# Size Effects on the Entropy Production in Oscillatory Flow between Parallel Plates

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## Abstract

**:**

## 1. Introduction

## 2. Transport Problem

#### 2.1. Basic Assumptions

#### 2.2. Fundamental Equations

#### 2.3. Velocity Field

#### 2.4. Temperature Field

## 3. Results and Discussion

**Figure 1.**Flux velocity vs. position in y for different times. Solid line: $t=0.00$ s, dashed: $t=2\times {10}^{-4}$ s, dotdashed: $t={10}^{-3}$ s. $\omega =100$ Hz, $a={10}^{-3}$ m.

**Figure 2.**Flux velocity vs. position in y for different frequencies of the external pressure gradient. Solid line: $\omega =200$ Hz, dashed: $\omega =210$ Hz, dotdashed: $\omega =220$ Hz. $a=\phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ m.

**Figure 3.**Flux temperature vs. position in y for different times. Solid line: $t=0.0$ s, dashed: $t=0.2$ s, dotdashed: $t=0.4$ s. $a=1.7\times {10}^{-4}$ m, ${h}_{1}=5000$ J/m${}^{2}$K, ${h}_{2}=1000$ J/m${}^{2}$K.

**Figure 4.**Local entropy production vs. position in y for different times. Solid line: $t=\phantom{\rule{3.33333pt}{0ex}}0.00$ s, dashed: $t=2\times {10}^{-4}$ s, dotdashed: $t=3\times {10}^{-4}$ s. $\omega =100$ Hz, $a=1.7\times {10}^{-4}$ m.

**Figure 5.**Global entropy production vs. plates separation. $\omega =100$ Hz, ${h}_{1}=5000$ J/m${}^{2}$K, ${h}_{2}=1000$ J/m${}^{2}$K.

**Figure 6.**Nusselt number at upper plate vs. plates separation. $\omega =100$ Hz, ${h}_{1}=\phantom{\rule{3.33333pt}{0ex}}5000$ J/m${}^{2}$K, ${h}_{2}=1000$ J/m${}^{2}$K.

**Figure 7.**Global entropy production vs. frequency of the external pressure gradient. $a=\phantom{\rule{3.33333pt}{0ex}}0.0001$ m, ${h}_{1}=5000$ J/m${}^{2}$K, ${h}_{2}=1000$ J/m${}^{2}$K.

