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The Fractal Calculus for Fractal Materials

Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran
Physics and Accelerators Research School, Nuclear Science and Technology Research Institute, P.O. Box 14395-836, Tehran, Iran
Author to whom correspondence should be addressed.
Fractal Fract 2019, 3(1), 8;
Received: 10 February 2019 / Revised: 2 March 2019 / Accepted: 5 March 2019 / Published: 6 March 2019
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The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefore, fractal structures and fractal reactors find importance. Using F α -fractal calculus, in this paper, we derive exact F α -differential forms of an ideal gas. Depending on the dimensionality of space, we should first obtain the integral staircase function and mass function of our geometry. When gases expand inside the fractal structure because of changes from the i + 1 iteration to the i iteration, in fact, we are faced with fluid mixing inside our fractal structure, which can be described by physical quantities P, V, and T. Finally, for the ideal gas equation, we calculate volume expansivity and isothermal compressibility. View Full-Text
Keywords: fractal; fractal dimension; fractal calculus; staircase function fractal; fractal dimension; fractal calculus; staircase function

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Jafari, F.K.; Asgari, M.S.; Pishkoo, A. The Fractal Calculus for Fractal Materials. Fractal Fract 2019, 3, 8.

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