# The Effect of Streaming Potential and Viscous Dissipation in the Heat Transfer Characteristics of Power-Law Nanofluid Flow in a Rectangular Microchannel

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{3}with higher thermal conductivity, and the bulk fluid is called base fluid. The thermal behavior of mixed convection flow for power-law nanofluid is studied in [30,31], in which, Carboxymethyl cellulose-water (CMC-water) with aluminum oxide (AI

_{2}O

_{3}) and Polyvinyl chloride (PVC) solution with Cu are represented by power-law nanofluid model. In addition, the researchers investigated the transient thermal characteristics of Newtonian nanofluid EOF [29] and PDF with streaming potential effect [32]. Since non-Newtonian fluids have been manipulated in micropumps and microreactors, the heat transfer characteristics of EOF for power-law nanofluid in a parallel plate microchannel were investigated by analytically solving the one-dimensional momentum equation and energy equation [33]. Due to the variety of microchannels applied in microdevices, the heat transfer of EOF of power-law nanofluid in a rectangular microchannel was discussed by Deng [34] based on the Brinkman model [35].

## 2. Mathematical Modeling

_{s}. It is also assumed that the electrolyte solution considered here is ionically symmetric, EDLs on the solid surface do not overlap, and the constant thermophysical properties are applied due to the low temperature variation. As an axial pressure gradient is imposed on the rectangular microchannel, the PDF with the streaming potential effect, namely, the electrokinetic flow occurs. Since the channel length is much longer than the width and height, the electric potential and velocity distribution can be seen as functions of x and y. Because of the symmetry, the volumetric domain Ω for the mathematical modeling below is confined to a quarter cross-section of the rectangular microchannel.

#### 2.1. Electric Potential Field

_{e}= −2z

_{v}en

_{0}sinh[z

_{v}eψ/(k

_{B}T

_{a})], and with the assumptions above, the Poisson–Boltzmann (P–B) equation governing the electric potential ψ and the corresponding boundary conditions can be given as

_{x=0}= 0, ∂ψ/∂y|

_{y=0}= 0, ψ|

_{x=b}= ξ, ψ|

_{y=a}= ξ

_{h}= 4ab/(a+b), the reciprocal of EDL thickness $\kappa $ = [2z

_{v}e

^{2}n

_{0}/(εε

_{0}k

_{B}T

_{a})]

^{1/2}, and the electrokinetic width $K=\kappa {D}_{h}$ where ε is the relative permittivity, ε

_{0}denotes the permittivity in vacuum, z

_{v}is the valence of ions, e is the elementary charge, k

_{B}is the Boltzmann constant, n

_{0}is the ionic number concentration of the bulk at neutral condition, and T

_{a}is the absolute temperature, respectively. As a result, the dimensionless P–B equation under Debye–Hückel approximation $(\mathrm{sinh}\overline{\psi}\approx \overline{\psi})$ [5] and the corresponding boundary conditions are obtained as

#### 2.2. Hydrodynamic Field

_{ij}= [(∂w

_{i}/∂x

_{j})+(∂w

_{j}/∂x

_{i})]/2 [7,17]. According to the model developed by Brinkman [35], the effective viscosity of the power-law nanofluid μ

_{eff}= μ

_{f}/(1−ϕ)

^{5/2}[31]. The viscosity of base fluid is expressed as μ

_{f}= m(|∂w/∂x|

^{n−}

^{1}, |∂w/∂y|

^{n−}

^{1}) based on the power-law model and the assumptions of PDF, which shows dependence on the strain rate, flow consistency index m of dimension [N·m

^{−2}·s

^{n}], and the flow behavior index n [7,17]. The second term in the LHS of Equation (5) denotes the axial pressure gradient, namely, the driving force of PDF. The third term in the LHS of Equation (5) represents the resistance force arising from the presence of streaming potential, namely the measurement of the streaming potential effect, where E

_{s}is the strength of induced electric field in EDL.

