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Research Papers: Fundamental Issues and Canonical Flows

# Modified Time-Dependent Penetration Length and Inlet Pressure Field in Rectangular and Cylindrical Channel Flows Driven by Non-Mechanical Forces

[+] Author and Article Information
Martin Ndi Azese

Mechanical Engineering Department, Texas Tech University, 7th Street and Boston Avenue, Lubbock, TX 79409 e-mail: martin.azese@ttu.edu

J. Fluids Eng 133(11), 111205 (Nov 11, 2011) (12 pages) doi:10.1115/1.4005135 History: Received April 22, 2011; Accepted September 18, 2011; Published November 11, 2011; Online November 11, 2011

## Abstract

In this paper, we derive the governing equation for the time dependent penetration length of a fluid column in rectangular and cylindrical channels under the action of nonmechanical forces like capillary or electro-osmotic force. For this purpose, first we obtain the velocity profile for unidirectional unsteady flow by satisfying momentum equation in differential form. Then, we relate the rate of change of penetration length with volume flux to obtain the governing equation of the penetration length. As the velocity profile is exact, the analysis is devoid of any mathematical error. As a result, the theoretical results are valid irrespective of the Reynolds number of the system as long as the flow inside the cylindrical or rectangular conduit is laminar. We then use the new expressions of velocity fields of respective conduits to derive a more accurate expression of the entrance pressure by using a hemispherical model for the control volume for finite aspect ratio. As these channels are very common, our governing equations for penetration length will have a wide range of applicability. These applications especially include creeping flow in micro fluidic domain for which we have a simplified version of the derived equation.

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## Figures

Figure 1

Rectangular channel of dimensions −a ≤ x ≤ a, −b ≤ y ≤ b, and z ≥ 0

Figure 2

Cylindrical channel of dimensions −R ≤ r ≤ R and z ≥ 0

Figure 3

Schematic representation of pressure field in both rectangular and cylindrical channel where (a) and (b) are, respectively, the front view and the side view of rectangular duct (c) is the cylindrical conduit

Figure 4

Schematic representation of the hemispherical control volume

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