Transport in Porous Media

, Volume 129, Issue 3, pp 673–699 | Cite as

Heat Transfer in Porous Microchannels with Second-Order Slipping Boundary Conditions

  • A. A. Avramenko
  • Y. Y. Kovetska
  • I. V. ShevchukEmail author
  • A. I. Tyrinov
  • V. I. Shevchuk


The paper presents results of a study of forced convection in a vertical flat and circular microchannels incorporating porous medium under slip boundary conditions of the first and second orders. The problem was solved analytically and compared with numerical simulations based on the lattice Boltzmann method. Effects of porosity and slip velocity on velocity and temperature profiles were investigated. Behavior of the normalized Nusselt number as a function of the Knudsen and Prandtl numbers, as well as the parameter M characterizing porosity of the medium in the microchannel, was also elucidated. Computations indicated that a decrease in porosity (an increase in the parameter M) causes a decrease in the velocity and temperature jumps on the wall, which contributes to the increase in the Nusselt number. Considering the effect of the second-order slip boundary conditions leads to the decrease in the velocity jump on the wall, when the coefficient A2 (this coefficient considers the second-order slip boundary conditions) changes from negative to positive values. The heat transfer rate at high Prandtl numbers increases with the increasing Knudsen number, because of the improved thermal interaction of the flow with the channel wall. Given the second-order boundary conditions, the effect of parameter A2 was not observed at small Prandtl numbers (Pr ≤ 1). For A2 < 0, the second-order boundary conditions cause an increase in the normalized Nusselt number, whereas for A2 > 0 the normalized Nusselt number decreases in comparison with the case of A2 = 0 (the first-order boundary conditions). The results by the analytical solution and the lattice Boltzmann method are in a good agreement with each other for A2 ≥ 0, with the differences ≤ 1%. For A2 < 0, the differences between two models are more noticeable. For A2 ≥ 0, predictions by the analytical model lie higher than the numerical results, whereas for A2 < 0, numerical simulations exceed the analytical solution.


Slip flow Heat transfer Nusselt number Knudsen number Second-order boundary conditions 

List of Symbols


Thermal diffusivity


Molecular velocity


Specific heat at constant pressure


Space dimension


Hydraulic (equivalent) diameter


External force term


Distribution function


Axial wall temperature gradient


Half-height of the flat channel


Unit vector directed streamwise




Free path of a gas molecule




Heat flux

r, z

Cylindrical coordinate


Radius of circular channel


Gas constant






Streamwise velocity component


Velocity vector

x, y

Cartesian coordinates

Greek Symbols


Dynamic viscosity


Channel perimeter




Relaxation time


Relative temperature

Dimensionless Values


Darcy number


Dimensionless distribution function


Parameter of pressure gradient


Number of elements in the lattice column across the channel


Nusselt number


Knudsen number


Parameter of porosity of the medium


Prandtl number


Dimensionless radial coordinate


Lattice dimensionless velocity


Dimensionless axial (streamwise) velocity


Dimensionless coordinate


Dimensionless temperature

\(\tilde{\rho }\)

Dimensionless density

\(\tilde{\tau }\)

Dimensionless relaxation time



Incompressible medium


Equilibrium distributions


Velocity directions









Computational fluid dynamics


Lattice Boltzmann method



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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institute of Engineering ThermophysicsNational Academy of SciencesKievUkraine
  2. 2.Institute of General Mechanical Engineering, Faculty of Computer Science and Engineering ScienceTH Köln – University of Applied SciencesGummersbachGermany
  3. 3.Ruetz System Solutions GmbHMunichGermany

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