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MICRO/NANOSCALE HEAT TRANSFER—PART I

# Flow and Stability of Rivulets on Heated Surfaces With Topography

[+] Author and Article Information
Tatiana Gambaryan-Roisman, Peter Stephan

Chair of Technical Thermodynamics, Technische Universität Darmstadt, Petersenstrasse 30, 64287 Darmstadt, Germany

J. Heat Transfer 131(3), 033101 (Jan 13, 2009) (6 pages) doi:10.1115/1.3056593 History: Received November 04, 2007; Revised October 09, 2008; Published January 13, 2009

## Abstract

Surfaces with topography promote rivulet flow patterns, which are characterized by a high cumulative length of contact lines. This property is very advantageous for evaporators and cooling devices, since the local evaporation rate in the vicinity of contact lines (microregion evaporation) is extremely high. The liquid flow in rivulets is subject to different kinds of instabilities, including the long-wave falling film instability (or the kinematic-wave instability), the capillary instability, and the thermocapillary instability. These instabilities may lead to the development of wavy flow patterns and to the rivulet rupture. We develop a model describing the hydrodynamics and heat transfer in flowing rivulets on surfaces with topography under the action of gravity, surface tension, and thermocapillarity. The contact line behavior is modeled using the disjoining pressure concept. The perfectly wetting case is described using the usual $h−3$ disjoining pressure. The partially wetting case is modeled using the integrated 6–12 Lennard-Jones potential. The developed model is used for investigating the effects of the surface topography, gravity, thermocapillarity, and the contact line behavior on the rivulet stability. We show that the long-wave thermocapillary instability may lead to splitting of the rivulet into droplets or into several rivulets, depending on the Marangoni number and on the rivulet geometry. The kinematic-wave instability may be completely suppressed in the case of the rivulet flow in a groove.

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## Figures

Figure 4

Disturbance growth rate for the kinematic-wave instability of isothermal rivulet for B=0.01, β=π/2, and Re2/S=0.298

Figure 5

Disturbance growth rate for the kinematic-wave instability of isothermal rivulet for Ψref=0.02, β=π/2, and Re2/S=0.298. Dashed line: asymptotic solution of Weiland and Davis (13) for fixed contact line position, flat surfaces, and small values of K.

Figure 6

Disturbance growth rate for the simultaneous action of gravity and thermocapillarity on the rivulet instability. Parameters: Ψref=0.02, B=0.01, β=π/2, and Re2/S=0.298.

Figure 1

The rivulet geometry

Figure 2

Basic rivulet shapes for B=0.01, Bi=2.59×10−3, κ/κs→0, and Re cot β/S→0. M=0.178 corresponds to ΔT=Tw−Tg=50 K for water, and M=−1.78×10−2 corresponds to ΔT=Tw−Tg=−5 K.

Figure 3

Disturbance growth rate for the thermocapillary instability of rivulet in the absence of gravity. Parameters: B=0.01, Bi=2.59×10−3, κ/κs→0, Re=0, A3=2.46×10−8, and A9=9.82×10−18 (or Ψref=5.0×10−3 and Hads=2.7×10−2).

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