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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

Abstract

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.

Keywords

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

Abbreviations

Cp

Specific heat, J kg−1 K−1

Bn

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

Bng

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

Bng,C

Critical generalized Bingham number

Br

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

Gr

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

H

Dimension of the cavity, m

k

Thermal conductivity, W m−1 K−1

K

Fluid consistency, kg m−1 s−2−n

m

Exponential growth parameter, s

M

Reduced exponential growth parameter

n

Flow index

NuAv

Average Nusselt number within the enclosure

Nuc

Local Nusselt number along the cold wall

Nuc,Av

Average Nusselt number along the cold wall

Nuh

Local Nusselt number along the hot wall

Nuh, Av

Average Nusselt number along the hot wall

p

Pressure, Pa

P

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

Pr

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

Re

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

Ri

Richardson number

Tc

Cold wall temperature, K

Th

Hot wall temperature, K

T0

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

u

Horizontal velocity component, m s−1

U

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

v

Vertical velocity component, m s−1

V

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

Vw

Lid-driven plate velocity, m s−1

x

Horizontal coordinate, m

X

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

y

Vertical coordinate, m

Y

Dimensionless vertical coordinate, = y/H

Greek letters

β

Thermal expansion coefficient, K−1

\(\dot{\gamma }\)

Strain rate, s−1

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

Dimensionless strain rate

μp

Plastic viscosity, kg m−1 s−1

η

Apparent viscosity, kg m−1 s−1

ηapp

Dimensionless apparent viscosity

θ

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

ρ

Fluid density, kg m−3

ρ0

Fluid density at a reference temperature, kg m−3

τ0

Yield stress, Pa

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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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