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Heat transfer analysis for the Cattaneo–Cristov heat flux model using integral transform technique with isothermal and isoflux wall conditions

  • Zeeshan Asghar EMAIL logo , Muhammad Waris Saeed Khan ORCID logo , Wasfi Shatanawi EMAIL logo and Muhammad Asif Gondal
Published/Copyright: April 22, 2025
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 80 Issue 6

Abstract

The present theoretical analysis elaborates on the thermal entrance problem using the Cattaneo–Cristov heat flux model. The modified heat equation for the Cattaneo–Cristov model is solved via an integral transform technique. The entering fluid has a uniform velocity and constant physical characteristics. Temperature jump boundary conditions and axial conduction are also considered. A hybrid scheme is utilized to achieve the solution of the modified heat equation with the aid of Mathematica 13 platform. The results of bulk temperature, local, and mean Nussetle number are plotted for different embedded parameters. Applications for this type of study include heat exchangers, plug flow reactors, low Prandtl number liquid flows, and duct-flowing liquid metals with low Reynolds numbers.

Keywords: Graetz problem; Cattaneo–Cristov model; axial conduction; temperature jump condition; eigenvalue problem; integral transform approach
Corresponding authors: Zeeshan Asghar, Department of Mathematics and Sciences, College of Humanities and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia, E-mail: ; and Wasfi Shatanawi, Department of Mathematics and Sciences, College of Humanities and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia; and Department of Mathematics, Faculty of Science, The Hashemite University, P.O. Box 330127, Zarqa 13133, Jordan, E-mail: 

Nomenclature

b 1 *

used in (Eq. (28)) […]

R t

coefficient of thermal accommodation […]

x

horizontal coordinate [m]

L

half-width or radius [m]

y

vertical coordinate [m]

T

temperature [K]

∪0

axial velocity [m/s]

X

nondimensional horizontal coordinate […]

Y

nondimensional vertical coordinate […]

θ

nondimensional temperature […]

θ b

mean temperature […]

Nu

Nusselt number […]

λ *

time relaxation parameter [s]

λ 3 *

molecular mean free path [……]

∇

gradient operator []

K

thermal conductivity [W/m K]

α*

thermal diffusivity [m2/s]

ReD

Reynolds number […]

pr

Prandtl number […]

l

1, 2, 3…

Kn

Knudsen number […]

∆T

temperature difference [K]

ϒ*

specific heat ratio […]

h

convective heat transfer coefficient [W/(m2 K)]

T s

surface temperature [K]

q w

wall heat flux [W/m2]

Acknowledgments

The authors would like to thank Prince Sultan University for their support through the TAS research lab.

  1. Research ethics: Not applicable. This study does not involve human participants or animals.

  2. Informed consent: Not applicable. This study does not involve human participants.

  3. Author contributions: Zeeshan Asghar: Contributed to the conception and design of the study, developed the mathematical models, and drafted the manuscript. Muhammad Waris Saeed Khan: Conducted the analysis, interpreted the results, and reviewed the manuscript for critical intellectual content. Wasfi Shatanawi: Provided technical support, assisted with the mathematical computations, and contributed to the final revision of the manuscript. M. Asif Gondal: Validated and developed the mathematical models.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The authors declare that there are no conflicts of interest regarding the publication of this paper.

  6. Research funding: This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

  7. Data availability: No data were generated or analyzed during this study.

  8. Statement of novelty: This study provides a novel theoretical exploration of the thermal entrance problem using the Cattaneo–Cristov heat flux model under isothermal and isoflux boundary conditions. Unlike previous works, it incorporates the time relaxation parameter into the model to analyze its effects on temperature distribution and Nusselt number in microchannel flow. The research leverages a hybrid analytical technique, combining the integral transform method and high-accuracy computations, to deliver new insights into the influence of Knudsen number, Peclet number, and time relaxation effects. This approach bridges a significant gap in the literature by extending classical solutions of the Graetz problem to include advanced heat conduction models relevant to microchannel systems and nonideal geometries. The findings have direct implications for modern thermal design and applications in heat exchangers, liquid-metal flows, and microreactors.

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Received: 2025-01-16
Accepted: 2025-04-02
Published Online: 2025-04-22
Published in Print: 2025-06-26

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/zna-2025-0018
Keywords for this article
Graetz problem; Cattaneo–Cristov model; axial conduction; temperature jump condition; eigenvalue problem; integral transform approach

Articles in the same Issue

  1. Frontmatter
  2. Dynamical Systems & Nonlinear Phenomena
  3. Controlling the convective heat transfer of shear thinning and shear thickening fluids in parallel plates with magnetic force
  4. Multistability, chaos and hyperchaos in a novel 3D discrete memristive system: microcontroller implementation and cryptography
  5. Comparative study of thermally intense cilia and non-cilia generated motion of Ellis’s fluid by using MATHEMATICA 14.1
  6. Fundamental Concepts of Physical Science
  7. Nonlinear electrodynamics and its possible connection to relativistic superconductivity: instance of a time-dependent system
  8. Galilean decoherence and quantum measurement
  9. Solid State Physics & Materials Science
  10. A dynamically tunable polarization-insensitive broadband vanadium dioxide-assisted absorber for terahertz applications
  11. Thermodynamics & Statistical Physics
  12. Heat transfer analysis for the Cattaneo–Cristov heat flux model using integral transform technique with isothermal and isoflux wall conditions
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