_{s}(t) along the flow direction is expressed as

_{c}(t) = σA

_{s}E

_{s}(t) where σ is the total electrical conductivity of the electrolyte solution and solid surface, and A

_{s}represents the rectangular cross-sectional area. Based on the ionic net current equilibrium condition in the rectangular microchannel, one has I

_{s}(t) + I

_{c}(t) = 0, and thus, the strength of induced electric field is expressed as

_{0}denotes the viscosity coefficient of Newtonian fluid, one eventually has the dimensionless modified Cauchy momentum equation and the strength of induced electric field

#### 2.3. Thermal Field

_{p}) denote thermal conductivity and heat capacity of power-law nanofluid at the reference pressure, respectively. The subscripts s, f, and eff stand for the solid nanoparticles, base fluid, and nanofluid, respectively. The boundary conditions that Equation (12) obeys are

_{w})/(T

_{m}-T

_{w})]/∂z = 0 where T

_{m}represents the mean temperature over the cross-sectional area of the rectangular microchannel, and T

_{w}stands for the wall temperature, which varies along the axial direction due to the axial thermal conduction on the wall. Consequently, when applying the constant wall heat flux, i.e., q

_{s}= const., it is derived that ∂T/∂z = dT

_{w}/dz = dT

_{m}/dz = const and ∂

^{2}T/∂z

^{2}= 0. Therefore, Equation (12) falls into the following simplified form

_{s}, the dimensionless mean velocity ${\overline{w}}_{m}$ and the Joule heating parameter $S=\sigma {E}_{s}{}^{2}{D}_{h}/{q}_{s}$ representing the ratio of Joule heating to the heat flux from the wall surface, one obtains the dimensionless version of Equation (14)

## 3. Solution Methodology

#### 3.1. In the Case of Newtonian Nanofluid Flow

#### 3.2. In the Ccase of Power-Law Nanofluid

_{1}= 1, D

_{2}= 1, ${D}_{3}=-({k}_{1}\overline{w}-{k}_{2}S-{k}_{2}\overline{\mathsf{\Phi}})$ and $f=T$. In terms of time variable, the time splitting method is used. In the first half time step, $\partial f/\partial \overline{t}={D}_{3}(\overline{x},\overline{y})$ is numerically solved based on the Runge–Kutta method and in the second half time step the Crank–Nicolson method is employed to solve $\partial f/\partial \overline{t}={D}_{1}{\partial}^{2}f/\partial {\overline{x}}^{2}+{D}_{2}{\partial}^{2}f/\partial {\overline{y}}^{2}$. The discretization procedure is found in details in [17,34]. A specified criterion Er is given to identify that if the velocity is fully developed, i.e., $\Vert {f}_{}^{l}-{f}_{}^{l+1}\Vert <Er$ because $\partial \overline{w}/\partial \overline{t}{|}_{\overline{t}\to \infty}=\partial \overline{T}/\partial \overline{t}{|}_{\overline{t}\to \infty}=0$. Eventually, the fully developed numerical velocity and numerical temperature are acquired.

## 4. Method Validation

## 5. Results and Discussion

_{f}/k

_{eff}in the expression of Nu namely Equation (17) caused by the increase in ϕ, outweighs the increase of $-1/{\overline{T}}_{m}$, no matter what type of nanofluid is considered. Therefore, one should have a second thought on choosing nanofluid as an approach to improve heat transfer performance.

