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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

## Abstract

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
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## Figures

Fig. 1

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

Fig. 2

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

Fig. 3

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

Fig. 4

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

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

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

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

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*

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*

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.

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

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

Fig. 5

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

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