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Research Papers

# Forced Convection With Laminar Pulsating Counterflow in a Saturated Porous Channel

[+] Author and Article Information
D. A. Nield

Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland 1142, New Zealandd.nield@auckland.ac.nz

A. V. Kuznetsov

Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC 27695-7910avkuznet@eos.ncsu.edu

J. Heat Transfer 131(10), 101005 (Jul 29, 2009) (8 pages) doi:10.1115/1.3180810 History: Received April 01, 2008; Revised July 23, 2008; Published July 29, 2009

## Abstract

An analytical solution is obtained for forced convection in a parallel-plate channel occupied by a layered saturated porous medium with counterflow produced by pulsating pressure gradients. The case of asymmetrical constant heat-flux boundary conditions is considered, and the Brinkman model is employed for the porous medium. A perturbation approach is used to obtain analytical expressions for the velocity, temperature distribution, and transient Nusselt number for convection produced by an applied pressure gradient that fluctuates with small amplitude harmonically in time about a nonzero mean. It is shown that the fluctuating part of the Nusselt number alters in magnitude and phase as the dimensionless frequency increases. The magnitude increases from zero, goes through a peak, and then decreases to zero. The height of the peak depends on the values of various parameters. The phase (relative to that of the steady component) decreases as the frequency increases. The phase angle at very low frequency can be $π/2$ or $−π/2$ depending on the degree of asymmetry of the heating and the values of other parameters.

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## Figures

Figure 1

Definition sketch

Figure 2

Plots of (a) the modulus and (b) the argument as a fraction of π/2 of ν where Nu=Nu0(1+νεeiωt) as functions of the frequency ω for various values of βT for the case of the Darcy limit, ξ=0.5 and N2/N1=0.1. The values of Nu0 are 4.35, 7.01, and 30.97 for the βT values 0, 1, and 10, respectively.

Figure 3

Plots of (a) the modulus and (b) the argument as a fraction of π/2 of ν where Nu=Nu0(1+νεeiωt) as functions of the frequency ω for various values of βT for the case of the Darcy limit, ξ=0.5 and N2/N1=10. The values of Nu0 are 4.36, 1.69, and −22.32 for the βT values 0, 1, and 10, respectively.

Figure 4

Plots of (a) the modulus and (b) the argument as a fraction of π/2 of ν where Nu=Nu0(1+νεeiωt) as functions of the frequency ω for various values of βT for the case of the clear fluid limit, ξ=0.5 and M2/M1=0.1. The values of Nu0 are 3.62, 5.83, and 25.73 for the βT values 0, 1, and 10, respectively.

Figure 5

Plots of (a) the modulus and (b) the argument as a fraction of π/2 of ν where Nu=Nu0(1+νεeiωt) as functions of the frequency ω for various values of βT for the case of the clear fluid limit, ξ=0.5 and M2/M1=10. The values of Nu0 are 3.62, 1.41, and –18.50.

Figure 6

Plots of (a) the modulus and (b) the argument as a fraction of π/2 of ν where Nu=Nu0(1+νεeiωt) as functions of the frequency ω for some general cases ξ=0.5, β=0, M1=1, and M2=1, and for (N1,N2)=(1,0.1), (10,1), and (100,10). The values of Nu0 are 0.00203, 0.156, and 2.175, respectively.

Figure 7

As for Fig. 6, but now for (N1,N2)=(0.1,1), (1,10), and (10,100), the values of Nu0 are 0.00203, 0.156, and 2.175, respectively

Figure 8

As for Fig. 6, but now with βT=1, the values of Nu0 are 0.0934, 0.947, and 4.502, respectively

Figure 9

As for Fig. 7, but now with βT=1, the values of Nu0 are −0.0893, −0.635, and −0.152, respectively

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