## 6. Conclusions

- For electrokinetic flow of power-law nanofluid, the streaming potential effect not only reduces and retards velocity distribution, but also narrows temperature difference between the bulk flow and channel wall, which in further reduces the Nusselt number. Thus, when considering the streaming potential effect on PDF in microchannels, increasing the electrokinetic width K is an effective approach to improve heat transfer performance of PDF.
- The bulk mean temperature rises as the volume fraction of nanoparticle ϕ increases no matter what fluid type is considered. However, a slight decrease of Nusselt number Nu with ϕ is observed and thus one should have a second thought when adding nanoparticles to liquid to enhance the heating transfer rate.
- Regarding the nanofluid type, it is notable that temperature distribution is a weak function of flow behavior index n. Compared to the Newtonian nanofluid and especially the shear thickening nanofluid, the shear thinning nanofluid exhibits greater heat transfer rate, indicating it to be more sensitive to the introduction of nanoparticles, the effects of streaming potential, and viscous dissipation. Therefore, to obtain higher heat transfer rate in engineering application, the working liquid can be chosen as shear thinning power-law nanofluid. Moreover, one should carefully consider the heat transfer characteristics when treating biofluids and other liquids with long chain molecules as Newtonian fluids.
- When the Brinkman number Br is augmented, the temperature distribution especially in the vicinity of channel wall increases and Nu is enhanced correspondingly. It reveals that the viscous dissipation effect plays a part on both temperature profile and Nusselt number, which is more pronounced in the case of shear thinning nanofluid. Therefore, the consideration of viscous dissipation for non-Newtonian fluids is worth the discussion above.
- The Nusselt number Nu shows a decreasing trend with Joule heating parameter S. The evident difference of Nu with and without consideration of Joule heating effect indicates that the Joule heating needs to be carefully considered when studying the heat transfer characteristics in electrokinetic flow of power-law nanofluid.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

_{NM}=(σ

_{N}+μ

_{M})

^{1/2}. As a result, replacing Equation (A6) into Equation (A3), and combining with the analytical electric potential $\overline{\psi}$ presented as Equation (18), one can acquire the dimensionless fully developed velocity expressed as Equation (20).

## Appendix B

_{1}and H

_{2}are obtained by using Equation (A10) and the boundary condition presented as Equation (16). Consequently, the further calculation yields ${Y}_{I}(\overline{y})$ as presented in Equation (24).

