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Technical Briefs

Effect of Second Order Velocity-Slip/Temperature-Jump on Basic Gaseous Fluctuating Micro-Flows

[+] Author and Article Information
M. A. Hamdan, Vladimir A. Hammoudeh

Department of Mechanical Engineering, University of Jordan, Amman 11942, Jordan

M. A. Al-Nimr

Department of Mechanical Engineering, J.U.S.T., Irbid 22110, Jordan

J. Fluids Eng 132(7), 074503 (Jul 22, 2010) (6 pages) doi:10.1115/1.4001970 History: Received June 26, 2009; Revised June 05, 2010; Published July 22, 2010; Online July 22, 2010

Abstract

In this work, the effect of the second-order term to the velocity-slip/temperature-jump boundary conditions on the solution of four cases in which the driving force is fluctuating harmonically was studied. The study aims to establish criteria that secure the use of the first order velocity-slip/temperature-jump model boundary conditions instead of the second-order ones. The four cases studied were the transient Couette flow, the pulsating Poiseuille flow, Stoke’s second problem, and the transient natural convection flow. It was found that at any given Kn number, increasing the driving force frequency, increases the difference between the first and second-order models. Assuming that a difference between the two models of over 5% is significant enough to justify the use of the more complex second-order model, the critical frequencies for the four different cases were found. For the cases for which the flow is induced by the fluctuating wall as in cases 1 and 3, we found that critical frequency at $Kn=0.1$ to be $ω=8.$ For the cases of flow driven by a fluctuating pressure gradient as in case 2, this frequency was found to be $ω=1$, at the same Kn number. In case 4, for the temperature-jump model, the critical frequency was found to be $ω=7$ and for the velocity-slip model the critical frequency at the same Kn number was found to be $ω=1.35$.

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Figures

Figure 1

Schematic diagram

Figure 2

Case 1—Normalized velocity-slip difference between the first- and second-order slips as a function of frequency for different Kn numbers

Figure 3

Case 1—Normalized velocity-slip difference between the first- and second-order slips as a function of frequency for τ=TP/4 and Kn=0.1

Figure 4

Case 2—Normalized velocity slip for the first- and second-order slips as a function of frequency for τ=TP/4 and Kn=0.1

Figure 5

Case 2—Normalized velocity-slip differences between first- and second-order slips as a function of frequency for τ=TP/4 and Kn=0.1

Figure 6

Case 3—Normalized differences of velocity slip as a function of the frequency for τ=TP/4 and Kn=0.1

Figure 7

Case 3—Normalized velocity-slip difference between first- and second-order slips as a function of frequency for τ=TP/4 and Kn=0.1

Figure 8

Case 4—Comparison between the slip velocities at the wall as a function of the frequency for τ=TP/4 and Kn=0.1

Figure 9

Case 4—Normalized temperature at the wall for the first- and second-order slips as a function of frequency for τ=TP/4 and Kn=0.1

Figure 10

Case 4—Normalized temperature difference at the wall for the first- and second-order slips as a function of frequency for τ=TP/4 and Kn=0.1

Figure 11

Case 4—Normalized velocity-slip difference at the wall between the first- and second-order slips at the wall as a function of frequency for different Kn numbers

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