An Exact Analytical Solution for Gaseous Flow and Heat Transfer in Microtubes with Constant Wall Temperature

Document Type : Full Lenght Research Article


1 Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran

2 School of Engineering, University of Liverpool, Brownlow Hill, Liverpool, L69 3GH, UK.

3 Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

4 Technical and Engineering Campus of Shahid Beheshti University, Tehran, Iran.


It is known that slip flow and temperature jump phenomena play a significant role in micro-scale investigations. In this paper, exact analytical solutions for the flow and the convective heat transfer of gaseous flow passing through microtubes are derived for the first time in form of the Whittaker function. Here, it is assumed that both flow and heat transfer is fully developed in a microtube with constant wall temperature. The solution is obtained by considering the Navier-slip conditions for flow and heat transfer at walls. Here, a modal analysis technique is employed to achieve possible solutions for this scenario. Due to the eigenvalue form of governing equations, obtaining the closed-form exact solution for this problem is too difficult from the mathematical point of view and previous studies have been restricted to numerical and approximate series expansion solutions. In this study, an additional constraint is introduced using the definition of the mean temperature and employed to obtain possible eigenvalues related to this problem. Finally, by implementing a scaling law of the Nusselt number of laminar flow in closed conduits, an exact analytical solution for temperature distribution and the heat transfer are derived. It was found that increasing the Prandtl number increases the Nusselt number and increasing the Knudson number decreases the Nusselt number. Based on the obtained solution, the effect of Prandtl number and Knudsen number on heat convection of microtubes are studied in detail.


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