## References

- Stone, H.A.; Stroock, A.D.; Ajdari, A. Engineering Flows in Small Devices. Ann. Rev. Fluid Mech.
**2004**, 36, 381–411. [Google Scholar] [CrossRef] [Green Version] - Chen, C.H.; Ding, C.Y. Study on the thermal behavior and cooling performance of a nanofluid-cooled microchannel heat sink. Int. J. Therm. Sci.
**2011**, 50, 378–384. [Google Scholar] [CrossRef] - Krishnan, M.; Mojarad, N.; Kukura, P.; Sandoghdar, V. Geometry-induced electrostatic trapping of nanometric objects in a fluid. Nature
**2010**, 467, 692–695. [Google Scholar] [CrossRef] - Bruus, H. Theoretical Microfluidics; Oxford University Press: New York, NY, USA, 2008. [Google Scholar]
- Das, S.; Chakraborty, S. Analytical solutions for velocity, temperature and concentration distribution in electroosmotic microchannel flows of a non-Newtonian bio-fluid. Anal. Chim. Acta
**2006**, 559, 15–24. [Google Scholar] [CrossRef] - Datta, S.; Ghosal, S.; Patankar, N.A. Electroosmotic flow in a rectangular channel with variable wall zeta-potential: Comparison of numerical simulation with asymptotic theory. Electrophoresis
**2006**, 27, 611–619. [Google Scholar] [CrossRef] - Deng, S.Y.; Jian, Y.J.; Bi, Y.H.; Chang, L.; Wang, H.J.; Liu, Q.S. Unsteady electroosmotic flow of power-law fluid in a rectangular microchannel. Mech. Res. Commun.
**2012**, 39, 9–14. [Google Scholar] [CrossRef] - Srinivas, B. Electroosmotic flow of a power law fluid in an elliptic microchannel. Colloids Surf. A Physicochem. Eng. Asp.
**2016**, 492, 144–151. [Google Scholar] [CrossRef] - Zhao, C.L.; Yang, C. Joule heating induced heat transfer for electroosmotic flow of power-law fluids in a microcapillary. Int. J. Heat Mass Transf.
**2012**, 55, 2044–2051. [Google Scholar] [CrossRef] - Li, F.Q.; Jian, Y.J.; Xie, Z.Y.; Liu, Y.B.; Liu, Q.S. Transient alternating current electroosmotic flow of a Jeffrey fluid through a polyelectrolyte-grafted nanochannel. Rsc Adv.
**2017**, 7, 782–790. [Google Scholar] [CrossRef] [Green Version] - Si, D.Q.; Jian, Y.J.; Chang, L.; Liu, Q.S. Unsteady Rotating Electroosmotic Flow through a Slit Microchannel. J. Mech.
**2016**, 32, 603–611. [Google Scholar] [CrossRef] - Qi, C.; Ng, C.O. Rotating electroosmotic flow of viscoplastic material between two parallel plates. Colloids Surf. A Physicochem. Eng. Asp.
**2017**, 513, 355–366. [Google Scholar] [CrossRef] [Green Version] - Mondal, A.; Shit, G.C. Transport of magneto-nanoparticles during electro-osmotic flow in a micro-tube in the presence of magnetic field for drug delivery application. J. Magn. Magn. Mater.
**2017**, 442, 319–328. [Google Scholar] [CrossRef] - Vakili, M.A.; Sadeghi, A.; Saidi, M.H. Pressure effects on electroosmotic flow of power-law fluids in rectangular microchannels. Theor. Comput. Fluid Dyn.
**2014**, 28, 409–426. [Google Scholar] [CrossRef] - Afonso, A.M.; Alves, M.A.; Pinho, F.T. Analytical solution of two-fluid electro-osmotic flows of viscoelastic fluids. J. Colloid Interface Sci.
**2013**, 395, 277–286. [Google Scholar] [CrossRef] [PubMed] - Soong, C.Y.; Wang, S.H. Theoretical analysis of electrokinetic flow and heat transfer in a microchannel under asymmetric boundary conditions. J. Colloid Interface Sci.
**2003**, 265, 202–213. [Google Scholar] [CrossRef] - Zhu, Q.Y.; Deng, S.Y.; Chen, Y.Q. Periodical pressure-driven electrokinetic flow of power-law fluids through a rectangular microchannel. J. Non-Newtonian Fluid Mech.
**2014**, 203, 38–50. [Google Scholar] [CrossRef] - Gong, L.; Wu, J.; Wang, L.; Cao, K. Streaming potential and electroviscous effects in periodical pressure-driven microchannel flow. Phys. Fluids
**2008**, 20, 063603. [Google Scholar] [CrossRef] - Chen, G.; Das, S. Streaming potential and electroviscous effects in soft nanochannels beyond Debye-Huckel linearization. J. Colloid Interface Sci.
**2015**, 445, 357–363. [Google Scholar] [CrossRef] - Shamshiri, M.; Khazaeli, R.; Ashrafizaadeh, M.; Mortazavi, S. Electroviscous and thermal effects on non-Newtonian liquid flows through microchannels. J. Non-Newtonian Fluid Mech.
