INTERFACE EQUATION AND VISCOSITY CONTRAST IN HELE-SHAW FLOW

1992 ◽  
Vol 06 (10) ◽  
pp. 1647-1656 ◽  
Author(s):  
J. CASADEMUNT ◽  
DAVID JASNOW ◽  
A. HERNÁNDEZ-MACHADO
Keyword(s):  
Computational Algorithm ◽  
Normal Component ◽  
Interface Velocity ◽  
Vortex Sheet ◽  
Viscous Fluids ◽  
Normal Velocity ◽  
Channel Geometry ◽  
Viscosity Contrast ◽  
Zero Viscosity ◽  
Hele Shaw Cell

We derive an integro-differential equation for the evolution of the interface separating two immiscible viscous fluids in a Hele-Shaw cell with a channel geometry, for arbitrary viscosity contrast. Our equation differs from a previous one obtained by a vortex-sheet formulation of the problem, in that the normal component of the interface velocity is formally decoupled from the gauge-dependent tangential part. The result is thus a closed integral equation for the normal velocity. We briefly comment on the advantages of such a formulation and implement an alternative computational algorithm based on it. Preliminary numerical results confirm a highly inefficient finger competition in the zero viscosity contrast limit.

scholarly journals The vortex-entrainment sheet in an inviscid fluid: theory and separation at a sharp edge

2019 ◽  
Vol 866 ◽  
pp. 660-688 ◽  
Author(s):  
A. C. DeVoria ◽  
K. Mohseni
Keyword(s):  
Mass Density ◽  
Normal Component ◽  
Vortex Sheet ◽  
Three Dimensions ◽  
Sharp Edge ◽  
Pressure Jump ◽  
Inviscid Fluid ◽  
Fluid Theory ◽  
Viscous Boundary ◽  
Singularity Removal

In this paper a model for viscous boundary and shear layers in three dimensions is introduced and termed a vortex-entrainment sheet. The vorticity in the layer is accounted for by a conventional vortex sheet. The mass and momentum in the layer are represented by a two-dimensional surface having its own internal tangential flow. Namely, the sheet has a mass density per-unit-area making it dynamically distinct from the surrounding outer fluid and allowing the sheet to support a pressure jump. The mechanism of entrainment is represented by a discontinuity in the normal component of the velocity across the sheet. The velocity field induced by the vortex-entrainment sheet is given by a generalized Birkhoff–Rott equation with a complex sheet strength. The model was applied to the case of separation at a sharp edge. No supplementary Kutta condition in the form of a singularity removal is required as the flow remains bounded through an appropriate balance of normal momentum with the pressure jump across the sheet. A pressure jump at the edge results in the generation of new vorticity. The shedding angle is dictated by the normal impulse of the intrinsic flow inside the bound sheets as they merge to form the free sheet. When there is zero entrainment everywhere the model reduces to the conventional vortex sheet with no mass. Consequently, the pressure jump must be zero and the shedding angle must be tangential so that the sheet simply convects off the wedge face. Lastly, the vortex-entrainment sheet model is demonstrated on several example problems.

Inviscid separated flow over a non-slender delta wing

1995 ◽  
Vol 305 ◽  
pp. 307-345 ◽  
Author(s):  
D. W. Moore ◽  
D. I. Pullin
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Delta Wing ◽  
Nonlinear Eigenvalue Problem ◽  
Vortex Sheet ◽  
Slender Body ◽  
Normal Velocity ◽  
Slender Body Theory ◽  
Closed Set ◽  
Body Theory

We consider inviscid incompressible flow about an infinite non-slender flat delta wing with leading-edge separation modeled by symmetrical conical vortex sheets. A similarity solution for the three dimensional steady velocity potential Φ is sought with boundary conditions to be satisfied on the line which is the intersection of the wing sheet surface with the surface of the unit sphere. A numerical approach is developed based on the construction of a special boundary element or ‘winglet’ which is effectively a Green function for the projection of ∇2Φ = 0 onto the spherical surface under the similarity ansatz. When the wing semi-apex angle γo is fixed satisfaction of the boundary conditions of zero normal velocity on the wing and zero normal velocity and pressure continuity across the vortex sheet then leads to a nonlinear eigenvalue problem. A method of ensuring a condition of zero lateral force on a lumped model of the inner part of the rolled-up vortex sheet gives a closed set of a equations which is solved numerically by Newton's method. We present and discuss the properties of solutions for γ0 in the range 1.30 < γ <89.50. The dependencies of these solutions on γ0 differs qualitatively from predictions of slender-body theory. In particular the velocity field is in general not conical and the similarity exponent must be calculated as part of the global eigenvalue problem. This exponent, together with the detailed flow field including the position and structure of the separated vortx sheet, depend only on γ0. In the limit of small γ0, a comparison with slender-body theory is made on the basis of an effective angle of incidence.

Magnetic field driven micro-convection in the Hele-Shaw cell

2013 ◽  
Vol 714 ◽  
pp. 612-633 ◽  
Author(s):  
K. Ērglis ◽  
A. Tatulcenkov ◽  
G. Kitenbergs ◽  
O. Petrichenko ◽  
F. G. Ergin ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Magnetic Fluid ◽  
Interface Velocity ◽  
Threshold Value ◽  
Homogeneous Magnetic Field ◽  
The Self ◽  
Initial Stage ◽  
Image Velocimetry ◽  
Ponderomotive Forces ◽  
Hele Shaw Cell

