Convection heat transfer inside a lid-driven cavity filled with a shear-thinning Herschel–Bulkley fluid

  • Nabila Labsi
  • Youb Khaled Benkahla
  • Abdelkader Boutra
  • Meriem Titouah
Technical Paper


The present numerical study, based on the finite volume method, deals with the characterization of flow and heat transfer convection inside a lid-driven square cavity filled with a shear-thinning Herschel–Bulkley fluid. The upper and bottom walls of the enclosure are thermally insulated, while the vertical ones are mobile and differentially heated. The study focuses on the effect of the fluid’s rheological properties, i.e., the fluid’s viscoplasticity (0.50 ≤ Bng ≤ 5000) and the flow index (0.2 ≤ n ≤ 1.0), on both flow and heat transfer within the cavity on one hand and on the modifications involved by the introduction of viscous dissipation (0 ≤ Br ≤ 10) on the other hand. The results show that the increase of the generalized Bingham number leads to the increase of the unyielded regions inside the enclosure. In addition, heat transfer is more pronounced for weak values of the generalized Bingham number and great values of the fluid’s flow index. Viscous dissipation modifies significantly both flow and heat transfer structures, especially for mixed and dominant natural convection. To sum up the obtained results, useful abacuses predicting the heat exchange within the enclosure are given.


Shear-thinning Herschel–Bulkley fluid Mixed convection Lid-driven cavity Generalized Bingham number Flow index Viscous dissipation Finite volume method 



Specific heat, J kg−1 K−1


Bingham number, \(= {{\tau_{0} \,H} \mathord{\left/ {\vphantom {{\tau_{0} \,H} {\mu_{\text{p}} \,V_{\text{w}} }}} \right. \kern-0pt} {\mu_{\text{p}} \,V_{\text{w}} }}\)


Generalized Bingham number, \(= {{\tau_{0} \,H^{n} } \mathord{\left/ {\vphantom {{\tau_{0} \,H^{n} } {K\,V_{w}^{n} }}} \right. \kern-0pt} {K\,V_{w}^{n} }}\)


Critical generalized Bingham number


Brinkman number, \(= {{KV_{\text{w}}^{2} } \mathord{\left/ {\vphantom {{KV_{\text{w}}^{2} } {k(T_{\text{h}} - T_{\text{c}} )}}} \right. \kern-0pt} {k(T_{\text{h}} - T_{\text{c}} )}}\)


Grashof number, \(= {{g\,\beta \,\left( {T_{\text{h}} - T_{\text{c}} } \right)\,\rho_{0}^{2} \,H^{1 + 2n} } \mathord{\left/ {\vphantom {{g\,\beta \,\left( {T_{\text{h}} - T_{\text{c}} } \right)\,\rho_{0}^{2} \,H^{1 + 2n} } {K^{2} }}} \right. \kern-0pt} {K^{2} }}\,V_{\text{w}}^{2}\)


Dimension of the cavity, m


Thermal conductivity, W m−1 K−1


Fluid consistency, kg m−1 s−2−n


Exponential growth parameter, s


Reduced exponential growth parameter


Flow index


Average Nusselt number within the enclosure


Local Nusselt number along the cold wall


Average Nusselt number along the cold wall


Local Nusselt number along the hot wall

Nuh, Av

Average Nusselt number along the hot wall


Pressure, Pa


Dimensionless pressure, \(= p/\rho V_{\text{w}}^{2}\)


Prandtl number, \(= {{K\,C_{\text{p}} V_{\text{w}}^{n - 1} } \mathord{\left/ {\vphantom {{K\,C_{\text{p}} V_{\text{w}}^{n - 1} } {kH^{n - 1} }}} \right. \kern-0pt} {kH^{n - 1} }}\)


Reynolds number, \(= {{\rho_{0} V_{\text{w}}^{2 - n} H^{n} } \mathord{\left/ {\vphantom {{\rho_{0} V_{\text{w}}^{2 - n} H^{n} } K}} \right. \kern-0pt} K}\)


Richardson number


Cold wall temperature, K


Hot wall temperature, K


Reference temperature (the temperature of the cold wall in the study), K


Horizontal velocity component, m s−1


Dimensionless horizontal velocity component, \(= {u \mathord{\left/ {\vphantom {u {V_{\text{w}} }}} \right. \kern-0pt} {V_{\text{w}} }}\)


Vertical velocity component, m s−1


Dimensionless vertical velocity component, \(= {v \mathord{\left/ {\vphantom {v {V_{\text{w}} }}} \right. \kern-0pt} {V_{\text{w}} }}\)


Lid-driven plate velocity, m s−1


Horizontal coordinate, m


Dimensionless horizontal coordinate, \(= \, {x \mathord{\left/ {\vphantom {x H}} \right. \kern-0pt} H}\)


Vertical coordinate, m


Dimensionless vertical coordinate, = y/H

Greek letters


Thermal expansion coefficient, K−1

\(\dot{\gamma }\)

Strain rate, s−1

\(\dot{\gamma }^{*}\)

