hartman number. They reported, as can be seen in Figure 10, thatan increment in Deborah number causes a decreasing manner for temperature and concentration distribution. Further, the Nu number reduces for larger Nt, but it remains constant for Nb.

Aziz and Shams [78] used water-based Cu and Al₂O₃ nanofluids on stretching sheet with entropy generation, heat source, and thermal conductivity plus thermal radiation. Via the Maxwell model and numerical simulation, they found that increasing values of the Maxwell parameter cause a decrease in heat transfer for both alumina–water and copper–water nanofluids. However, skin friction has the opposite behavior. Figure 11 shows the effects of nanofluid volume fraction on velocity and temperature.

Hashemi and Shokouhmand [79] investigated the effects of Re and Ra on the heat transfer of Al₂O₃Propellant Dough nanofluid. They used the Herschel–Buckley model for the non-Newtonian model in the vertical plate shown in Figure 12 via a numerical method.

Constant wall temperature boundary condition in heat exchangers was their view and goal in this investigation. In a paper by Hashemi and Shokouhmand, after introducing the rmophysical characteristics, they continue by presenting equations and the Herschel–Bulkley model; their most important results are presented below.

Figure 13 and Figure 14 show that by increasing the Reynolds number, the surface friction coefficient will decrease, and with our expectation, the Nu will increase. On the other hand, via increasing Ra, the Nu number will also increase. Elahi et al. [80] provided a series solution for MHD in the coaxial cylinder for non-Newtonian nanofluid considering slip condition. They chose constant and variable viscosity and nonlinear equations; they used the homotopic asymptotic method. They started with continuity and Navier–Stokes equations by considering incompressibility in the conservation of total mass, momentum, thermal energy, and nanoparticles. Figure 15 shows the thermophoresis effects on temperature and concentration distribution. They also studied the effects of nanoparticle Brownian motion on temperatures and velocity, as shown in Figure 16 and Figure 17, respectively.

Table 2. Geometry methodology and results of articles.

