Rheologica Acta

, Volume 58, Issue 11–12, pp 755–769 | Cite as

On the miscible thermo-viscous fingering instability of non-Newtonian fluids in heterogeneous porous media

  • Hosna Shokri
  • Mohammad Hassan Kayhani
  • Mahmood NorouziEmail author
Original Contribution


In this paper, the thermo-viscous fingering instability in the displacement of Newtonian fluid by a shear-thinning non-Newtonian fluid in the heterogeneous porous media is studied. Here, the heterogeneity of medium is assumed to be layered. Two different types of displacements are considered: a cold low viscous fluid displaces a hot high viscous one and vice versa. The numerical solution is obtained using a spectral method based on Hartley transforms and fourth-order Adams–Bashforth technique. By interoperation of concentration and temperature contours, transversely averaged profiles, mixing length, and sweep efficiency, it is shown that heterogeneity of porous media has a significant effect on the thermo-viscous fingering instability. The decreasing thermal-lag coefficient leads to more unstable flow and Lewis number has different effects on displacements. When a cold less viscous fluid is displacing fluid, this parameter has destabilizing effect while for displacing hot more viscous fluid, and it acts in favor of stability.


Thermo-viscous fingering instability Non-Newtonian fluid Heterogeneous porous media Spectral method 



Sspect ratio of Hele–Shaw cell


Size of gap in the Hele–Shaw (m)




Dispersion tensor (m2/s)


Wave number of the disturbances


Permeability of the porous medium (m2)


Average permeability of the medium (m2)


Length of the Hele–Shaw cell (m)


Lewis number


Power law index


Pressure (Pa)


Peclect number


Frequency of layers in y direction


Variance of permeability


Temperature (°C)


Time (s)


Average velocity of fluid injection (m/s)


Velocity (m/s)


Streamwise component of the velocity field (m/s)


Magnitude of the velocity vector (m/s)


Width of the Hele–Shaw cell (m)


Coordinate in the streamwise direction (m)


Coordinate in the transverse direction (m)

Greek symbols


Natural logarithm of the viscosity ratio

\( \dot{\gamma} \)

Generalized shear rate (1/s)


Characteristic time of Careau model (s)


Apparent viscosity (Pa s)


Infinite shear rate viscosity (Pa s)


Zero shear rate viscosity (Pa s)


Dimensionless temperature


Thermal lag coefficient


Growth rate of the disturbance




Stream function (m2/s)


Vorticity (1/s)



Dimensionless parameter





Displacing fluid


Displaced fluid


Base state profile






Supplementary material

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Hosna Shokri
    • 1
  • Mohammad Hassan Kayhani
    • 1
  • Mahmood Norouzi
    • 1
    Email author
  1. 1.Faculty of Mechanical EngineeringShahrood University of TechnologyShahroodIran

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