AbstractMicro-convection caused by ponderomotive forces of the self-magnetic field of a magnetic fluid in the Hele-Shaw cell under the action of a vertical homogeneous magnetic field is studied both experimentally and numerically. It is shown that a non-potential magnetic force at magnetic Rayleigh numbers greater than the critical value causes fingering at the interface between the miscible magnetic and non-magnetic fluids. The threshold value of the magnetic Rayleigh number depends on the smearing of the interface between fluids. Fingering with its subsequent decay due to diffusion of particles significantly increases the mixing at the interface. Velocity and vorticity fields at fingering are determined by particle image velocimetry measurements and qualitatively correspond well to the results of numerical simulations of the micro-convection in the Hele-Shaw cell carried out in the Darcy approximation, which account for ponderomotive forces of the self-magnetic field of the magnetic fluid. Gravity plays an important role at the initial stage of the fingering observed in the experiments.

scholarly journals Low viscosity contrast fingering in a rotating Hele-Shaw cell

2004 ◽  
Vol 16 (4) ◽  
pp. 908-924 ◽  
Author(s):  
E. Alvarez-Lacalle ◽  
J. Ortı́n ◽  
J. Casademunt
Keyword(s):  
Low Viscosity ◽  
Viscosity Contrast ◽  
Hele Shaw Cell

scholarly journals Geometry and dynamics of vortex sheets in 3 dimension

2002 ◽  
pp. 55-76 ◽  
Author(s):  
D.A. Burton ◽  
R.W. Tucker
Keyword(s):  
Fluid Velocity ◽  
Normal Component ◽  
Vortex Sheet ◽  
The Self ◽  
Vortex Filament ◽  
Vortex Sheets ◽  
Filament Dynamics ◽  
De Rham ◽  
Induced Velocity

We consider the properties and dynamics of vortex sheets from a geometrical, coordinate-free, perspective. Distribution-valued forms (de Rham currents) are used to represent the fluid velocity and vorticity due to the vortex sheets. The smooth velocities on either side of the sheets are solved in terms of the sheet strengths using the language of double forms. The classical results regarding the continuity of the sheet normal component of the velocity and the conservation of vorticity are exposed in this setting. The formalism is then applied to the case of the self-induced velocity of an isolated vortex sheet. We develop a simplified expression for the sheet velocity in terms of representative curves. Its relevance to the classical Localized Induction Approximation (LIA) to vortex filament dynamics is discussed. .

scholarly journals Doubly periodic array of bubbles in a Hele-Shaw cell

2010 ◽  
Vol 467 (2126) ◽  
pp. 346-360 ◽  
Author(s):  
Antônio M. P. Silva ◽  
Giovani L. Vasconcelos
Keyword(s):  
Surface Tension ◽  
Integral Form ◽  
Periodic Array ◽  
Channel Geometry ◽  
Flow Domain ◽  
Mapping Techniques ◽  
Doubly Periodic Array ◽  
Simply Connected Region ◽  
Simply Connected ◽  
Hele Shaw Cell

Exact solutions are presented for a doubly periodic array of steadily moving bubbles in a Hele-Shaw cell when surface tension is neglected. It is assumed that the bubbles are either symmetrical with respect to the channel centreline or have fore-and-aft symmetry, or both, so that the relevant flow domain can be reduced to a simply connected region. By using conformal-mapping techniques, a general solution with any number of bubbles per unit cell is obtained in integral form. Several examples are given, including solutions for multi-file arrays of bubbles in the channel geometry and doubly periodic solutions in an unbounded cell.

Viscous fingering with a single fluid

2005 ◽  
Vol 83 (5) ◽  
pp. 551-564 ◽  
Author(s):  
Kristi E Holloway ◽  
John R de Bruyn
Keyword(s):  
Numerical Simulations ◽  
Viscous Fingering ◽  
High Flow ◽  
Cell Width ◽  
Flow Rates ◽  
Viscosity Contrast ◽  
Fingering Instability ◽  
Hele Shaw Cell ◽  
Initial Viscosity

We study fingering that occurs when hot glycerine displaces cooler, more viscous glycerine in a radial Hele-Shaw cell. We find that fingering occurs for a sufficiently large initial viscosity contrast and for sufficiently high flow rates of the displacing fluid. The wavelength of the fingering instability is proportional to the cell width for thin cells, but the ratio of wavelength to cell width decreases for our thickest cell. Similar fingering is seen in numerical simulations of this system.PACS Nos.: 47.54.+r, 68.15.+e, 47.20.–k

scholarly journals Extension of Kelvin’s minimum energy theorem to incompressible fluid domains with open regions

2017 ◽  
Vol 825 ◽  
pp. 208-212
Author(s):  
Tony Saad ◽  
Joseph Majdalani
Keyword(s):  
Incompressible Fluid ◽  
Rotational Velocity ◽  
Minimum Energy ◽  
Normal Component ◽  
Normal Velocity ◽  
Surface Integral ◽  
Potential Motion ◽  
The Difference ◽  
Simply Connected Region ◽  
Classic Theorem

Kelvin’s minimum energy theorem predicts that the irrotational motion of a homogeneously incompressible fluid in a simply connected region will carry less kinetic energy than any other profile that shares the same normal velocity conditions on the domain’s boundary. In this work, Kelvin’s analysis is extended to regions with boundaries on which the normal velocity requirements are relaxed. Given the ubiquity of practical configurations in which such boundaries exist, the question of whether Kelvin’s theorem continues to hold is one of significant interest. In reconstructing Kelvin’s proof, we find it useful to define a net rotational velocity as the difference between the generally rotational flow and the corresponding potential motion. In Kelvin’s classic theorem, the normal component of the net rotational velocity at all domain boundaries is zero. In contrast, the present analysis derives a sufficient condition for ensuring the validity of Kelvin’s theorem in a domain where the normal component of net rotational velocity at some or all of the boundaries is not zero. The corresponding criterion requires the evaluation of a simple surface integral over the boundary.

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