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Research Papers: Forced Convection

Solution of the Extended Graetz–Nusselt Problem for Liquid Flow Over Isothermal Parallel Ridges

[+] Author and Article Information
Georgios Karamanis

Department of Mechanical Engineering,
Tufts University,
Medford, MA 02155
e-mail: Georgios.Karamanis@tufts.edu

Marc Hodes

Department of Mechanical Engineering,
Tufts University,
Medford, MA 02155
e-mail: Marc.Hodes@tufts.edu

Toby Kirk

Department of Mathematics,
Imperial College London,
London SW7 2AZ, UK
e-mail: toby.kirk12@imperial.ac.uk

Demetrios T. Papageorgiou

Department of Mathematics,
Imperial College London,
London SW7 2AZ, UK
e-mail: d.papageorgiou@imperial.ac.uk

1In general, the cavities beneath the menisci are filled with inert gas and vapor on account of the vapor pressure of the liquid phase, and for brevity, we refer to this mixture as “gas.”

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received July 5, 2017; final manuscript received January 18, 2018; published online March 9, 2018. Assoc. Editor: George S. Dulikravich.

J. Heat Transfer 140(6), 061703 (Mar 09, 2018) (15 pages) Paper No: HT-17-1395; doi: 10.1115/1.4039085 History: Received July 05, 2017; Revised January 18, 2018

We consider convective heat transfer for laminar flow of liquid between parallel plates. The configurations analyzed are both plates textured with symmetrically aligned isothermal ridges oriented parallel to the flow, and one plate textured as such and the other one smooth and adiabatic. The liquid is assumed to be in the Cassie state on the textured surface(s) to which a mixed boundary condition of no-slip on the ridges and no-shear along flat menisci applies. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). We solve for the developing three-dimensional temperature profile resulting from a step change of the ridge temperature in the streamwise direction assuming a hydrodynamically developed flow. Axial conduction is accounted for, i.e., we consider the extended Graetz–Nusselt problem; therefore, the domain is of infinite length. The effects of viscous dissipation and (uniform) volumetric heat generation are also captured. Using the method of separation of variables, the homogeneous part of the thermal problem is reduced to a nonlinear eigenvalue problem in the transverse coordinates which is solved numerically. Expressions derived for the local and the fully developed Nusselt number along the ridge and that averaged over the composite interface in terms of the eigenvalues, eigenfunctions, Brinkman number, and dimensionless volumetric heat generation rate. Estimates are provided for the streamwise location where viscous dissipation effects become important.

