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Research Papers: Fundamental Issues and Canonical Flows

Stability of Plane Channel Flow With Viscous Heating

[+] Author and Article Information
K. C. Sahu1

Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Yeddumailaram 502205, Andhra Pradesh, Indiaksahu@iith.ac.in

O. K. Matar

Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UKo.matar@imperial.ac.uk

1

Corresponding author.

J. Fluids Eng 132(1), 011202 (Jan 12, 2010) (7 pages) doi:10.1115/1.4000847 History: Received June 01, 2009; Revised December 07, 2009; Published January 12, 2010; Online January 12, 2010

The linear stability analysis of pressure-driven flow undergoing viscous heating through a channel is considered. The walls of the channel are maintained at different constant temperatures and Nahme’s law is applied to model the temperature dependence of the fluid viscosity. A modified Orr–Sommerfeld equation coupled with a linearized energy equation is derived and solved using an efficient spectral collocation method. Our results indicate that increasing the influence of viscous heating is destabilizing. It is also shown that the critical Reynolds number decreases by one order of magnitude with increase in the Nahme number. An energy analysis is conducted to understand the underlying physical mechanism of the instability.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic diagram of the channel; bottom and top walls are maintained at different, constant temperatures, Tl and Tu, respectively. Also shown here are profiles of the steady, horizontal velocity component generated with Na=0.86, rT=1 (solid line), Na=0.86, rT=−0.5 (dotted line), and Na=0, rT=0 (dashed line).

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Figure 2

Basic state profiles of the second derivative of the horizontal component of the velocity (U″) and temperature (T0) for different Na when rT=1 and rT=−0.5 are shown in (a) and (b) and (c) and (d), respectively

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Figure 3

The effect of increasing the order of Chebyshev polynomials (N) on the neutral stability curve with rT=1, Pr=1, Gr=100, and Na=0.86. The dotted line shows the neutral stability curve for rT=0, Gr=0, and Na=0, which corresponds to isothermal flow through a channel; the critical Reynolds number for this case is 5772.2.

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Figure 4

Stability boundaries for different values of rT. The rest of the parameter values are Pr=0, Gr=0, and Na=0. In Fig. 4, ReH≡UmρH/μhot. Note that the curves associated with rT=−0.5 and rT=0.5 in panel (b) are indistinguishable.

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Figure 5

The effect of varying the Na on the dispersion curves for rT=1 (a) and rT=−0.5 (b). The rest of the parameter values are Re=104, Pr=1, and Gr=100. The dotted line and the line with the open squares represent the dispersion curves for isothermal channel (rT=0, Na=0) and symmetrically heated channel flow (rT=0, Na=0.86). The labels A–E and F–H are used to designate the maxima in the dispersion curves in (a) and (b), respectively; the energy budgets associated with the points labeled A–E and F–H are provided in Tables  12, respectively.

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Figure 6

Variation in the critical Reynolds number with Na for different values of rT. The rest of the parameter values are Pr=1 and Gr=100.

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Figure 7

The effect of varying rT on the dispersion curves for Na=0.1 (a) and Na=1.1 (b). The rest of the parameter values are same as those used to generate Fig. 6. The labels A–H are used to designate the maxima in the dispersion curves; the energy budgets associated with these points are provided in Tables  34.

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