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Introduction to Types of Flows01:23

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Fluid flows are categorized by dimensionality and behavior, with one-dimensional flow being the simplest form, where properties like velocity and pressure change only along a single axis. Water moving through straight pipes exemplifies this flow type, as variations in other directions are minimal. One-dimensional analysis helps simplify understanding such flows, focusing solely on changes along the pipe's length.
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Fluid dynamics is the study of fluids in motion. Velocity vectors are often used to illustrate fluid motion in applications like meteorology. For example, wind—the fluid motion of air in the atmosphere—can be represented by vectors indicating the speed and direction of the wind at any given point on a map. Another method for representing fluid motion is a streamline. A streamline represents the path of a small volume of fluid as it flows. When the flow pattern changes with time, the...
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Gradually varying flow (GVF) in open channels describes situations where water depth changes slowly along the channel due to factors like non-uniform bed slope, channel shape variations, or obstructions. This flow type occurs when the depth adjusts gradually to balance gravitational forces, shear forces, and energy requirements, resulting in a low rate of depth change.Characteristics of Gradually Varying FlowGVF is commonly observed in natural streams, rivers, and canals, where flow depth...
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Turbulent flow is characterized by unpredictable fluctuations in velocity and pressure, which result in a chaotic fluid movement distinct from the orderly patterns of laminar flow. While laminar flow is governed by smooth, parallel layers with minimal mixing, turbulent flow exhibits highly irregular, three-dimensional patterns. This behavior arises due to instabilities in the fluid's velocity profile, and amplifies as the flow velocity increases. Minor disturbances, known as turbulent...
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Consider a control volume, such as a pipe with solid boundaries, through which fluid flows and changes direction due to the impulse exerted by the resulting force from the pipe walls. In steady flow, the mass of fluid entering the control volume at a given time, t, with velocity v1, is equal to the mass leaving after infinitesimal time dt, with velocity v2.
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Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking.

Mike R Jeffrey1

  • 1Engineering Mathematics, University of Bristol, Merchant Venturer's Building, Bristol BS8 1UB, United Kingdom.

Chaos (Woodbury, N.Y.)
|April 3, 2016
PubMed
Summary

Researchers explored how systems exit sliding modes, revealing mechanisms like tangency and spiraling. This study clarifies complex dynamics in discontinuous vector fields, crucial for robust control systems.

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Area of Science:

  • Control Theory
  • Dynamical Systems
  • Nonlinear Dynamics

Background:

  • Sliding modes are robust states arising from discontinuous vector fields.
  • Understanding the exit dynamics from sliding modes is crucial but poorly understood.
  • Complex behaviors emerge as flows leave sliding modes.

Purpose of the Study:

  • To investigate the fundamental mechanisms of flow exit from sliding modes.
  • To analyze exit dynamics along single switching surfaces and intersections of surfaces.
  • To understand sliding and exit phenomena involving multiple switches.

Main Methods:

  • Analysis of flow behavior on switching surfaces and their intersections.
  • Examination of tangency conditions for flow exit.
  • Investigation of spiraling dynamics and geometric divergence.
  • Resolution of determinacy-breaking singularities using switching layers.
  • Exploration of dynamics via preliminary simulations.

Main Results:

  • Exit from a single switching surface occurs via flow tangency.
  • Exit along an intersection can involve tangency or spiraling with geometric divergence.
  • Determinacy-breaking singularities can lead to set-valued flows.
  • Switching layers are used to resolve these singularities.
  • Simulations explore determinacy-breaking events as dynamic organizing centers.

Conclusions:

  • The study elucidates key mechanisms governing flow exit from sliding modes.
  • It provides a framework for understanding complex dynamics in systems with discontinuous vector fields.
  • The resolution of determinacy-breaking events offers insights into robust system behavior.