Range of effectiveness of hydraulic diameter model as an analytical solution for rectangular microchannels

Spontaneous capillary flow in rectangular microfluidic channels is employed in microfluidic devices for various applications. The exact solution for flow in a rectangular cross-sectional channel has a complex point that contains an infinite sum term. The flow depends on the depth-width ratio of the rectangular channel's cross-section, <i>ε</i>. In previous studies, several approximations from exact solutions were useful for <i>ε</i> values smaller than 1 or 2. In this study, we propose a conversion equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>J</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> that turns the hydraulic-diameter (HD) model into a rectangular-channel (RC) model. Experiments: Rectangular type flow experiments with ethanol solution were conducted to confirm the difference between the RC and HD models when the analysis was performed based on each model. Findings: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>J</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> depends on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ε</mml:mi></mml:mrow> </mml:math> . At <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ε</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.441</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2.27</mml:mn></mml:mrow> </mml:math> , the two models coincide. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>0.247</mml:mn> <mml:mo><</mml:mo> <mml:mi>ε</mml:mi> <mml:mo><</mml:mo> <mml:mn>4.04</mml:mn></mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>J</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> deviates by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>±</mml:mo> <mml:mn>12.5</mml:mn> <mml:mspace></mml:mspace> <mml:mo>%</mml:mo></mml:mrow> </mml:math> .