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Irreversibility analysis for ion size-dependent electrothermal transport of micropolar fluid in a microtube

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Abstract

This paper is devoted to studying the thermal characteristics of a completely developed electrokinetic flow of micropolar fluid through a cylindrical microtube when a static electric field is applied to it. Due to the constant heat flux applied, the microtube wall is supposed to get heated continuously. In addition to this, the local thermal equilibrium (LTE) model is taken into account while analyzing the heat transfer phenomenon. Under low Reynolds numbers and long channel length approximations, the partial differential equations that describe the electrothermal flow of non-Newtonian micropolar fluid have been switched to ordinary differential equations. The finite difference method (FDM) is used to calculate velocity and temperature with second-order precision using uniform grids along the microtube’s radial direction. The Cavalieri–Simpson technique for numerical integration was used to get numerical values for the mean velocity, bulk mean temperature, and mean entropy/Bejan number. Variations in the Nusselt number for changes in velocity and temperature fields and fluctuations in the Bejan number due to heat transfer irreversibility have been presented. Moreover, a comprehensive study has been performed to discuss the impact of pertinent factors on the optimization of the system’s irreversibility through mean entropy generation analysis. Thermofluidic micropumps for chemical mixing/separation and biomicrofluidic devices for diagnostics may be designed using the results obtained from this study.

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Abbreviations

\(\textbf{U}\) :

Velocity vector

\(\textbf{F}\) :

External body force vector

\(\textbf{W}\) :

Micro-rotational vector

\(\textbf{I}\) :

Gyration coefficient vector

\(\textbf{E}\) :

Electric field vector

T :

Temperature

R :

Radius of microtube

\({E}_0\) :

Characteristic electric field

p :

Pressure

t :

Time

k :

Thermal conductivity

e :

Protonic charge

n :

Micropolarity concentration parameter

\({n}_0\) :

Bulk ionic concentration

\({n}^\pm\) :

Ionic densities

a :

Micropolarity parameter

b :

Couple stress parameter

\({k}_{\mathrm{b}}\) :

Boltzmann constant

\({c}_\mathrm{p}\) :

Specific heat capacity

\({u}_{\mathrm{s}}\) :

Helmholtz–Smoluchowski velocity

\({q}_{\mathrm{w}}\) :

Heat flux

\({u}_{\mathrm{a}}^*\) :

Average/mean velocity

\({T}_{\mathrm{a}}\) :

Average/mean Temperature

\({T}_{\mathrm{m}}\) :

Bulk mean temperature

\({I}_{\mathrm{s}}^*\) :

Streaming current

\({I}_{\mathrm{c}}^*\) :

Conduction current

\({I}_{\mathrm{T}}^*\) :

Stern layer current

\(\text{Du}\) :

Dukhin number

\({T}_{\mathrm{s}}\) :

Microtube’s surface temperature

\(\text{Pe}\) :

Peclet number

\(\text{Br}\) :

Brinkman number

S :

Joule heat parameter

\(\text{Nu}\) :

Nusselt number

\(\text{Be}\) :

Bejan number

\(\rho\) :

Fluid density

\(\mu\) :

Fluid viscosity

\(\eta\) :

Material constant of fluid

\(\gamma\) :

Angular viscosity coefficient

\(\Phi\) :

Viscous dissipation factor

\(\Psi\) :

Electrical potential

\(\rho _\mathrm{e}\) :

Net ionic charge density

\(\wp\) :

Steric factor

\(\varsigma\) :

Ionic valency

\(\epsilon\) :

Medium permittivity

\(\lambda\) :

Electrokinetic parameter

\(\rho _\mathrm{e}\) :

Net charge density

\(\Gamma\) :

Velocity scale ratio

\(\psi ^*\) :

Impulsively produced electric potential

\(\zeta ^*\) :

Zeta potential

\(\beta ^*\) :

Slip length

\(\sigma _{\mathrm{S}}\) :

Stern layer electrical conductivity

\(\sigma _{\mathrm{B}}\) :

Bulk electrical conductivity

\(\sigma _\mathrm{e}\) :

Electrical conductivity of the fluid

\(\xi\) :

Conductivity parameter

\(\theta _{\mathrm{m}}\) :

Dimensionless bulk mean temperature

\(\Omega\) :

Temperature difference parameter

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Acknowledgements

The authors are thankful to the esteemed reviewers for their valuable comments, based on which the manuscript has been amply revised.

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Authors and Affiliations

  1. Department of Mathematics, University of Allahabad, Prayagraj, Uttar Pradesh, 211002, India

    B. Mallick

  2. Department of Mathematics, Directorate of Open and Distance Learning, University of Kalyani, Kalyani, Nadia, West Bengal, 741235, India

    A. Choudhury

  3. Honorary Visiting Professor, Department of Mathematics, Ramakrishna Mission Vidyamandira (Autonomous Postgraduate College), Belur Math, Howrah, West Bengal, 711202, India

    J. C. Misra

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Appendix

Appendix

$$\begin{aligned}{} & {} A_1^\text{i}=1,~A_2^\text{i}=\frac{1}{r^\text{i}},~A_3^\text{i}=\frac{\lambda ^2 \sinh (\psi ^\text{i})/\psi ^\text{i}}{1+4\wp \sinh ^2(\psi ^\text{i}/2)},~A_4^\text{i}=0,\\{} & {} B_1^\text{i}=(1 +a),~B_2^\text{i}=\frac{1+a}{r^\text{i}},~B_3^\text{i}=0, \\{} & {} B_4^\text{i}=-\Bigg (\Gamma +\frac{\lambda ^2E}{\zeta }\frac{\sinh (\psi ^\text{i})}{1+4\wp \sinh ^2(\psi ^\text{i}/2)}\Bigg )\\{} & {} \quad\quad -2a\left\{ \left( \frac{\partial w_\upvarphi }{\partial r}\right) ^\text{i}+\frac{w^\text{i}_\upvarphi }{r^\text{i}}\right\} ,\\{} & {} C_1^\text{i}=1,~C_2^\text{i}=\frac{1}{r_\mathrm{i}},~C_3^\text{i}=-4b,~C_4^\text{i}=2b\left( \frac{\partial u}{\partial r}\right) ^\text{i},\nonumber \\{} & {} D_1=1,~~D_2=\frac{D_1}{r_\mathrm{i}},~~D_3=0,\\{} & {} D_4=\frac{u_{\mathrm{z}}^\text{i}}{u_{\mathrm{a}}}A-E^2S-\left( 1+a\right) Br\left[ \left( \frac{\partial u_{\mathrm{z}}}{\partial r}\right) ^2\right] ^\text{i} \end{aligned}$$

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Mallick, B., Choudhury, A. & Misra, J.C. Irreversibility analysis for ion size-dependent electrothermal transport of micropolar fluid in a microtube. J Therm Anal Calorim 148, 12017–12035 (2023). https://doi.org/10.1007/s10973-023-12538-x

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