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Fractal-induced flow dynamics: Viscous flow around Mandelbrot and Julia sets

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  • Mohammad, Mutaz
  • Trounev, Alexander

Abstract

This study explores viscous flow dynamics around fractal geometries, specifically the Mandelbrot and Julia sets, using finite element simulations. We analyze the impact of fractal roughness on flow characteristics across different Reynolds numbers Re0. At low Reynolds numbers, the influence of fractal roughness is minimal. However, as the Reynolds number increases, Kármán vortex shedding emerges, exhibiting distinct fractal-dependent patterns at specific thresholds (Re0=342.87 for the Mandelbrot set, Re0=289.553 for the San Marco fractal, and Re0=178.25,356.5,891.248 for the Siegel disk fractal). At sufficiently high Reynolds numbers, chaotic flow structures detach from the fractal boundary, destabilizing the boundary layer. To better capture the transition from laminar to turbulent regimes, we propose an alternative modeling approach using fractional derivatives in time. These findings provide new insights into flow behavior over complex geometries, with implications for turbulence modeling and engineering applications.

Suggested Citation

  • Mohammad, Mutaz & Trounev, Alexander, 2025. "Fractal-induced flow dynamics: Viscous flow around Mandelbrot and Julia sets," Chaos, Solitons & Fractals, Elsevier, vol. 199(P1).
    • Handle: RePEc:eee:chsofr:v:199:y:2025:i:p1:s0960077925006320
    DOI: 10.1016/j.chaos.2025.116619
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    References listed on IDEAS

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    1. Roberto Garrappa, 2018. "Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial," Mathematics, MDPI, vol. 6(2), pages 1-23, January.
    2. Shuai-Jia Kou & Chun-Hui He & Xing-Chen Men & Ji-Huan He, 2022. "Fractal Boundary Layer And Its Basic Properties," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 30(09), pages 1-9, December.
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    1. Mohammad, Mutaz & Trounev, Alexander, 2025. "Numerical simulation and nonlinear dynamics in rotating magnetoconvection: Chaos, Attractors, and Stability Transitions," Chaos, Solitons & Fractals, Elsevier, vol. 201(P2).

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