# Free Surface Flow in a Microfluidic Corner and in an Unconfined Aquifer with Accretion: The Signorini and Saint-Venant Analytical Techniques Revisited

I give no sources, because it is indifferent to me whether what I have thought has already been thought before me by another.

Ludwig Wittgenstein, Tractatus Logico-Philosophicus

## Abstract

Steady, laminar, fully developed flows of a Newtonian fluid driven by a constant pressure gradient in (1) a curvilinear constant cross section triangle bounded by two straight no-slip segments and a circular meniscus and (2) a wedge bounded by two rays and an adjacent film bulging near the corner are studied analytically by the theory of holomorphic functions and numerically by finite elements. The analytical solution of the first problem is obtained by reducing the Poisson equation for the longitudinal flow velocity to the Laplace equation, conformal mapping of the corresponding transformed physical domain onto an auxiliary half-plane and solving there the Signorini mixed boundary value problem (BVP). The numerical solution is obtained by meshing the circular sector and solving a system of linear equations ensuing from the Poisson equation. Comparisons are made with known solutions for flows in a rectangular conduit, circular annulus and Philip’s circular duct with a no-shear sector. Problem (2) is treated by the Saint-Venant semi-inverse method: the free surface (quasi-meniscus) is reconstructed by a one-parametric family, which specifies a holomorphic function of the first derivative of the physical coordinate with respect to an auxiliary variable. The latter maps the flow domain onto a quarter of a unit disc where a mixed BVP for a characteristic function is solved by the Zhukovsky–Chaplygin method. Velocity distributions in a cross section perpendicular to the flow direction are obtained. It is shown that the change of the type of the boundary condition from no slip to perfect slip (along the meniscus) causes a dramatic increase of the total flow rate (conductance). For example, the classical Saint-Venant formulae for a sector, with all three boundaries being no-slip segments, predict up to four times smaller rate as compared to a free surface meniscus. Mathematically equivalent problems of unconfined flows in aquifers recharged by a constant-intensity infiltration are also addressed.

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### Applying the immersed boundaries to the incompressible fluid flows using inverse source problem

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## Notes

1. Moffatt and Duffy (1980), a century after Saint-Venant, re-obtained his full solution in an approximate manner. This example illustrates the Wittgenstein approach (see the epigraph).

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## Acknowledgments

The work has been supported by the SQU under Grant IG-AGR/SWAE/14/02 and the Russian Foundation for Basic Research under Grants 15-01-06029-a, 15-41-02433 (RFBR-Tatarstan). Helpful comments by three anonymous referees are highly appreciated.

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Kacimov, A.R., Maklakov, D.V., Kayumov, I.R. et al. Free Surface Flow in a Microfluidic Corner and in an Unconfined Aquifer with Accretion: The Signorini and Saint-Venant Analytical Techniques Revisited. Transp Porous Med 116, 115–142 (2017). https://doi.org/10.1007/s11242-016-0767-y