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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