0
Research Papers: Forced Convection

Nusselt Numbers for Thermally Developing Couette Flow With Hydrodynamic and Thermal Slip

[+] Author and Article Information
Lisa Steigerwalt Lam

Mechanical Engineering Department,
Tufts University,
Medford, MA 02155
e-mail: lisa_lam@alum.mit.edu

Corey Melnick, Marc Hodes, Gennady Ziskind

Mechanical Engineering Department,
Tufts University,
Medford, MA 02155

Ryan Enright

Stokes Institute,
University of Limerick,
Limerick, Ireland

According to Kennard [19].

The Peclet number referred to here is Pec = RecPr, where Rec=ρu¯cl/μ is the Reynolds number at the composite interface, u¯c is the mean velocity at the composite interface, l is the pitch of the structures, and Pr is the Prandtl number of the liquid.

Po=fReDh

We use um rather than Uo in the Reynolds number. This has the effect of normalizing the mass flow rate for various hydrodynamic slip boundary conditions and allows one to compare temperature profiles at a given dimensionless channel length to determine the relative amount of thermal energy absorbed by the fluid for different boundary conditions.

We computed a value of 3.918 for the fully developed Nusselt number for case D.1, 0.7% below that reported by Sesták and Rieger. If one solves for the eigenvalue using the eigenfunction provided by Sesták and Rieger and then uses their equation for Nusselt number, the same value of 3.918 is obtained.

Several studies replicated the Sesták and Rieger results. The case A.1 dimensionless temperature profile is replicated by Hudson and Bankoff [7]; Bruin [8] shows the case B.1 dimensionless temperature profile and Num,fd in agreement with Sesták and Rieger, and Davis [9] replicates Nus,fd for case B.1.

1On sabbatical leave from Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 7, 2013; final manuscript received December 12, 2013; published online February 26, 2014. Assoc. Editor: James A. Liburdy.

J. Heat Transfer 136(5), 051703 (Feb 26, 2014) (11 pages) Paper No: HT-13-1006; doi: 10.1115/1.4026305 History: Received January 07, 2013; Revised December 12, 2013