Ref	Geometry/Problem Description	Remarks and Methodology	Variable Parameters	Highlight Results	Type of Non-Newtonian Nano fluid
Hashemi and shokouhmand [79]		Numerical FLUENT Laminar	Re = 1.3,1.4,4.5 * 10−4 Φw=1–5%	Nuave ↥ via Φ = 4 and ↧ by more than 4. Brinkman↑Nu↑	Al2O3 Propellant Dough (35/45) Herschel-Bulkley (model)
Ellahi et al. [80]	Non-Newtonian nanofluid flow through a porous media between 2 coaxial cylinder	HAM (Homotopy analysis method) Porous medium	R = 1.0–2.0 Nb = Nt = 1.0–4.0	Nanoconcentration ↥ Nt and thermophoresis is the opposite of Brownian motion for thermal boundary.	Blood modified Darcy’s law model
Pahlevaninejad et al. [81]	wavy microchannel	FVM Fluent constant heat flux	Re = 5–300 φ = 0.005–0.050	volume fraction$↥$ → Nuave$↥$ nanoparticle diameter$↥$→ Nuave ↧ Re$↥$and volume fraction$↥$→ Pressure drop$↥$	Water + CMC (non-Newtonian base flow), Al2O3, (nano particles) power-law
Maghsoudi et al. [82]	Two infinite vertical flat plates +magnetic field + thermal radiation	Galerkin method (GM) Natural Convective Magnetic field Thermal Radiation	Ha = 0–100 Ec =2–92 φ = 0.005–0.010 N = 0–100 (Radiation dimensionless parameter)	Magnetic parameter$↥$velocity ↧ skin friction ↧ Nu ↧ non-Newtonian viscosity$↥$Nu ↧ and skin friction ↧ Ec$↥$skin friction ↧ Nu$↥$	Water CuO Comment: (non-Newtonian model is not mentioned)
Yang et al. [83]	sinusoidal minitube	FVM Sinusoidal-wavy Mini-tube	Re = 5000–15,000 θ = 0–180 particle size= 16–90nm	water thermal properties better than EG/water. EG/water @ dnp = 90 nm the maximum mean Nu	ethylene-glycol/water (Base fluid) Silicon-Carbide (Nano particle) power-law
Singah and Kishore [84]	Non-Newtonian Nanofluids Flowing Vertically Upward Across a Confined Circular Cylinder	FVM with Fluent Laminar Mixed	Re = 1–40 φ = 0.005–0.045 Ri = 0–40 λ = 0.0625–0.5 (confinement ratio of circular cylinder)	Re$↥$heat transfer$↥$ φ$↥$heat transfer$↥$ Ri$↥$heat transfer$↥$ K$↥$heat transfer$↥$ At φ >0.005 and Ri < 2Nuave ↧ with ↥ Ri	shear-thinning power-law (TYPE OF NANOFLUID IS NOT MENTIONED)
Rao et al. [85]		KBM (MHD)	ζ = 0–90 Nt = 0.02–0.2 St = 0.5–3.0 Nb = 0.01–0.4 Le = 5–50 Pr = 7–100	↥ (Casson) parameter decelerates the flow and also ↧ thermal and nanoparticle concentration boundary layer thickness. ↥ thermal slip strongly ↧ velocities, temperatures and nano-particle concentrations.	Casson viscoplastic model
Rajkotwala and Banerjee [86]		FVM Natural	Ra = 104–106 φ = 0.05%–5%	↥ φ, ↧ heat transfer for non-Newtonian models and ↥ for Newtonian model.	Water Copper Ostwald–de Waele model
Mahdy and Chamkha [87]	non-Newtonian nanofluid over an unsteady contracting cylinder	Numerical MATLAB Buongiorno Unsteady	Nt = 0.3–1.0, fw = (−0.2, 0.2), β = 0.4–5.0 Le = 1.0–50 Pr = 0.3–20	The heat and mass transfer rates ↧ via ↥ unsteadiness parameters and Brownian motion.	Casson fluid model
Agbajeet al. [88]		(MD-BSQLM) METHOD Heat Generation, Thermal Radiation	Pr = 0.3–7.0; Fw= −1.0–1.0 Nb = 0.3–0.9	ε and δ momentum boundary-layer thickness and The heat generationand thermal radiation enhancetemperature and thermal boundary-layer thickness	Powell-Eyring,
Akinshilo et al. [89]		Adomian decomposition method (ADM)	δ = Ec = 0.5–1, pr = 0.6–2.0 φ = 0.01%–0.05%	Silver provides ↥ heat transfer rates with ↥ thermal conductivity on the other hand ↧ temperature distribution compared with Alumina.	water Sodium Alginate (SA) silver Ag and Al2O3
Abdelsalam and Bhattic [90]		Homotopy Perturbation Technique Non-uniform Channel Peristaltic	Nb = 0.5–1.5 Nt = 0.5–2.0 Β = 0.1–3.0 M = 0.0–0.8	↥ Brownian parameter thermal conductivity will ↥ , but ↧ in concentration profile. and without magnet, the pressure ↥ attains the highest value, and the friction force attains the ↧ value	Blood power-law
Loenko et al. [91]		FDM Natural Convection Heat-Generating	Ra = 104–106 N = 0.8–1.4 K = 1–1000	↥ Ra the average Nusselt ↥ .	Power-Law
Ling et al. [92]		(FVM) FLUENT k-ε model Elliptic tube Experimental	H = 35–78 φ = 0.2%–1% Re =200–800 Nu = 26–50 Eu = 14–23	By adding nanoparticles heat transfer ↥ and for better heat transfer for xgbase fluid and heat transfer ↥ by MWCNTs.	xanthan gum (XG) (Base Fluid) MWCNT (Nano particles) power-law
Shamsi et al. [93]	Rectangular microchannel with triangular ribs	Laminar flow Microchannel Triangular ribs	5 < Re < 300 φ = 0%–2% L1 = 4 Nuave=0–30 Θ = 30–60	↥ nanoparticles lead to ↥ heat transfer ↧ diameter of nanoparticles lead to ↥ heat transfer angle of 30° lead to max Nusselt number and the min pressure drop.	Water Carboxy methyl cellulose (CMC) Al2O3 Power-law
Hojjat [94]		An optimal artificial neural network (ANN) turbulent regime	Pe = 190,000–350,000 Nu = 90–230 39 <Pr< 71	ANN predicts the Nusselt number of nanofluids better than the previously correlation. The maxand average absolute relative deviations of the ANN are almost half and one fifth of the values predicted by the correlation in the literature	Al2O3 TiO2 CuO
JahanbakhshinandNadooshan [95]	square enclosure with central heating source	FDM Natural Convection Heat Generation	Ra = 104–106 φ = 0.05%–5% n = 0.75–1.4	Ra for the start of natural convection in the square enclosure is ↧ by ↥ the power law indexstrongernatural convection causes ↥ Nusselt.	Power-Law
SHARIFI ASL et al. [96]		(CFD) TURBULENT FLOW	Nu = 100–180 φ = 0. 5%–1.5%	↥ heat transfer coefficient and Nusselt number using non-Newtonian nanofluid. with direct relationship with the volume fraction of the nanoparticles and the Re.	carboxymethyl cellulose (CMC) power-law A12O3
Zhang, et al. [97]		FLUENT steady-state laminar flow Microchannel Heat sink	D = 0.2–0.3 V = 1.41–8.69 Vfr = 0.5%–3.5%. Re < 450	Nu/Nu0 ↥ as inlet velocity ↥ . Al2O3-water nanofluids have superior heat transfer performance than water flowing insmoothmicrochannel	Al2O3 Water power-law shear-thinning model
Hussain et al. [98]	Conjugate natural convection +hybrid nanofluid + wavy-shaped enclosure	Galerkin-based finite element method	φ = 1%	The best overall heat transfer rate is achieved at AR = 0.4 for the pseudo-plastic fluid (n=0.6), at AR = 0.5 for the Newtonian fluid (n=1), and at AR = 0.6 for dilatant fluid (n ˃1).	Ag-MgO
Sajadifar et al. [99]	Microtube considering slip velocity +temperature jump boundary conditions	FVM non-uniform Forced convection	Φ = 0.0%–5% Re = 0–1600 Nu= 3–12	↥ slip coefficient causes ↥ Nu at ↥ Re which meant averaged Nusselt number ↥ due to slip flow regime.	CMC–aluminum oxide under the slip flow
Akbari et al. [100]	two-dimensional microchannel with hydrodynamic and temperature fixed boundary conditions	FVM Laminar forced flow	10 ≤ Re ≤ 1000 Φ = 0.0%–0.5% T = 300–330 Pe = 80–2000	↥ volume fraction of the solid nanofluid lead to ↥ in heat transfer rate, Nu and pressure loss.	carboxy methyl cellulose (CMC) power-law
Ternik et al. [101]		FVM Natural Convection	101< Ra< 106 0% <φ < 10%	At a fixed value of the base-fluid Rabf, the nanofluid Ra nf ↧ with the volume fraction of the nanoparticles. Nu ↥ with the ↥ values of the Rabf	carboxymethyl cellulose (CMC) based gold (Au) Al2O3, Cu, TiO power-law
Zhang et al. [102]	stretching sheet with variable magnetic field and power-law velocity slip effect	Numerical DTM–NIM	S (unsteadiness) = 0.8–1.4 Pr = 0.1–5.0 M (Hartmann) = 0.1–2 N (power law index) = 0.76–1.0	Hartmann number ↑ velocity ↥ CuO–EVA nanofluid has better enhancement on heat transfer than TiO2/Al2O3–EVA.	Al2O3 CuO TiO2 EVA (Base fluid) Power law

