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Research Papers: Fundamental Issues and Canonical Flows

The Instability of Flow Through a Slowly Diverging Pipe With Viscous Heating

[+] Author and Article Information
Kirti Chandra Sahu

 Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Yeddumailaram 502 205, Andhra Pradesh, Indiaksahu@iith.ac.in

J. Fluids Eng 133(7), 071201 (Jul 05, 2011) (8 pages) doi:10.1115/1.4004299 History: Received October 31, 2010; Accepted May 19, 2011; Published July 05, 2011; Online July 05, 2011

The nonparallel linear stability analysis of flow through a slowly diverging pipe undergoing viscous heating is considered. The pipe wall is maintained at constant temperatures and Nahme’s law is applied to model the temperature dependence of the fluid viscosity. A one-parameter family of velocity profiles for the basic state is obtained for small angles of divergence. The nonparallel stability equations for the disturbance velocity coupled to a linearized energy equation are derived and solved using a spectral collocation method. Our results indicate that increasing viscous heating, characterized by increasing Nahme number, is destabilizing. The Prandtl number has a negligible effect on the linear stability characteristics. The Grashof number stablizes the flow for Gr>106, below which it has a negligible effect.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Typical basic state profiles of the axial velocity (U), wall-normal velocity (V), and viscosity (μ0) for different Na values. The rest of the parameter values are Re = 100, a=0.02, and rT=1.

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Figure 2

The effect of increasing the order of Chebyshev polynomials N on the neutral stability curves (α versus Re) for a=0.02, rT=0.5, Na=0.3, Gr = 0, Pr = 1, and n = 1

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Figure 3

Comparison of the eigenvalue distribution generated for the isothermal, fully developed flow through a straight pipe with that of Schmid [28]. The parameter values are Re=2000, a = 0, rT=0, Na = 0, α=0.5, and n = 1.

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Figure 4

The effect of varying Na on (a) the neutral stability curves (α versus Re) and (b) dispersion curves (Ωi versus α) for rT=1. The rest of the parameter values are n = 1, a=10-3, Gr = 0, and Pr = 1; a typical value of Re = 4000 is considered in (b).

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Figure 5

The effect of varying Na on (a) the neutral stability curves (α versus Re) and (b) dispersion curves (Ωi versus α) for rT=-0.1; a typical value of Re=104 is considered in (b). The rest of the parameter values are the same as those used to generate Fig. 4.

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Figure 6

Variation of critical Reynolds number with slope of the pipe a for different values of Na with (a) rT=1 and (b) rT=-0.1. The rest of the parameter values are n = 1, Gr = 0, and Pr = 1. The dotted lines in (a) and (b) are associated with isothermal flow which are obtained by setting rT=0 and Na = 0.

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Figure 7

Inviscid instability function I for (a) rT=1 and (b) rT=-0.1 for different values of Na. The rest of the parameter values are S=0.5, Pr = 1, and Gr = 0. A typical value of α=1.26 corresponding to the critical instability in Fig. 4 is used.

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Figure 8

The effect of Gr on the critical Reynolds number. The rest of the parameter values are n = 1, a=10-3, Na = 1, and Pr = 1.

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