### 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.

Original language | English |
---|---|

Pages (from-to) | 115-142 |

Number of pages | 28 |

Journal | Transport in Porous Media |

Volume | 116 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2017 |

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### Keywords

- Meniscus
- Poisson equation
- Signorini formula
- Viscous film
- Zhukovsky–Chaplygin method
- Zunker’s pendular water slug

### ASJC Scopus subject areas

- Catalysis
- Chemical Engineering(all)

### Cite this

*Transport in Porous Media*,

*116*(1), 115-142. https://doi.org/10.1007/s11242-016-0767-y

**Free Surface Flow in a Microfluidic Corner and in an Unconfined Aquifer with Accretion : The Signorini and Saint-Venant Analytical Techniques Revisited.** / Kacimov, A. R.; Maklakov, D. V.; Kayumov, I. R.; Al-Futaisi, A.

Research output: Contribution to journal › Article

*Transport in Porous Media*, vol. 116, no. 1, pp. 115-142. https://doi.org/10.1007/s11242-016-0767-y

}

TY - JOUR

T1 - Free Surface Flow in a Microfluidic Corner and in an Unconfined Aquifer with Accretion

T2 - The Signorini and Saint-Venant Analytical Techniques Revisited

AU - Kacimov, A. R.

AU - Maklakov, D. V.

AU - Kayumov, I. R.

AU - Al-Futaisi, A.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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.

AB - 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.

KW - Meniscus

KW - Poisson equation

KW - Signorini formula

KW - Viscous film

KW - Zhukovsky–Chaplygin method

KW - Zunker’s pendular water slug

UR - http://www.scopus.com/inward/record.url?scp=84989169861&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84989169861&partnerID=8YFLogxK

U2 - 10.1007/s11242-016-0767-y

DO - 10.1007/s11242-016-0767-y

M3 - Article

VL - 116

SP - 115

EP - 142

JO - Transport in Porous Media

JF - Transport in Porous Media

SN - 0169-3913

IS - 1

ER -