A bifurcation study of convective heat transfer in a hele‐shaw cell

D. K. Ryland; K. Nandakumar

doi:10.1002/cjce.5450720311

A bifurcation study of convective heat transfer in a hele‐shaw cell

D. K. Ryland, K. Nandakumar^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Steady‐state multiplicity characteristics of convective heat transfer within a Hele‐Shaw cell are investigated. The Navier‐Stokes equations and the energy equation are averaged across the narrow gap, d, of the cell. The resulting two‐dimensional, stationary equations depend on the following parameters: (i) the length to height aspect ratio γ, (ii) the tilt anle ϕ (iii) the Prandtl number Pr, (iv) an inertia parameter ξ = d²/ 12a², and (v) the Grashof number. Gr = Q_gβga⁵/kv². Here a is the height of the cell and Q, is the heat generation rate per unit volume. The complete structure of symmetric and asymmetric stationary solutions are traced using recent algorithms from bifurcation theory. In the double limit of ξ → 0 and Gr → ∞ such that Ra = 4GrPrξ remains finite (where Ra is the Rayleigh number for the Darcy model) the Hele‐Shaw model reduces to that of the Darcy model.

Original language	English
Pages (from-to)	457-467
Number of pages	11
Journal	Canadian Journal of Chemical Engineering
Volume	72
Issue number	3
DOIs	https://doi.org/10.1002/cjce.5450720311
State	Published - Jun 1994
Externally published	Yes

Keywords

bifurcation study, Hele‐Shaw cell, convective heat transfer

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10.1002/cjce.5450720311

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title = "A bifurcation study of convective heat transfer in a hele‐shaw cell",

abstract = "Steady‐state multiplicity characteristics of convective heat transfer within a Hele‐Shaw cell are investigated. The Navier‐Stokes equations and the energy equation are averaged across the narrow gap, d, of the cell. The resulting two‐dimensional, stationary equations depend on the following parameters: (i) the length to height aspect ratio γ, (ii) the tilt anle ϕ (iii) the Prandtl number Pr, (iv) an inertia parameter ξ = d2/ 12a2, and (v) the Grashof number. Gr = Qgβga5/kv2. Here a is the height of the cell and Q, is the heat generation rate per unit volume. The complete structure of symmetric and asymmetric stationary solutions are traced using recent algorithms from bifurcation theory. In the double limit of ξ → 0 and Gr → ∞ such that Ra = 4GrPrξ remains finite (where Ra is the Rayleigh number for the Darcy model) the Hele‐Shaw model reduces to that of the Darcy model.",

keywords = "bifurcation study, Hele‐Shaw cell, convective heat transfer",

author = "Ryland, {D. K.} and K. Nandakumar",

year = "1994",

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AU - Ryland, D. K.

AU - Nandakumar, K.

PY - 1994/6

Y1 - 1994/6

N2 - Steady‐state multiplicity characteristics of convective heat transfer within a Hele‐Shaw cell are investigated. The Navier‐Stokes equations and the energy equation are averaged across the narrow gap, d, of the cell. The resulting two‐dimensional, stationary equations depend on the following parameters: (i) the length to height aspect ratio γ, (ii) the tilt anle ϕ (iii) the Prandtl number Pr, (iv) an inertia parameter ξ = d2/ 12a2, and (v) the Grashof number. Gr = Qgβga5/kv2. Here a is the height of the cell and Q, is the heat generation rate per unit volume. The complete structure of symmetric and asymmetric stationary solutions are traced using recent algorithms from bifurcation theory. In the double limit of ξ → 0 and Gr → ∞ such that Ra = 4GrPrξ remains finite (where Ra is the Rayleigh number for the Darcy model) the Hele‐Shaw model reduces to that of the Darcy model.

AB - Steady‐state multiplicity characteristics of convective heat transfer within a Hele‐Shaw cell are investigated. The Navier‐Stokes equations and the energy equation are averaged across the narrow gap, d, of the cell. The resulting two‐dimensional, stationary equations depend on the following parameters: (i) the length to height aspect ratio γ, (ii) the tilt anle ϕ (iii) the Prandtl number Pr, (iv) an inertia parameter ξ = d2/ 12a2, and (v) the Grashof number. Gr = Qgβga5/kv2. Here a is the height of the cell and Q, is the heat generation rate per unit volume. The complete structure of symmetric and asymmetric stationary solutions are traced using recent algorithms from bifurcation theory. In the double limit of ξ → 0 and Gr → ∞ such that Ra = 4GrPrξ remains finite (where Ra is the Rayleigh number for the Darcy model) the Hele‐Shaw model reduces to that of the Darcy model.

KW - bifurcation study, Hele‐Shaw cell, convective heat transfer

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A bifurcation study of convective heat transfer in a hele‐shaw cell

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