Physics in space–time with scale-dependent metrics

The aim of this Reply is to elucidate the difference between the fractal continuum models used in the preceding Comment and the models of fractal continuum flow which were put forward in our previous articles [Phys. Rev. E 85, 025302(R) (2012); 85, 056314 (2012)]. In this way, some drawbacks of the former models are highlighted. Specifically, inconsistencies in the definitions of the fractal derivative, the Jacobian of transformation, the displacement vector, and angular momentum are revealed. The proper forms of the Reynolds' transport theorem and angular momentum principle for the fractal continuum are reaffirmed in a more illustrative manner. Consequently, we emphasize that in the absence of any internal angular momentum, body couples, and couple stresses, the Cauchy stress tensor in the fractal continuum should be symmetric. Furthermore, we stress that the approach based on the Cartesian product measured and used in the preceding Comment cannot be employed to study the path-connected fractals, such as a flow in a fractally permeable medium. Thus, all statements of our previous works remain unchallenged.

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“…[1-7, [15][16][17][18][19][20][21]). Alternative models lead to very different solutions of the same problems for physical fractal media.…”

Section: Discussionmentioning

confidence: 99%

“…Such an approximation permits one to study physical phenomena in fractal media within a continuum framework (for reviews, see Refs. [1][2][3][4][5][6][7][11][12][13][14][15][16][17][18][19][20][21][22][23][24]). …”

Section: Fractal Continuum Approximation Of Fractal Mediummentioning

confidence: 99%

Reply to “Comment on ‘Hydrodynamics of fractal continuum flow’ and ‘Map of fluid flow in fractal porous medium into fractal continuum flow’ ”

Balankin

Elizarraraz

2013

Phys. Rev. E

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“…We note that an alternative way to interpret the form of the fractal derivative (5) could be possibly from a fractal metric [3]. Very recently Balankin [11] formally proposed a fractal metric in terms of the fractional integral (9) to define the fractal derivative (5). However, we want to clarify two issues:…”

Section: M(w)mentioning

confidence: 99%

Comment on “Hydrodynamics of fractal continuum flow” and “Map of fluid flow in fractal porous medium into fractal continuum flow”

Ostoja–Starzewski

2013

Phys. Rev. E

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In two recent papers [Phys. Rev. E 85, 025302(R) (2012) and Phys. Rev. E 85, 056314 (2012)], the authors proposed fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. While in general providing a certain advancement of continuum mechanics modeling of fractal media to fluid flows, some results and statements to previous works need clarification. We first show that the nonlocal character those authors alleged in our paper [Proc. R. Soc. A 465, 2521 (2009)] actually does not exist; instead, all those works are in the same general representation of derivative operators differing by specific forms of the line coefficient c 1 . Next, the claimed generalization of the volumetric coefficient c 3 is, in fact, equivalent to previously proposed product measures when considering together the separate decomposition of c 3 on each coordinate. Furthermore, the modified Jacobian proposed in the two commented papers does not relate the volume element between the current and initial configurations, which henceforth leads to a correction of the Reynolds' transport theorem. Finally, we point out that the asymmetry of the Cauchy stress tensor resulting from the conservation of the angular momentum must not be ignored; this aspect motivates a more complete formulation of fractal continuum models within a micropolar framework.

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“…The bicomplex BC is a generalization of complex which has close relations with Fractal geometry, Minkowski Space-Time, Maxwell's equations, Schrödinger equation, and Gaussian pulse wave. [1][2][3][4][5][6][7] The theory of bicomplex numbers is a matter of active research in recent times, and there are many further discussions in this direction. Banerjee studied bicomplex Fourier transform, 8 bicomplex Laplace transform, 9 bicomplex indefinite inner product modules, 10 and bicomplex Harmonic and Isotonic Oscillators.…”

Section: Introductionmentioning

confidence: 99%

The Schwarz lemma in bicomplex analysis

Dai

2023

Math Methods in App Sciences

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We investigate Schwarz lemma in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero-divisors. The bicomplex is a generalization of complex which has close relations with Fractal geometry, Minkowski Space-Time, Maxwell's equations, Schrödinger equation, and Gaussian pulse wave. In this paper, we first construct a type of bicomplex Möbius transformation and obtain some results: the mapping properties on bicomplex sphere and bicomplex ball, preserving the inverse points with respect to the bicomplex sphere B(0, 1). Then we obtain the Poisson integral formula in bicomplex setting, and by using the Poisson integral formula, we give the Schwarz lemma for bicomplex holomorphic functions in bicomplex setting.Finally, we shall give the Schwarz lemma and the Schwarz-Pick type lemma for holomorphic functions in bicomplex analysis. These results may give new energy for the development of quantum mechanics.

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Physics in space–time with scale-dependent metrics

Cited by 17 publications

References 73 publications

Reply to “Comment on ‘Hydrodynamics of fractal continuum flow’ and ‘Map of fluid flow in fractal porous medium into fractal continuum flow’ ”

Reply to “Comment on ‘Hydrodynamics of fractal continuum flow’ and ‘Map of fluid flow in fractal porous medium into fractal continuum flow’ ”

Comment on “Hydrodynamics of fractal continuum flow” and “Map of fluid flow in fractal porous medium into fractal continuum flow”

The Schwarz lemma in bicomplex analysis

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