Dimensionless strain rate


Plastic viscosity, kg m−1 s−1


Apparent viscosity, kg m−1 s−1


Dimensionless apparent viscosity


Dimensionless temperature, = (T − Tc)/(Th − Tc)


Fluid density, kg m−3


Fluid density at a reference temperature, kg m−3


Yield stress, Pa


  1. 1.
    Cianfrini C, Corcione M, Dell’Omo PP (2005) Natural convection in tilted square cavities with differentially heated opposite walls. Int J Thermal Sci 44:441–451CrossRefGoogle Scholar
  2. 2.
    Cheng M, Hung KC (2006) Vortex structure of steady flow in a rectangular cavity. Comput Fluids 35:1046–1062CrossRefzbMATHGoogle Scholar
  3. 3.
    Patil DV, Lakshmisha KN, Rogg B (2006) Lattice Boltzmann simulation of lid-driven flow in deep cavities. Comput Fluids 35:1116–1125CrossRefzbMATHGoogle Scholar
  4. 4.
    Shah P, Rovagnati B, Mashayek F, Jacobs GB (2007) Subsonic compressible flow in two-sided lid-driven cavity. Part I: equal walls temperatures. Int J Heat Mass Trans 50:4206–4218CrossRefzbMATHGoogle Scholar
  5. 5.
    Al Amiri A, Khanafer KM, Pop I (2007) Numerical simulation of combined thermal and mass transport in a square lid-driven cavity. Int J Thermal Sci 46(7):662–671CrossRefGoogle Scholar
  6. 6.
    Tiwari RK, Das MK (2007) Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int J Heat Mass Trans 50:2002–2018CrossRefzbMATHGoogle Scholar
  7. 7.
    Ouertatani N, Ben Cheikh N, Ben Baya B, Lili T, Campo A (2009) Mixed convection in a double lid-driven cubic cavity. Int J Thermal Sci 48:1265–1272CrossRefGoogle Scholar
  8. 8.
    Basak T, Roy S, Sharma PK, Pop I (2009) Analysis of mixed convection flows within a square cavity with linearly heated side wall(s). Int J Heat Mass Trans 52:2224–2242CrossRefzbMATHGoogle Scholar
  9. 9.
    Oztop HF, Al-Salem K, Varol Y, Pop I (2011) Natural convection heat transfer in a partially opened cavity filled with porous media. Int J Heat Mass Trans 54(11–12):2253–2261CrossRefzbMATHGoogle Scholar
  10. 10.
    Boutra A, Ragui K, Labsi N, Benkahla YK (2015) Lid-driven and inclined square cavity filled with a nanofluid: optimum heat transfer. Open Eng 5(1):248–255CrossRefGoogle Scholar
  11. 11.
    Ouyahia S, Benkahla YK, Labsi N (2016) Numerical study of the hydrodynamic and thermal proprieties of titanium dioxide nanofluids trapped in a triangular geometry. Arab J Sci Eng. Google Scholar
  12. 12.
    Gangawane KM (2017) Computational analysis of mixed convection heat transfer characteristics in lid-driven cavity containing triangular block with constant heat flux: effect of Prandtl and Grashof numbers. Int J Heat Mass Trans 105:34–57CrossRefGoogle Scholar
  13. 13.
    Gangawane KM, Manikandan B (2017) Mixed convection characteristics in lid-driven cavity containing heated triangular block. Chinese J Chem Eng. Google Scholar
  14. 14.
    Vikhansky A (2010) On the onset of natural convection of Bingham liquid in rectangular enclosures. J Non-Newtonian Fluid Mech 165:1713–1716CrossRefzbMATHGoogle Scholar
  15. 15.
    Zhang J (2010) An augmented Lagrangian approach to Bingham fluid flows in a lid-driven square cavity with piecewise linear equal-order finite elements. Computer Methods Appl Mech Eng 199(45–48):3051–3057MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Frey S, Silveira FS, Zinani F (2010) Stabilized mixed approximations for inertial viscoplastic fluid flows. Mech Res Commun 37:145–152CrossRefzbMATHGoogle Scholar
  17. 17.
    Turan O, Chakraborty N, Poole RJ (2010) Laminar natural convection of bingham fluids in a square enclosure with differentially heated side walls. J Non-Newtonian Fluid Mech 165:901–913CrossRefzbMATHGoogle Scholar
  18. 18.
    Turan O, Poole RJ, Chakraborty N (2011) Aspect ratio effects in laminar natural convection of Bingham fluids in rectangular enclosures with differentially heated side walls. J Non-Newtonian Fluid Mech 166:208–230CrossRefzbMATHGoogle Scholar
  19. 19.
    Dos Santos DDO, Frey S, Naccache MF, De Souza Mendes PR (2011) Numerical approximations for flow of viscoplastic fluids in a lid-driven cavity. J Non-Newtonian Fluid Mech 166:667–679CrossRefzbMATHGoogle Scholar
  20. 20.
