A better nondimensionalization scheme for slender laminar flows: The Laplacian operator scaling method

Abstract

A scaling of the two-dimensional Laplacian operator is demonstrated for certain solutions (at least) to Poisson’s equation. It succeeds by treating the operator as a single geometric scale entity. The belated and rather subtle method provides an efficient assessment of the geometrical dependence of the problem and is preferred when practicable to the hydraulic diameter or term-by-term scaling for slender fully developed laminar flows. The improved accuracy further reduces the reliance of problems on widely varying numerical data or cumbersome theoretical forms and improves the prospects of exact or approximate theoretical analysis. Simple example problems are briefly described that demonstrate the application and potential of the method.

Capillary imbibition in a square tube

“…In this paper, we will consider the problem of liquid imbibition in a square tube, accounting for the coupling explicitly between the bulk part and the finger part. Similar system has been studied by Weislogel [24] using the Laplacian scaling method [25]. We shall show that both bulk part and the finger part grows in time obeying the Lucas-Washburn law, but differ in the numerical coefficients.…”

Suspended microflows between vertical parallel walls

“…Dynamics of SCFs has been documented in the literature for many shapes of channels to the exception of suspended channels (Weislogel et al 2008;Rye et al 1996;Ouali et al 2013). Let us first consider the forces acting on the fluid.…”

Compound capillary rise

“…where A q is the quadrant section area (see shaded regions in figure 3), and F n ≈ 1 ± 0.1 (Weislogel et al 2008). An exact analytic or numeric value for F n (or F i ) may always be employed if higher accuracy is demanded, but in many cases (2.4) is suitable for efficient design and analysis.…”