The effects of hydrodynamic and thermal slip on heat transfer in a thermally developing, steady, laminar Couette flow are investigated. Fluid temperature at the inlet to a parallel plate channel is prescribed, as various combinations of isothermal and adiabatic boundary conditions are along its surfaces. Analytical expressions incorporating arbitrary slip are developed for temperature profiles, and developing and fully developed for Nusselt numbers. The results are relevant to liquid and gas flows in the presence of apparent and molecular slip, respectively.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Quéré, D., 2005, “Non-Sticking Drops,” Rep. Prog. Phys., 68, pp. 2495–2532. [CrossRef]
Wang, E. N., Bucaro, M. A., Taylor, J. A., Kolodner, P., Aizenberg, J., and Krupenkin, T., 2009, “Droplet Mixing Using Electrically Tunable Superhydrophobic Nanostructured Surfaces,” Microfluid. Nanofluid., 7(1), pp. 137–140. [CrossRef]
Lam, L. S., Hodes, M., and Enright, R., 2013, “Galinstan-Based Microgap Cooling Enhancement Using Structured Surfaces,” Proceedings of the ASME 2013 Summer Heat Transfer Conference.
Lifton, V., Taylor, J., Vyas, B., Kolodner, P., Cirelli, R., Basavanhally, N., Papazian, A., Frahm, R., Simon, S., and Krupenkin, T., 2008, “Superhydrophobic Membranes With Electrically Controllable Permeability and Their Application to Smart Microbatteries,” Appl. Phys. Lett., 93(4), p. 043112. [CrossRef]
Cassie, A. B. D., and Baxter, S., 1944, “Wettability of Porous Surfaces,” Trans. Faraday Soc., 40, pp. 546–551. [CrossRef]
Vogelpohl, G., 1951, “Die Temperaturverteilung in Schmierschichten zwishen parallen warmendurchlassigen Wanden,” Z. Angew. Math. Mech., 31, pp. 349–356. [CrossRef]
Hudson, J., and Bankoff, S., 1965, “Heat Transfer to a Steady Couette Flow With Pressure Gradient,” Chem. Eng. Sci., 20(5), pp. 415–423. [CrossRef]
Bruin, S., 1972, “Temperature Distributions in Couette Flow With and Without Additional Pressure,” Int. J. Heat Mass Transfer, 15, pp. 341–349. [CrossRef]
Davis, E. J., 1973, “Exact Solutions for a Class of Heat and Mass transfer problems,” Can. J. Chem. Eng., 51, pp. 562–572. [CrossRef]
El-Ariny, A., and Aziz, A., 1976, “A Numerical Solution of Entrance Region Heat Transfer in Plane Couette Flow,” ASME J. Heat Transfer, 98, pp. 427–431. [CrossRef]
Sesták, J., and Rieger, F., 1969, “Laminar Heat Transfer to a Steady Couette Flow Between Parallel Plates,” Int. J. Heat Mass Transfer, 12, pp. 71–80. [CrossRef]
Schamberg, R., 1947, “The Fundamental Differential Equations and the Boundary Conditions for High Speed Slip-Flow, and Their Application to Several Specific Problems,” Ph.D. thesis, California Institute of Technology, Pasadena, CA.
Marques, W., Jr., Kremer, G., and Sharipov, F., 2000, “Couette Flow With Slip and Jump Boundary Conditions,” Continuum Mech. Thermodyn., 12(6), pp. 379–386. [CrossRef]
Fang, Y., and Liou, W. W., 2002, “Computations of the Flow and Heat Transfer in Microdevices Using DSMC With Implicit Boundary Conditions,” ASME J. Heat Transfer, 124(2), pp. 338–345. [CrossRef]
Sharipov, F., and Strapasson, J. L., 2013, “Benchmark Problems for Mixtures of Rarefied Gases. I. Couette Flow,” Phys. Fluids, 25(2), p. 027101. [CrossRef]
Milicev, S. S., and Stevanovic, N. D., 2013, “A Non-Isothermal Couette Slip Gas Flow,” Sci. Chin. Phys., Mech. Astron., 56(9), pp. 1782–1797. [CrossRef]
Navier, C., 1823, “Mémoire sur les du mouvement des fluids,” Mémoires de l'Académie Royale des Sciences de l'Institut de France, 6, pp. 389–440.
Maxwell, J. C., 1965, Scientific Papers, Dover Publications, New York.
Kennard, E. H., 1938, Kinetic Theory of Gases, 1st ed., McGraw-Hill, New York.
Sparrow, E., and Lin, S., 1962, “Laminar Heat Transfer in Tubes Under Slip Flow Conditions,” ASME J. Heat Transfer, pp. 362–369.
Barron, R. F., Wang, X., Ameel, T. A., and Warrington, R. O., 1997, “The Graetz Problem Extended to Slip-Flow,” Int. J. Heat Mass Transfer, 40(8), pp. 1817–1823. [CrossRef]
Jiji, L. M., 2009, Heat Convection, 2nd ed., Springer-Verlag, Berlin.
Colin, S., 2012, “Gas Microflows in the Slip Flow Regime: A Critical Review on Convective Heat Transfer,” ASME J. Heat Transfer, 134(2), p. 020908. [CrossRef]