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References

Quéré, D. , 2005, “ Non-Sticking Drops,” Rep. Prog. Phys., 68(11), pp. 2495–2532. [CrossRef]
Cassie, A. , and Baxter, S. , 1944, “Wettability of Porous Surfaces,” Trans. Faraday Soc., 40, pp. 546–551. [CrossRef]
Lam, L. S. , Hodes, M. , and Enright, R. , 2015, “Analysis of Galinstan-Based Microgap Cooling Enhancement Using Structured Surfaces,” ASME J. Heat Transfer, 137(9), p. 091003. [CrossRef]
Huang, D. M. , Sendner, C. , Horinek, D. , Netz, R. R. , and Bocquet, L. , 2008, “Water Slippage versus Contact Angle: A Quasiuniversal Relationship,” Phys. Rev. Lett., 101(22), p. 226101. [CrossRef] [PubMed]
Cottin-Bizonne, C. , Steinberger, A. , Cross, B. , Raccurt, O. , and Charlaix, E. , 2008, “Nanohydrodynamics: The Intrinsic Flow Boundary Condition on Smooth Surfaces,” Langmuir, 24(4), pp. 1165–1172. [CrossRef] [PubMed]
Lobaton, E. , and Salamon, T. , 2007, “Computation of Constant Mean Curvature Surfaces: Application to the Gas-Liquid Interface of a Pressurized Fluid on a Superhydrophobic Surface,” J. Colloid Interface Sci., 314(1), pp. 184–198. [CrossRef] [PubMed]
Enright, R. , Hodes, M. , Salamon, T. , and Muzychka, Y. , 2014, “Isoflux Nusselt Number and Slip Length Formulae for Superhydrophobic Microchannels,” ASME J. Heat Transfer, 136(1), p. 012402. [CrossRef]
Philip, J. R. , 1972, “Flows Satisfying Mixed No-Slip and No-Shear Conditions,” Z. Angew. Math. Phys., 23(3), pp. 353–372. [CrossRef]
Sbragaglia, M. , and Prosperetti, A. , 2007, “A Note on the Effective Slip Properties for Microchannel Flows With Ultrahydrophobic Surfaces,” Phys. Fluids, 19(4), p. 043603. [CrossRef]
Maynes, D. , Jeffs, K. , Woolford, B. , and Webb, B. W. , 2007, “Laminar Flow in a Microchannel With Hydrophobic Surface Patterned Microribs Oriented Parallel to the Flow Direction,” Phys. Fluids, 19(9), p. 093603.
Teo, C. , and Khoo, B. , 2009, “Analysis of Stokes Flow in Microchannels With Superhydrophobic Surfaces Containing a Periodic Array of Micro-Grooves,” Microfluid. Nanofluidics, 7(3), pp. 353–382. [CrossRef]
Teo, C. J. , and Khoo, B. C. , 2010, “Flow Past Superhydrophobic Surfaces Containing Longitudinal Grooves: Effects of Interface Curvature,” Microfluid. Nanofluid., 9(2–3), pp. 499–511. [CrossRef]
Rothstein, J. P. , 2010, “Slip on Superhydrophobic Surfaces,” Annu. Rev. Fluid Mech., 42(1), pp. 89–109. [CrossRef]
Ng, C.-O. , Chu, H. C. W. , and Wang, C. Y. , 2010, “On the Effects of Liquid-Gas Interfacial Shear on Slip Flow Through a Parallel-Plate Channel With Superhydrophobic Grooved Walls,” Phys. Fluids, 22(10), p. 102002.
Marshall, J. , 2017, “Exact Formulae for the Effective Slip Length of a Symmetric Superhydrophobic Channel With Flat or Weakly Curved Menisci,” SIAM J. Appl. Math., 77(5), pp. 1606–1630.
Ng, C.-O. , and Wang, C. Y. , 2014, “Temperature Jump Coefficient for Superhydrophobic Surfaces,” ASME J. Heat Transfer, 136(6), p. 064501. [CrossRef]
Lam, L. S. , Hodes, M. , Karamanis, G. , Kirk, T. , and MacLachlan, S. , 2016, “Effect of Meniscus Curvature on Apparent Thermal Slip,” ASME J. Heat Transfer, 138(12), p. 122004. [CrossRef]
Hodes, M. , Lam, L. S. , Cowley, A. , Enright, R. , and MacLachlan, S. , 2015, “Effect of Evaporation and Condensation at Menisci on Apparent Thermal Slip,” ASME J. Heat Transfer, 137(7), p. 071502. [CrossRef]
Lam, L. S. , Melnick, C. , Hodes, M. , Ziskind, G. , and Enright, R. , 2014, “Nusselt Numbers for Thermally Developing Couette Flow With Hydrodynamic and Thermal Slip,” ASME J. Heat Transfer, 136(5), p. 051703 . [CrossRef]
Maynes, D. , and Crockett, J. , 2014, “Apparent Temperature Jump and Thermal Transport in Channels With Streamwise Rib and Cavity Featured Superhydrophobic Walls at Constant Heat Flux,” ASME J. Heat Transfer, 136(1), p. 011701. [CrossRef]
Kirk, T. , Hodes, M. , and Papageorgiou, D. T. , 2017, “Nusselt Numbers for Poiseuille Flow Over Isoflux Parallel Ridges Accounting for Meniscus Curvature,” J. Fluid Mech., 811(1), pp. 315–349. [CrossRef]
Karamanis, G. , Hodes, M. , Kirk, T. , and Papageorgiou, D. , 2017, “Solution of the Graetz–Nusselt Problem for Liquid Flow Over Isothermal Parallel Ridges,” ASME J. Heat Transfer, 139(9), p. 091702.
Nusselt, W. , 1923, “Der Wärmeaustausch Am Berieselungskühler,” VDI-Z, 67(9), pp. 206–216.
Brown, G. M. , 1960, “Heat or Mass Transfer in a Fluid in Laminar Flow in a Circular or Flat Conduit,” AIChE J., 6(2), pp. 179–183. [CrossRef]
Mikhaĭlov, M. D. , and Öz iş ik, M. N. , 1994, Unified Analysis and Solutions of Heat and Mass Diffusion, Dover Publications, Mineola, NY.
Agrawal, H. , 1960, “Heat Transfer in Laminar Flow Between Parallel Plates at Small Peclet Numbers,” Appl. Sci. Res., 9(1), pp. 177–189. [CrossRef]
Deavours, C. , 1974, “An Exact Solution for the Temperature Distribution in Parallel Plate Poiseuille Flow,” ASME J. Heat Transfer, 96(4), pp. 489–495. [CrossRef]
Pahor, S. , and Strnad, J. , 1961, “A Note on Heat Transfer in Laminar Flow Through a Gap,” Appl. Sci. Res., 10(1), pp. 81–84. [CrossRef]
Dang, V.-D. , 1983, “Heat Transfer of Power Law Fluid at Low Peclet Number Flow,” ASME J. Heat Transfer, 105(3), pp. 542–549. [CrossRef]
Jambal, O. , Shigechi, T. , Davaa, G. , and Momoki, S. , 2005, “Effects of Viscous Dissipation and Fluid Axial Heat Conduction on Heat Transfer for Non-Newtonian Fluids in Ducts With Uniform Wall Temperature—Part I: Parallel Plates and Circular Ducts,” Int. Commun. Heat Mass Transfer, 32(9), pp. 1165–1173. [CrossRef]
Sparrow, E. , Novotny, J. , and Lin, S. , 1963, “Laminar Flow of a Heat-Generating Fluid in a Parallel-Plate Channel,” AIChE J., 9(6), pp. 797–804. [CrossRef]
Hodes, M. , Kirk, T. , Karamanis, G. , and MacLachlan, S. , 2017, “Effect of Thermocapillary Stress on Slip Length for a Channel Textured With Parallel Ridges,” J. Fluid Mech., 814, pp. 301–324. [CrossRef]
Panton, R. L. , 2006, Incompressible Flow, Wiley, Hoboken, NJ.
Maynes, D. , Webb, B. , Crockett, J. , and Solovjov, V. , 2013, “Analysis of Laminar Slip-Flow Thermal Transport in Microchannels With Transverse Rib and Cavity Structured Superhydrophobic Walls at Constant Heat Flux,” ASME J. Heat Transfer, 135(2), p. 021701. [CrossRef]
MathWorks, 2013, “Partial Differential Equation Toolbox User's Guide,” The MathWorks Inc., Natick, MA.

