Generalized formulation for estimating pressure drop in fully-developed laminar flow in singly and doubly connected channels of non-circular cross-sections

摘要

Fully developed internal laminar flow through both, singly- and doubly-connected ducts of arbitrary cross-sections, is investigated using a two-dimensional semi-analytical technique in which the condition at the inner/outer arbitrarily defined peripheral boundary of the ducts is matched by a collocation technique. This boundary collocation technique has been applied and validated for (a) singly-connected ducts of standard cross-sectional shapes such as square, circular, etc., (b) a variety of complicated singly-connected duct cross-sectional geometries and, (c) two principal variations of the doubly-connected ducts (ⅰ) annular duct having an arbitrary outside perimeter while the internal core perimeter is circular, and (ⅱ) annular duct having a circular outside perimeter while the internal core perimeter is arbitrarily shaped. The proposed model is only a function of geometrical parameters of the duct cross-section. Fluid velocity contours, shear stress distribution along the duct walls and the friction factor (Poiseuille number) has been estimated with this technique and validated against existing solutions for several standard cross-sections and found to be in excellent agreement. Several arbitrary shapes have been considered for analysis. As most micro-fabrication techniques normally do not produce mathematically well-defined or ideal geometries, an experimental test case is also presented wherein, the application of the present methodology in accurately predicting the fully-developed velocity profile, and therefore the net pressure drop, in a microchannel etched on a silicon substrate, is discussed. This simple and universally applicable semi-analytical technique substantially improves the overall predictions for real-time geometries. The critical nuances of successful application of this technique are outlined.