Cheng, Y., Teo, C., and Khoo, B., 2009, “Microchannel Flows With Superhydrophobic Surfaces: Effects of Reynolds Number and Pattern Width to Channel Height Ratio,” Phys. Fluids, 21, p. 122004. [CrossRef]
Rothstein, J. P., 2010, “Slip on Superhydrophobic Surfaces,” Annu. Rev. Fluid Mech., 42(1), pp. 89–109. [CrossRef]
Philip, J., 1972, “Flows Satisfying Mixed No-Slip and No-Shear Conditions,” J. Appl. Math. Phys., 23, pp. 353–372. [CrossRef]
Philip, J., 1972, “Integral Properties of Flows Satisfying Mixed No-Slip and No-Shear Conditions,” J. Appl. Math. Phys., 23, pp. 960–968. [CrossRef]
Lauga, E., and Stone, H., 2003, “Effective Slip in Pressure-Driven Stokes Flow,” J. Fluid Mech., 489, pp. 55–77. [CrossRef]
Ou, J., Perot, B., and Rothstein, J. P., 2004, “Laminar Drag Reduction in Microchannels Using Ultrahydrophobic Surfaces,” Phys. Fluids, 16(12), pp. 4635–4643. [CrossRef]
Ou, J., and Rothstein, J., 2005, “Direct Velocity Measurements of the Flow Past Drag-Reducing Ultrahydrophobic Surfaces,” Phys. Fluids, 17, p. 103606. [CrossRef]
Priezjev, N. V., Darhuber, A. A., and Troian, S. M., 2005, “Slip Behavior in Liquid Films on Surfaces of Patterned Wettability,” Phys. Rev. E, 71, p. 041608. [CrossRef]
Truesdell, R., Mammoli, A., Vorobieff, P., van Swol, F., and Brinker, C., 2006, “Drag Reduction on a Patterned Superhydrophobic Surface,” Phys. Rev. Lett., 97(4), pp. 1–4. [CrossRef]
Lee, C., Choi, C.-H., and Kim, C.-J., 2008, “Structured Surfaces for a Giant Liquid Slip,” Phys. Rev. Lett., 101(6), pp. 1–4.
Ybert, C., Barentin, C., Cottin-Bizonne, C., Joseph, P., and Bocquet, L., 2007, “Achieving Large Slip With Superhydrophobic Surfaces: Scaling Laws for Generic Geometries,” Phys. Fluids, 19(12), p. 123601. [CrossRef]
Salamon, T., Lee, W., Hodes, M., Kolodner, P., Enright, R., and Salinger, A., 2005, “Numerical Simulation of Fluid Flow in Microchannels With Superhydrophobic Walls,” IMECE Conference Proceedings, ASME, pp. 819–829, Paper No. 42215.
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. [CrossRef]
Davies, J., Maynes, D., Webb, B. W., and Woolford, B., 2006, “Laminar Flow in a Microchannel With Superhydrophobic Walls Exhibiting Transverse Ribs,” Phys. Fluids, 18(8), p. 087110. [CrossRef]
Teo, C. J., and Khoo, B. C., 2010, “Flow Past Superhydrophobic Surfaces Containing Longitudinal Grooves: Effects of Interface Curvature,” Microfluidics Nanofluidics, 9(2–3), pp. 499–511. [CrossRef]
Steinberger, A., Cottin-Bizonne, C., Kleimann, P., and Charlaix, E., 2007, “High Friction on a Bubble Mattress,” Nature Mater., 6(9), pp. 665–668. [CrossRef]
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, p. 012402. [CrossRef]
Enright, R., Hodes, M., Salamon, T., Krupenkin, T., Kolodner, P., Dalton, T., and Eason, C., 2006, “Friction Factors and Nusselt Numbers in Microchannels With Superhydrophobic Walls,” Proceedings of the Fourth International Conference on Nanochannels, Microchannels and Minichannels, Limerick Ireland, ASME, New York, pp. 599–609, Paper No. ICNMM2006-96134.
Williams, A. D., Vorobieff, P., and Mammoli, A., 2012, “Effect of Slip Flow on Heat Transfer: Numerical Analysis,” 50th AIAA Aerospace Sciences Meeting, p. 7726.
Maynes, D., Webb, B. W., and Davies, J., 2008, “Thermal Transport in a Microchannel Exhibiting Ultrahydrophobic Microribs Maintained at Constant Temperature,” ASME J. Heat Transfer, 130(2), p. 022402. [CrossRef]
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]
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, p. 011701 [CrossRef].
Inman, R. M., 1964, “Laminar Slip Flow Heat Transfer in a Parallel-Plate Channel or a round Tube With Uniform Wall Heating,” NASA Technical Note D-2393.
Teo, C. J., and Khoo, B. C., 2009, “Analysis of Stokes Flow in microchannels With Superhydrophobic Surfaces Containing a Periodic Array of Micro-Grooves,” Microfluid. Nanofluid., 7, pp. 352–382. [CrossRef]
Weigand, B., 2004, Analytical Methods for Heat Transfer and Fluid Flow Problems, Springer, New York.
Haberman, R., 2004, Applied Partial Differential Equations With Fourier Series and Boundary Value Problems, 4th ed., Prentice-Hall, Englewood Cliffs, NJ.
Greenberg, M., 1998, Advanced Engineering Mathematics, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ.

