Fractional vector calculus and fluid mechanics

AbstractBasic fluid mechanics equations are studied and revised under the prism of fractional continuum mechanics (FCM), a very promising research field that satisfies both experimental and theoretical demands. The geometry of the fractional differential has been clarified corrected and the geometry of the fractional tangent spaces of a manifold has been studied in Lazopoulos and Lazopoulos (Lazopoulos KA, Lazopoulos AK. Progr. Fract. Differ. Appl. 2016, 2, 85–104), providing the bases of the missing fractional differential geometry. Therefore, a lot can be contributed to fractional hydrodynamics: the basic fractional fluid equations (Navier Stokes, Euler and Bernoulli) are derived and fractional Darcy’s flow in porous media is studied.

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Effective models of the Navier–Stokes flow in porous media with a thin fissure

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2011.09.005 ◽

2012 ◽

Vol 387 (2) ◽

pp. 542-555 ◽

Cited By ~ 4

Author(s):

Hongxing Zhao ◽

Zheng-an Yao

Keyword(s):

Porous Media ◽

Stokes Flow ◽

Flow In Porous Media ◽

Navier Stokes ◽

Effective Models

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Numerical Investigation of the “Poor Man’s Navier–Stokes Equations” with Darcy and Forchheimer Terms

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416500863 ◽

2016 ◽

Vol 26 (05) ◽

pp. 1650086

Author(s):

Tingting Tang ◽

Zhiyong Li ◽

J. M. McDonough ◽

P. D. Hislop

Keyword(s):

Porous Media ◽

Numerical Investigation ◽

Stokes Equations ◽

Discrete Dynamical System ◽

Flow In Porous Media ◽

Phase Portraits ◽

Navier Stokes ◽

Flow Through Porous Media ◽

Navier Stokes Equations ◽

Flow Through

In this paper, a discrete dynamical system (DDS) is derived from the generalized Navier–Stokes equations for incompressible flow in porous media via a Galerkin procedure. The main difference from the previously studied poor man’s Navier–Stokes equations is the addition of forcing terms accounting for linear and nonlinear drag forces of the medium — Darcy and Forchheimer terms. A detailed numerical investigation focusing on the bifurcation parameters due to these additional terms is provided in the form of regime maps, time series, power spectra, phase portraits and basins of attraction, which indicate system behaviors in agreement with expected physical fluid flow through porous media. As concluded from the previous studies, this DDS can be employed in subgrid-scale models of synthetic-velocity form for large-eddy simulation of turbulent flow through porous media.

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Buoyancy-induced flow in porous media generated near a drilled oil well. Part 1. The accumulation of filtrate at a horizontal impermeable boundary

Journal of Fluid Mechanics ◽

10.1017/s0022112093002137 ◽

1993 ◽

Vol 254 ◽

pp. 283-311 ◽

Cited By ~ 16

Author(s):

E. B. Dussan V. ◽

François M. Auzerais

Keyword(s):

Porous Media ◽

Fluid Mechanics ◽

Drilling Fluid ◽

Flow In Porous Media ◽

Porous Solids ◽

Oil Well ◽

Great Success ◽

Ct Scanner ◽

X Ray ◽

Impermeable Barrier

A substantial amount of drilling fluid can invade a permeable bed during the drilling of an oil well. The presence of this fluid, often referred to as filtrate, can greatly influence the performance of instruments lowered into the wellbore for the purpose of locating these permeable beds. The invaded filtrate can also substantially alter the physical properties of the porous rock. For these reasons, it is of great interest to known where the filtrate goes upon entering the bed. The objective of this study is to quantify the influence of the difference in density between the filtrate and the naturally occurring formation fluid on the shape of the filtrate front as the filtrate invades the formation. This type of phenomenon is often referred to as buoyancy or gravity segregation. In this study, Part 1, we determine the behaviour of the filtrate as it accumulates (and spreads out) at a horizontal impermeable barrier within the formation. This is a combined theoretical and experimental study in which an X-ray CT scanner is extensively used to determine the appropriateness and limitations of the simplifying assumptions used in the theory. In Part 2, the flow of the invading filtrate within the entire bed will be presented. The problem addressed in Part 1 may be viewed from the broader, more fundamental, perspective, as a well-defined model fluid mechanics problem for flow in porous media. One fundamental issue infrequently addressed concerns the consequence on the dynamics of the fluids of heterogeneities, always present to some degree, in consolidated porous solids. The X-ray CT scanner enables the assessment of the appropriateness of modelling such porous solids as spatially homogeneous, a very popular assumption. This study also addresses the limitation of the small-slope approximation when a fluid–fluid interface occurs in a porous solid, an approximation which has enjoyed great success in free-surface fluid mechanics problems when no porous media is present.

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