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

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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.

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Abbreviations

C p :

Specific heat, J kg−1 K−1

Bn :

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

Bn g :

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

Bn g,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

Nu Av :

Average Nusselt number within the enclosure

Nu c :

Local Nusselt number along the cold wall

Nu c,Av :

Average Nusselt number along the cold wall

Nu h :

Local Nusselt number along the hot wall

Nu h, 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

T c :

Cold wall temperature, K

T h :

Hot wall temperature, K

T 0 :

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

V w :

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

β :

Thermal expansion coefficient, K−1

\(\dot{\gamma }\) :

Strain rate, s−1

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

Dimensionless strain rate

μ p :

Plastic viscosity, kg m−1 s−1

η :

Apparent viscosity, kg m−1 s−1

η app :

Dimensionless apparent viscosity

θ :

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

ρ :

Fluid density, kg m−3

ρ 0 :

Fluid density at a reference temperature, kg m−3

τ 0 :

Yield stress, Pa

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Authors and Affiliations

  1. Equipe Rhéologie et Simulation Numérique des Ecoulements, Laboratoire des Phénomènes de Transfert, Faculté de Génie Mécanique et de Génie des Procédés, Université des Sciences et de la Technologie Houari Boumediene, BP. 32 El Alia, Bab Ezzoouar, 16111, Algiers, Algeria

    Nabila Labsi, Youb Khaled Benkahla, Abdelkader Boutra & Meriem Titouah

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  1. Nabila Labsi
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  2. Youb Khaled Benkahla
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Correspondence to Nabila Labsi.

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Technical Editor: Cezar Negrao.

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Labsi, N., Benkahla, Y.K., Boutra, A. et al. Convection heat transfer inside a lid-driven cavity filled with a shear-thinning Herschel–Bulkley fluid. J Braz. Soc. Mech. Sci. Eng. 40, 123 (2018). https://doi.org/10.1007/s40430-018-1051-6

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