is a review of non-Newtonian nanofluid articles that have been conducted until now and illustrates different aspects, comparing them in the table. In advance, the reference is in the first column, then detailed geometry, method, models, the remarkable parameters with their ranges, and finally, the non-Newtonian fluid and base fluid are shown and compared.

3. Statistics and Results

The following section exhibits a statistical report through the investigated articles [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102] with four questions:

3.1. Which Nanoparticles Were Used the Most?

The diagram shown here is about the most popular nanoparticles. As shown in Figure 18, the concentration ofAl₂O₃ was more than the others, and CuO and TiO2 are as follows. The chief reason to use Al₂O₃ is its efficiency in thermal conductivity and heat transfer compared to other nanoparticles.

3.2. Which BaseFluidswere Used the Most?

It is important to investigate the various base fluids through the articles, especially in non-Newtonian fluids. Using water and CMC (by specific volume fraction) was the most used base fluid, and approximately 50% of the articles use this mixture as a base non-Newtonian fluid.

It is also important to know what percentage of nanofluids that are considered non-Newtonian have a Newtonian base fluid such as water. Approximately 25% of the articles considered nanofluids in Newtonian base fluid as a non-Newtonian model as shown in Figure 18 and Figure 19.

3.3. Which Non-Newtonian Models Were Used the Most?

In Table 1, different models for non-Newtonian fluid are investigated and listed. The power law model was used in 65% of papers for modeling the non-Newtonian approach of nanofluid. After that, the Casson model was more important than others as shown in Figure 20.

3.4. Which Numerical/Analytical Methods Were Used the Most?

Despite the other issues, methodologies for solving equations were varied, and the numerical method was the most popular. In the diagram, the methods are shown among non-Newtonian nanofluid articles. The FVM is used more for numerical modeling of nanofluid heat transfer as shown in Figure 21.

4. Conclusions

This article provides a comprehensive analysis of non-Newtonian models, which have applied flow modeling and nanofluid heat transfer. This article discusses non-Newtonian nanofluids; also, a statistical analysis regarding the commonality of nanoparticles, base fluids, and methodology is presented in this article. The results demonstrate that:

• Al₂O₃was the most frequently used nanoparticle.
• 25% of nanofluids modeled as non-Newtonian have a Newtonian base fluid.
• The FVM numerical method is the most widely applied method.
• The water CMC base fluid is further investigated compared with other base fluids.
• The power-law model has been the most frequently applied non-Newtonian modeling of nanofluid heat transfer.
• In all non-Newtonian models, when nanofluid volume fraction increases, the forced convection heat transfer will be increased.
• Increasing non-Newtonian parameters, such as power-law index, Maxwell parameter, etc., leads to a decrease in heat transfer.