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Related Concept Videos

Dimensionless Groups in Fluid Mechanics01:15

Dimensionless Groups in Fluid Mechanics

Dimensionless groups in fluid mechanics provide simplified ratios that help analyze fluid behavior without relying on specific units. The Reynolds number (Re), which represents the ratio of inertial to viscous forces, distinguishes between laminar and turbulent flows, making it essential in the design of pipelines and aerodynamic surfaces. The Froude number (Fr), the ratio of inertial to gravitational forces, is particularly useful in predicting wave formation and hydraulic jumps in...
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The thermodynamic processes can be classified into reversible and irreversible processes. The processes that can be restored to their initial state are called reversible processes. It is only possible if the process is in quasi-static equilibrium, i.e., it takes place in infinitesimally small steps, and the system remains at equilibrium However, these are ideal processes and do not occur naturally. An ideal system undergoing a reversible process is always in thermodynamic equilibrium within...
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Reynolds Transport Theorem01:24

Reynolds Transport Theorem

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Analyzing Mixing Inhomogeneity in a Microfluidic Device by Microscale Schlieren Technique
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Two-scale renormalization-group classification of diffusive processes.

Daniel O'Malley1, John H Cushman

  • 1Department of Earth, Atmospheric, and Planetary Sciences, Purdue University, West Lafayette, Indiana 47906, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
PubMed
Summary
This summary is machine-generated.

Renormalization-group operators classify stochastic processes across different time scales. This robust method, applicable to complex systems like diffusion, aids in understanding long-term and short-term behaviors.

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Area of Science:

  • Statistical Physics
  • Complex Systems Analysis

Background:

  • Stochastic processes are fundamental in modeling natural phenomena.
  • Classifying these processes based on temporal behavior is crucial for understanding complex systems.
  • Existing methods may struggle with nonstationary increments and infinite moments.

Purpose of the Study:

  • To introduce a novel classification scheme for stochastic processes using renormalization-group operators.
  • To demonstrate the robustness of this scheme across various conditions, including nonstationary increments and infinite second moments.
  • To apply the scheme to classify specific physical and biological processes.

Main Methods:

  • Utilizing renormalization-group operators to analyze stochastic processes on two distinct time scales.
  • Investigating the long-time and short-time behavior through repeated operator application.
  • Examining the fixed points of these operators for process subclassification.
  • Applying the developed scheme to models of advection-diffusion and bronchial tree diffusion.

Main Results:

  • The renormalization-group operator approach effectively classifies stochastic processes based on their temporal dynamics.
  • The classification scheme is robust, handling nonstationary increments and infinite second moments.
  • Fixed points of the operators provide a means for further subclassification when applicable.
  • Successful classification of advection-diffusion and a human bronchial tree diffusion model was achieved.

Conclusions:

  • Renormalization-group operators offer a powerful and robust framework for classifying stochastic processes.
  • This method enhances the understanding of both short-term and long-term behaviors in complex systems.
  • The approach has broad applicability, demonstrated by its use in physical and biological modeling.