**2012**, 173–174, 1–12. [Google Scholar] [CrossRef] - Tan, D.K.; Liu, Y. Combined effects of streaming potential and wall slip on flow and heat transfer in microchannels. Int. Commun. Heat Mass
**2014**, 53, 39–42. [Google Scholar] [CrossRef] - Habibi Matin, M. Electroviscous effects on thermal transport of electrolytes in pressure driven flow through nanoslit. Int. J. Heat Mass Transf.
**2017**, 106, 473–481. [Google Scholar] [CrossRef] - Vakili, M.A.; Saidi, M.H.; Sadeghi, A. Thermal transport characteristics pertinent to electrokinetic flow of power-law fluids in rectangular microchannels. Int. J. Therm. Sci.
**2014**, 79, 76–89. [Google Scholar] [CrossRef] - Babaie, A.; Saidi, M.H.; Sadeghi, A. Heat transfer characteristics of mixed electroosmotic and pressure driven flow of power-law fluids in a slit microchannel. Int. J. Therm. Sci.
**2012**, 53, 71–79. [Google Scholar] [CrossRef] - Sarkar, S.; Ganguly, S.; Dutta, P. Electrokinetically induced thermofluidic transport of power-law fluids under the influence of superimposed magnetic field. Chem. Eng. Sci.
**2017**, 171, 391–403. [Google Scholar] [CrossRef] - Shit, G.C.; Mondal, A.; Sinha, A.; Kundu, P.K. Electro-osmotic flow of power-law fluid and heat transfer in a micro-channel with effects of Joule heating and thermal radiation. Phys. A Stat. Mech. Its Appl.
**2016**, 462, 1040–1057. [Google Scholar] [CrossRef] - Chen, C.-H. Fully-developed thermal transport in combined electroosmotic and pressure driven flow of power-law fluids in microchannels. Int. J. Heat Mass Transf.
**2012**, 55, 2173–2183. [Google Scholar] [CrossRef] - Choi, U.S. Enhancing thermal conductivity of fluids with nanoparticles. ASME FED
**1995**, 231, 99–103. [Google Scholar] - Ganguly, S.; Sarkar, S.; Kumar Hota, T.; Mishra, M. Thermally developing combined electroosmotic and pressure-driven flow of nanofluids in a microchannel under the effect of magnetic field. Chem. Eng. Sci.
**2015**, 126, 10–21. [Google Scholar] [CrossRef] - Ellahi, R.; Hassan, M.; Zeeshan, A. A study of heat transfer in power law nanofluid. Therm. Sci.
**2016**, 20, 2015–2026. [Google Scholar] [CrossRef] [Green Version] - Si, X.; Li, H.; Zheng, L.; Shen, Y.; Zhang, X. A mixed convection flow and heat transfer of pseudo-plastic power law nanofluids past a stretching vertical plate. Int. J. Heat Mass Transf.
**2017**, 105, 350–358. [Google Scholar] [CrossRef] - Zhao, G.P.; Jian, Y.J.; Li, F.Q. Heat transfer of nanofluids in microtubes under the effects of streaming potential. Appl. Therm. Eng.
**2016**, 100, 1299–1307. [Google Scholar] [CrossRef] - Shehzad, N.; Zeeshan, A.; Ellahi, R. Electroosmotic Flow of MHD Power Law Al2O3-PVC Nanouid in a Horizontal Channel: Couette-Poiseuille Flow Model. Commun. Theor. Phys.
**2018**, 69, 655. [Google Scholar] [CrossRef] - Deng, S. Thermally Fully Developed Electroosmotic Flow of Power-Law Nanofluid in a Rectangular Microchannel. Micromachines
**2019**, 10, 363. [Google Scholar] [CrossRef] [Green Version] - Brinkman, H.C. The Viscosity of Concentrated Suspensions and Solutions. J. Chem. Phys.
**1952**, 20, 571–581. [Google Scholar] [CrossRef] - Liu, K.; Ding, T.; Li, J.; Chen, Q.; Xue, G.; Yang, P.; Xu, M.; Wang, Z.L.; Zhou, J. Thermal-Electric Nanogenerator Based on the Electrokinetic Effect in Porous Carbon Film. Adv. Energy Mater.
**2018**, 8, 1702481. [Google Scholar] [CrossRef] - Hemmat Esfe, M.; Saedodin, S.; Mahian, O.; Wongwises, S. Thermal conductivity of Al
_{2}O_{3}/water nanofluids. J. Therm. Anal. Calorim.**2014**, 117, 675–681. [Google Scholar] [CrossRef] - Yu, W.; Choi, S.U.S. The role of interfacial layer in the enhanced thermal conductivity of nanofluids a renovated Maxwell model. J. Nanopart. Res.
**2004**, 6, 355–361. [Google Scholar] [CrossRef] - Jian, Y.J.; Liu, Q.S.; Yang, L.G. AC electroosmotic flow of generalized Maxwell fluids in a rectangular microchannel. J. Non Newton. Fluid Mech.
**2011**, 166, 1304–1314. [Google Scholar] [CrossRef]

