On nonlocal fractal laminar steady and unsteady flows

El-Nabulsi, Rami Ahmad

doi:10.1007/s00707-020-02929-8

Cited by 34 publications

(4 citation statements)

References 67 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Carpinteri column [23] is a simple example of an anisotropic product-like fractal porous structure with a microstructure modelled by the Sierpi ński carpet in the cross-section and a Cantor set along the longitudinal axis. Natural examples of product-like fractals are found in [10][11][12][13] with applications in molecular physics, superconductivity, tumour growth and fluid flows. Also, the geometry and mechanics of planetary rings, like those of Saturn, can be handled by fractal products in polar coordinate systems [24].…”

Section: Scaling In Product Fractals and Basic Relationsmentioning

confidence: 99%

See 1 more Smart Citation

Micropolar mechanics of product fractal media

Ostoja–Starzewski

2022

Proc. R. Soc. A.

View full text Add to dashboard Cite

Motivated by the abundance of fractals in the natural world, this paper further develops continuum-type models for product-like fractals. The theory is based on a version of the non-integer dimensional space approach, in which global balance laws written for fractal media are expressed in terms of conventional (integer-order) integrals. Key relations of calculus for finite strain kinematics of fractal media are obtained, especially clarifying the fractal Jacobian and the fractal Reynolds transport theorem. The local forms are then written in terms of partial differential equations with derivatives of integer order. Hence, fractal versions of local continuity, linear and angular momenta and energy balance are derived. The angular momentum balance implies that the approximating continuum is micropolar rather than classical. Accordingly, the Cauchy postulate, lemma and theorem for Cauchy force-stress and couple-stress are re-formulated. The corresponding partial differential equations for finite as well as infinitesimal elasticity are given explicitly in both the displacement and stress formulations. The invariance of the stress field in planar fractal elastic media is shown to hold just like the one in planar micropolar elasticity.

show abstract

Section: Scaling In Product Fractals and Basic Relationsmentioning

confidence: 99%

“…Applications range over solid mechanics of fractal media (elasticity, beams, elastic-brittle fracture, viscoelasticity, among others) as well as fluid mechanics, molecular dynamics and quantum mechanics [e.g. 4,[10][11][12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Micropolar mechanics of product fractal media

Ostoja–Starzewski

2022

Proc. R. Soc. A.

View full text Add to dashboard Cite

show abstract

“…In fact, this new concept was introduced recently by Li and Ostoja-Starzewski in order to describe dynamics in anisotropic and continuum media [42][43][44] and was motivated by Tarasov fractal calculus arguments [45,46]. It is considered a successful approach, which has proved to have several successful implications in sciences and engineering at different scales [47][48][49][50][51][52][53][54][55][56][57][58]. In the Li and Ostoja-Starzewski approach (LOSA henceforth), the dynamic equations of motion hold mathematical forms involving integer-order integrals, whereas their local forms are expressed through partial differential equations with integerorder derivatives except that they contain coefficients involving fractal dimensions.…”

Section: Introductionmentioning

confidence: 99%

Emergence of lump-like solitonic waves in Heimburg–Jackson biomembranes and nerves fractal model

El-Nabulsi

2022

J. R. Soc. Interface.

View full text Add to dashboard Cite

The aim of this study is to extend the soliton propagation model in biomembranes and nerves constructed by Heimburg and Jackson for the case of fractal dimensions. Our analyses are based on the product-like fractal measure concept introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media and electromagnetic fields. The mathematical model presented in the paper is formulated for only a part of a single nerve cell (an axon). The analytical and numerical envelop soliton of this equation are reported. The results obtained prove the emergence of lump-type solitonic waves in nerves and biomembranes. In particular, these waves decay algebraically to the background wave in space direction. This scenario is viewed as a particular class of rational localized waves which are solutions of the integrable Ishimori I equation and the (2 + 1) Kadomtsev–Petviashvili I equation. The effects of fractal dimensions are discussed and were found to be significant to some extents.

show abstract

“…In the present study, we would like to construct an anisotropic kinetic theory in fractal dimensions starting from a completely dissimilar fractal perspective known by the concept of 'product-like fractal geometry' which was introduced by Li and Ostoja-Starzewski in [75] and entitled Li and Ostoja-Starzewski approach (LOSA). This approach describes fruitfully the nonlinear dynamics and the physics in anisotropic fractal-dimensional media [76][77][78][79][80][81][82] and has a series of motivating implications in various fields of sciences and engineering [76,[83][84][85]. Given that LOSA is productively used in anisotropic media, it will be of interest to explore its implications in kinetic biology.…”

Section: Introductionmentioning

confidence: 99%

Fractal Pennes and Cattaneo–Vernotte bioheat equations from product-like fractal geometry and their implications on cells in the presence of tumour growth

El-Nabulsi

2021

J. R. Soc. Interface.

Self Cite

View full text Add to dashboard Cite

In this study, the Pennes and Cattaneo–Vernotte bioheat transfer equations in the presence of fractal spatial dimensions are derived based on the product-like fractal geometry. This approach was introduced recently, by Li and Ostoja-Starzewski, in order to explore dynamical properties of anisotropic media. The theory is characterized by a modified gradient operator which depends on two parameters: R which represents the radius of the tumour and R 0 which represents the radius of the spherical living tissue. Both the steady and unsteady states for each fractal bioheat equation were obtained and their implications on living cells in the presence of growth of a large tumour were analysed. Assuming a specific heating/cooling by a constant heat flux equivalent to the metabolic heat generation in the tissue, it was observed that the solutions of the fractal bioheat equations are robustly affected by fractal dimensions, the radius of the tumour growth and the dimensions of the living cell tissue. The ranges of both the fractal dimensions and temperature were obtained, analysed and compared with recent studies. This study confirms the importance of fractals in medicine.

show abstract

On nonlocal fractal laminar steady and unsteady flows

Cited by 34 publications

References 67 publications

Micropolar mechanics of product fractal media

Micropolar mechanics of product fractal media

Emergence of lump-like solitonic waves in Heimburg–Jackson biomembranes and nerves fractal model

Fractal Pennes and Cattaneo–Vernotte bioheat equations from product-like fractal geometry and their implications on cells in the presence of tumour growth

Contact Info

Product

Resources

About