## 4. Summary and Conclusions

## Acknowledgements

## References

- Chatwin, P.C. On the longitudinal dispersion of passive contaminant in oscillating flows in tubes. J. Fluid Mech.
**1975**, 71, 513–527. [Google Scholar] [CrossRef] - Kurzweg, U.H. Enhanced heat conduction in fluids subjected to sinusoidal oscillations. J. Heat Transfer
**1985**, 107, 459–462. [Google Scholar] - Kurzweg, U.H. Enhanced heat conduction in oscillating viscous flows within parallel-plate channels. J. Fluid Mech.
**1985**, 156, 291–300. [Google Scholar] [CrossRef] - López de Haro, M.; del Río, J.A.; Whitaker, S. Transport Porous Media
**1996**, 25, 167. - del Río, J.A.; López de Haro, M.; Whitaker, S. Enhancement in the dynamic response of a viscoelastic fluid flowing in a tube. Phys. Rev. E
**1998**, 58, 6323–6327. [Google Scholar] [CrossRef] - del Río, J.A.; López de Haro, M.; Whitaker, S. Erratum: Enhancement in the dynamic response of a viscoelastic fluid flowing in a tube. Phys. Rev. E
**2001**, 64, 039901. [Google Scholar] [CrossRef] - Tsiklauri, D.; Beresnev, I. Enhancement in the dynamic response of a viscoelastic fluid flowing through a longitudinally vibrating tube. Phys. Rev. E
**2001**, 63, 046304–046308. [Google Scholar] [CrossRef] - Castrejón-Pita, J.R.; del Río, J.A.; Castrejón-Pita, A.A.; Huelsz, G. Experimental observation of dramatic differences in the dynamic response of Newtonian and Maxwellian fluids. Phys. Rev. E
**2003**, 68, 046301–046305. [Google Scholar] [CrossRef] - Lambert, A.A.; Ibáñez, G.; Cuevas, S.; del Río, J.A. Optimal behavior of viscoelastic flow at resonant frequencies. Phys. Rev. E
**2004**, 70, 056302. [Google Scholar] [CrossRef] - Lambert, A.A.; Ibáñez, G.; Cuevas, S.; del Río, J.A. Erratum: Optimal behavior of viscoelastic flow at resonant frequencies. Phys. Rev. E
**2006**, 73, 049902. [Google Scholar] - Yakhot, V.; Colosqui, C. Stokes’ second flow problem in a high-frequency limit: Application to nanomechanical resonators. J. Fluid Mech.
**2007**, 586, 249–258. [Google Scholar] [CrossRef] - Bejan, A. Entropy Generation through Heat and Fluid Flow; Wiley: New York, NY, USA, 1994. [Google Scholar]
- Bejan, A. Minimization of Entropy Generation; CRC Press: Boca Raton, FL, USA, 1996. [Google Scholar]
- Sobhan, Ch.B.; Garimella, S.V. A comparative analysis of studies on heat transfer and fluid flow in microchannels. Microscale Thermophys. Eng.
**2001**, 5, 293–311. [Google Scholar] [CrossRef] - Guo, Z.-Y.; Li, Z.X. Size effects on microscale single-phase flow and heat transfer. Int. J. Heat Mass Tran.
**2003**, 46, 149–159. [Google Scholar] [CrossRef] - Guo, Z.-Y.; Li, Z.X. Size effects on single-phase channel flow and heat transfer at microscale. Int. J. Heat Fluid Flow
**2003**, 24, 284–298. [Google Scholar] [CrossRef] - Morini, J.L. Single-phase convective heat transfer in microchannels: A review of experimental results. Int. J. Therm. Sci.
**2004**, 43, 631–651. [Google Scholar] [CrossRef] - Bayraktar, T.; Pidugu, S.B. Characterization of liquid flows in microfluidic systems. Int. J. Heat Mass Tran.
**2006**, 49, 815–824. [Google Scholar] [CrossRef] - Hooman, K. Entropy generation for microscale forced convection: Effects of different thermal boundary conditions, velocity slip, temperature. Int. Comm. Heat Mass Tran.
**2007**, 34, 945–957. [Google Scholar] [CrossRef] - Liang, X. Some effects of interface on fluid flow and heat transfer on micro- and nanoscale, jump, viscous dissipation, and duct geometry. Chin. Sci. Bull.
**2007**, 52, 2457–2472. [Google Scholar] [CrossRef] - Yao, J.; Yao, Y.F.; Patel, M.K.; Mason, P.J. On Reynolds number and scaling effects in microchannel flows. Eur. Phys. J. Appl. Phys.
**2007**, 37, 229–235. [Google Scholar] [CrossRef] - de Groot, S.R.; Mazur, P. Non-Equilibrium Thermodynamics; Dover: New York, NY, USA, 1984. [Google Scholar]
- Fischer, P.; Rehage, H. Non-linear flow properties of viscoelastic surfactant solutions. Rheol. Acta
**1997**, 36, 13–27. [Google Scholar] [CrossRef] - Aydin, O.; Avci, M. Analysis of laminar heat transfer in micro-Poiseuille flow. Int. J. Therm. Sci.
**2007**, 46, 30–37. [Google Scholar] [CrossRef] - Beskok, A.; Karniadakis, G.E. Simulation of heat and momentum transfer in complex micro-geometries. J. Thermophys. Heat Tran.
**1994**, 8, 355–370. [Google Scholar] [CrossRef] - Reichl, L. A Modern Course in Statistical Physics; John Wiley & Sons, Inc.: New York, NY, USA, 1998; Chapter 11. [Google Scholar]
- Oelschlaeger, C.; Schopferer, M.; Scheffold, F.; Willenbacher, N. Linear-to-branched micelles transition: A rheometry and diffusing wave spectrometry (DWS) study. Langmuir
**2009**, 25, 716–723. [Google Scholar] [CrossRef] [PubMed] - Rehage, H.; Hoffman, R.H. Rheological properties of viscoelastic surfactant systems. J. Phys. Chem.
**1988**, 92, 4712–4719. [Google Scholar] [CrossRef] - Ibáñez, G.; Cuevas, S.; López de Haro, M. Minimization of entropy generation by asymmetric convective cooling. Int. J. Heat Mass Tran.
**2003**, 46, 1321–1328. [Google Scholar] [CrossRef] - Ibáñez, G.; Cuevas, S.; López de Haro, M. Thermodynamic optimization of radial MHD flow between parallel circular disks. J. Non-Equilib. Thermodyn.
**2004**, 29, 107–122. [Google Scholar] - Vázquez, F.; Olivares-Robles, M.A.; Cuevas, S. Viscoelastic effects on the entropy production in oscillatory flow between parallel plates with convective cooling. Entropy
**2009**, 11, 4–26. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Vazquez, F.; Olivares-Robles, M.A.; Medina, S.
Size Effects on the Entropy Production in Oscillatory Flow between Parallel Plates. *Entropy* **2011**, *13*, 542-553.
https://doi.org/10.3390/e13020542

**AMA Style**

Vazquez F, Olivares-Robles MA, Medina S.
Size Effects on the Entropy Production in Oscillatory Flow between Parallel Plates. *Entropy*. 2011; 13(2):542-553.
https://doi.org/10.3390/e13020542

**Chicago/Turabian Style**

Vazquez, Federico, Miguel Angel Olivares-Robles, and Sac Medina.
2011. "Size Effects on the Entropy Production in Oscillatory Flow between Parallel Plates" *Entropy* 13, no. 2: 542-553.
https://doi.org/10.3390/e13020542