**Figure 1.**The sketch of the rectangular microchannel with height 2a and width 2b. (

**a**) Three-dimensional sketch, (

**b**) Two-dimensional sketch.

**Figure 2.**The comparison of numerical solutions with the analytical solutions for (

**a**) velocity and (

**b**) temperature.

**Figure 3.**The comparison of velocity distributions across the rectangular microchannel at different flow behavior n, different particle volume fraction ϕ and different dimensionless electrokinetic width K.

**Figure 4.**The comparison of velocity profile at $\overline{y}=0$ for different flow behavior n, different particle volume fraction ϕ and different dimensionless electrokinetic width K. (

**a**) The influence of ϕ when K = 35, (

**b**) The influence of K when ϕ = 0.03.

**Figure 5.**The comparison of temperature distributions across the rectangular microchannel at different flow behavior n, different particle volume fraction ϕ, different dimensionless electrokinetic width K and different Brinkman number Br.

**Figure 6.**The comparison of temperature profiles at $\overline{y}=0$ for different flow behavior n, different particle volume fraction ϕ, different dimensionless electrokinetic width K and different Brinkman number (S = 0.5). (

**a**) The influence of ϕ when K = 35, Br = 0.01. (

**b**) The influence of K when ϕ = 0.03, Br = 0.01. (

**c**) The influence of K when K = 75, ϕ = 0.03.

**Figure 7.**The effect of dimensionless electrokinetic width on (

**a**) the induced electric field strength and (

**b**) the Nusselt number for different types of base fluid when ϕ = 0.03, Br = 0.01, S = 0.5.

**Figure 8.**The effect of Brinkman number on Nusselt number for different types of base fluid when ϕ = 0.03, K = 35, S = 1.

**Figure 9.**The effect of Joule heating parameter on Nusselt number for different types of base fluid when ϕ = 0.03, Br = 0.01, K = 35.

**Figure 10.**The effect of volume fraction of nanoparticle on (

**a**) the induced electric field strength and (

**b**) Nusselt number for different types of base fluid when K = 35, Br = 0.01, S = 0.5.

Parameters (notation) | Value (unit) |
---|---|

The permittivity in vacuum ε | 8.85 × 10^{−12} C·V^{−1}·m^{−1} |

Boltzmann constant k_{B} | 1.38 × 10^{−23} J·K^{−1} |

Absolute temperature T_{a} | 293 K |

Elementary charge e | 1.6 × 10^{−19} C |

Half channel height a | 1 × 10^{−6} m |

Half channel width b | 1.5 × 10^{−6} m |

Total electrical conductivity σ | 1.2639 × 10^{−7} S·m^{−1} |

Flow consistency index of power-law fluid m | 9 × 10^{−4} N·m^{−}^{2}·s^{n} |

Viscosity of Newtonian fluid μ_{0} | 9 × 10^{−4} N·m^{−2}·s |

Zeta potential ξ | 0.025 V |

Thermal conductivity of the solid nanoparticle k_{s} | 40 W·m^{−1}·K^{−1} |

Thermal conductivity of the base fluid k_{f} | 0.618 W·m^{−1}·K^{−1} |

The pressure gradient dp/dz | −1 × 10^{4} Pa |

The relative permittivity ε_{0} | 80 |

Valence of ions z_{v} | 1 |

Electrokinetic width K | 15–75 |

Flow behavior index n | 0.6–1.4 |

Nanoparticle volume fraction ϕ | 0.0–0.1 |

Joule heating Parameter S | −3 – 3 |

Brinkman number Br | 0–0.1 |

Ratio of nanolayer thickness to original particle radius ω | 1.1 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Deng, S.; An, Q.; Li, M.
The Effect of Streaming Potential and Viscous Dissipation in the Heat Transfer Characteristics of Power-Law Nanofluid Flow in a Rectangular Microchannel. *Micromachines* **2020**, *11*, 421.
https://doi.org/10.3390/mi11040421

**AMA Style**

Deng S, An Q, Li M.
The Effect of Streaming Potential and Viscous Dissipation in the Heat Transfer Characteristics of Power-Law Nanofluid Flow in a Rectangular Microchannel. *Micromachines*. 2020; 11(4):421.
https://doi.org/10.3390/mi11040421

**Chicago/Turabian Style**

Deng, Shuyan, Quan An, and Mingying Li.
2020. "The Effect of Streaming Potential and Viscous Dissipation in the Heat Transfer Characteristics of Power-Law Nanofluid Flow in a Rectangular Microchannel" *Micromachines* 11, no. 4: 421.
https://doi.org/10.3390/mi11040421