    De Souza Mendes PR, Dutra ESS (2004) Viscosity function for yield-stress liquids. Appl Rheol 14:296–302Google Scholar
  21. 21.
    Turan O, Chakraborty N, Poole RJ (2012) Laminar Rayleigh-Bénard convection of yield stress fluids in a square enclosure. J Non-Newtonian Fluid Mech 171–172:83–96CrossRefGoogle Scholar
  22. 22.
    Syrakos EA, Georgiou GC, Alexandrou AN (2013) Solution of the square lid-driven cavity flow of a Bingham plastic using the finite volume method. J Non-Newtonian Fluid Mech 195:19–31CrossRefGoogle Scholar
  23. 23.
    Ragui K, Benkahla YK, Labsi N, Boutra A (2015) Natural convection heat transfer in a differentially heated enclosure with adiabatic partitions and filled with a Bingham fluid. J Heat Trans Res 46(8):765–783CrossRefGoogle Scholar
  24. 24.
    Chong L, Magnin A, Métivier C (2016) Natural convection in shear-thinning yield stress fluids in a square enclosure. AIChE J 62:1347–1355CrossRefGoogle Scholar
  25. 25.
    Min T, Choi HG, Yoo JY, Choi H (1997) Laminar convective heat transfer of a Bingham plastic in a circular pipe-II. Numerical approach-hydrodynamically developing flow and simultaneously developing flow. Int J Heat Mass Trans 40:3689–3701CrossRefzbMATHGoogle Scholar
  26. 26.
    Labsi N, Benkahla YK, Boutra A, Ammouri A (2013) Heat and flow properties of a temperature dependent viscoplastic fluid including viscous dissipation. J Food Process Eng 36:450–461CrossRefGoogle Scholar
  27. 27.
    Labsi N, Benkahla YK (2016) Herschel–Bulkley fluid flow within a pipe by taking into account viscous dissipation. Mechanics and Industry 17(3) Paper No. 304.
  28. 28.
    Labsi N, Benkahla YK, Boutra A (2013) Convective heat transfer in a square cavity filled with a viscoplastic fluid by taking into account viscous dissipation. J Heat Trans Res 44(7):645–663CrossRefGoogle Scholar
  29. 29.
    Bose A, Nirmalkar N, Chhabra RP (2014) Forced Convection from a heated equilateral triangular cylinder in Bingham plastic fluids. Num Heat Trans Part A: Appl 66:107–129CrossRefGoogle Scholar
  30. 30.
    Gupta AK, Chhabra RP (2014) Spheroids in viscoplastic fluids: drag and heat transfer. Ind Eng Chem Res 53(49):18943–18965CrossRefGoogle Scholar
  31. 31.
    Nirmalkar CN, Bose A, Chhabra RP (2014) Mixed convection from a heated sphere in Bingham plastic fluids. Num Heat Trans Part A: Appl 66:1048–1075CrossRefGoogle Scholar
  32. 32.
    Patel SA, Chhabra RP (2013) Steady flow of Bingham plastic fluids past an elliptical cylinder. J Non-Newtonian Fluid Mech 202:32–53CrossRefGoogle Scholar
  33. 33.
    Roche PE (2007) Applicability of Boussinesq approximation in a turbulent fluid with constant properties. Phys Fluids., hal-00180267
  34. 34.
    Zeytounian RK (2003) Joseph Boussinesq and his approximation: a contemporary view. C R Mecanique 331:575–586CrossRefzbMATHGoogle Scholar
  35. 35.
    Papanastasiou TC (1987) Flow of materials with yield. J Rheol 31:385–404CrossRefzbMATHGoogle Scholar
  36. 36.
    Mitsoulis E (2004) On creeping drag flow of a viscoplastic fluid past a circular cylinder: wall effects. Chem Eng Sci 59:789–800CrossRefGoogle Scholar
  37. 37.
    Mitsoulis E, Galazoulas S (2009) Simulation of viscoplastic flow past cylinders in tubes. J Non-Newtonian Fluid Mech 158(1–3):132–141CrossRefzbMATHGoogle Scholar
  38. 38.
    Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere Publishing Co., New YorkzbMATHGoogle Scholar
  39. 39.
    Oztop HF, Dagtekin I (2004) Mixed convection in two-sided lid-driven differencially heated square cavity. Int J Heat Mass Trans 47:1761–1769CrossRefzbMATHGoogle Scholar
  40. 40.
    Mitsoulis E, Zisis T (2001) Flow of Bingham plastics in a lid-driven square cavity. J Non-Newtonian Fluid Mech 101:173–180CrossRefzbMATHGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  • Nabila Labsi
    • 1
  • Youb Khaled Benkahla
    • 1
  • Abdelkader Boutra
    • 1
  • Meriem Titouah
    • 1
  1. 1.Equipe Rhéologie et Simulation Numérique des Ecoulements, Laboratoire des Phénomènes de Transfert, Faculté de Génie Mécanique et de Génie des ProcédésUniversité des Sciences et de la Technologie Houari BoumedieneAlgiersAlgeria

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