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

Abstract

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.

A continuum framework for mechanics of fractal materials I: from fractional space to continuum with fractal metric

“…(23) and (24), the kinematics of these models is quite different due to the difference of metrics (see Ref. [44]). (20), metric (19), (21), and measure (22).…”

“…whereas the deformations of fractal continua should satisfy the laws of momentum conservation allied with the Newton's first and second laws [44].…”

“…However, some fundamental questions regarding to the definition of fractal continuum still remain under debate (see Refs. [43,44]).…”