Figures

Grahic Jump Location
Fig. 1

Sketch of a composite surface. Liquid is suspended on the tips of the structures and vapor fills the space below.

Grahic Jump Location
Fig. 2

Representative fluid velocity profiles for a Couette flow with four possible combinations of hydrodynamic slip

Grahic Jump Location
Fig. 3

Sketch of temperature profiles for four thermal boundary conditions for cases A–D in the absence of slip

Grahic Jump Location
Fig. 4

Sketch of temperature profiles for four thermal boundary conditions for cases A–D in the presence of slip

Grahic Jump Location
Fig. 5

Nusselt number versus dimensionless channel length, x*, for case A.1, symmetric constant temperature boundary conditions with no-slip

Grahic Jump Location
Fig. 6

Fully developed Nusselt numbers at the stationary and moving plates, Nus,fd and Num,fd, versus thermal slip at the stationary plate, bt,s, for incremental values of hydrodynamic slip at the stationary plate, bs, for case A, symmetric constant temperature boundary conditions

Grahic Jump Location
Fig. 7

Fully developed Nusselt numbers at the stationary and moving plates, Nus,fd and Num,fd, versus thermal slip at the moving plate, bt,m, for incremental values of hydrodynamic slip at the stationary plate, bs, for case A, symmetric constant temperature boundary conditions

Grahic Jump Location
Fig. 8

Dimensionless temperature profiles at x* = 0.1, for cases A.1–4, symmetric constant temperature boundary conditions with no-slip and various values of hydrodynamic and thermal slip, bs, bt,s, and bt,m

Grahic Jump Location
Fig. 9

Dimensionless temperature profiles for case A.4, symmetric constant temperature boundary conditions with hydrodynamic and thermal slip on both surfaces when bs* = bt,s* = bt,m* = 0.5 for various values of x*

Grahic Jump Location
Fig. 10

Dimensionless temperature profiles for case B.4, asymmetric constant temperature boundary conditions with hydrodynamic and thermal slip at both surfaces when bs* = bt,s* = bt,m* = 0 for various values of x*

Grahic Jump Location
Fig. 11

Nusselt number versus dimensionless channel length, x*, case A.1–4, symmetric constant temperature boundary conditions with no-slip and varying values of hydrodynamic and thermal slip, bs, bt,s, and bt,m.

Grahic Jump Location
Fig. 12

Nusselt number versus dimensionless channel length, x*, for case B.1, asymmetric constant temperature boundary conditions with no-slip when bs* = bs* = bt,s* = bt,m* = 0

Grahic Jump Location
Fig. 13

Nusselt number versus dimensionless channel length, x*, for case B.1–4, asymmetric constant temperature boundary conditions with no-slip and varying values of hydrodynamic and thermal slip, bs, bt,s, and bt,m

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In