Figures

Grahic Jump Location
Fig. 1

Depiction of a structured microchannel etched into the upper portion of a microprocessor die

Grahic Jump Location
Fig. 3

Schematic of the periodic domain when one plate is textured with isothermal ridges and the other one is smooth and adiabatic; the computational domain is indicated with dotted line

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Fig. 2

Schematic of the periodic domain when both plates are textured; the computational domain is indicated with dotted line

Grahic Jump Location
Fig. 4

Nufd,Pe− versus ϕ for Pe=1 and selected H/d when both plates are textured with isothermal ridges

Grahic Jump Location
Fig. 6

Nul,fd,Pe+ versus the normalized coordinate along the ridge (x̃−ã)/(d̃−ã) for Pe=1, H/d=10 and selected values of ϕ when both plates are textured with isothermal ridges

Grahic Jump Location
Fig. 7

Nufd,Pe− versus ϕ for Pe=0.01,1,10 and H/d=1 when both plates are textured with isothermal ridges

Grahic Jump Location
Fig. 8

Nufd,Pe+ versus ϕ for Pe=0.01,1,10 and Pe→∞, and H/d=1 when both plates are textured with isothermal ridges

Grahic Jump Location
Fig. 9

Nufd,Pe− versus ϕ for Pe=0.01,1,10 and H/d=10 when both plates are textured with isothermal ridges

Grahic Jump Location
Fig. 10

Nufd,Pe+ versus ϕ for Pe=0.01,1,10 and Pe→∞, and H/d=10 when both plates are textured with isothermal ridges

Grahic Jump Location
Fig. 5

Nufd,Pe+ versus ϕ for Pe=1 and selected H/d when both plates are textured with isothermal ridges

Grahic Jump Location
Fig. 11

Nufd,Br± versus ϕ for selected H/d when both plates are textured with isothermal ridges

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Fig. 12

Nufd,q˙̃± versus ϕ for selected H/d when both plates are textured with isothermal ridges

Grahic Jump Location
Fig. 13

Nu+ versus z̃ for ϕ=0.01, H/d=10, Pe=1, and q˙̃=0 when both plates are textured with isothermal ridges; z̃Pe+=0.13, z̃Br+,1=2.81, and z̃Br+,2=4.65

Grahic Jump Location
Fig. 14

Nu+ versus z̃ for ϕ=0.01, H/d=10, Pe=10, and q˙̃=0 when both plates are textured with isothermal ridges; z̃Pe+=0.02, z̃Br+,1=0.71, and z̃Br+,2=1.17

Grahic Jump Location
Fig. 19

Nufd,Pe− versus ϕ for Pe=0.01,1,10 and H/d=10 when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Grahic Jump Location
Fig. 20

Nufd,Pe+ versus ϕ for Pe=0.01,1,10 and Pe→∞, and H/d=10 when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Grahic Jump Location
Fig. 21

Nufd,Br± versus ϕ for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Grahic Jump Location
Fig. 22

Nufd,q˙̃± versus ϕ for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Grahic Jump Location
Fig. 23

Nu+ versus z̃ for ϕ=0.01, H/d=10, Pe=1, and q˙̃=0 when one plate is textured with isothermal ridges and the other one is smooth and adiabatic; z̃Pe+=0.24, z̃Br+,1=5.03, and z̃Br+,2=8.34

Grahic Jump Location
Fig. 24

Nu+ versus z̃ for ϕ=0.01, H/d=10, Pe=10, and q˙̃=0 when one plate is textured with isothermal ridges and the other one is smooth and adiabatic; z̃Pe+=0.05, z̃Br+,1=1.65, and z̃Br+,2=2.74

Grahic Jump Location
Fig. 15

Nufd,Pe− versus ϕ for Pe=1 and selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

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Fig. 16

Nufd,Pe+ versus ϕ for Pe=1 selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Grahic Jump Location
Fig. 17

Nufd,Pe− versus ϕ for Pe=0.01,1,10 and H/d=1 when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Grahic Jump Location
Fig. 18

Nufd,Pe+ versus ϕ for Pe=0.01,1,10 and Pe→∞, H/d=1 when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

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