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Related Concept Videos

Velocity Potential01:20

Velocity Potential

423
In steady, incompressible flow through a long, straight pipe with a uniform cross-section, the flow in the central region (far from the pipe walls) is irrotational. This irrotational nature means that fluid particles do not rotate around their axes, and a scalar function called the velocity potential, represented by ϕ, can be used to describe their movement. In irrotational flows, the velocity field V is defined as the gradient of the velocity potential:
423
Newtonian Fluid: Problem Solving01:18

Newtonian Fluid: Problem Solving

357
Newtonian fluids exhibit a constant viscosity, meaning their shear stress and shear strain rate are directly proportional. This property ensures a predictable and stable response to applied forces, maintaining a linear relationship between force and flow. Examples include water, air, and light oils, consistently demonstrating this proportional behavior regardless of external conditions.
A velocity gradient forms within the fluid when a Newtonian fluid is placed between two parallel plates, with...
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Accelerating Fluids01:17

Accelerating Fluids

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When a fluid is in constant acceleration, the pressure and buoyant force equations are modified. Suppose a beaker is placed in an elevator accelerating upward with a constant acceleration, a. In the beaker, assume there is a thin cylinder of height h with an infinitesimal cross-sectional area, ΔS.
The motion of the liquid within this infinitesimal cylinder is considered to obtain the pressure difference. Three vertical forces act on this liquid:
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Plane Potential Flows01:23

Plane Potential Flows

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Plane potential flows simplify fluid motion by assuming the fluid to be irrotational and incompressible. These characteristics allow these flows to be described by a velocity potential function, ϕ, representing the flow speed in a given direction, and a stream function, ψ, that visualizes the flow path, both governed by Laplace's equation. These parameters help in estimating flow patterns, velocity distributions, and pressure fields around various hydraulic structures.
Uniform...
448
Turbulent Flow: Problem Solving01:09

Turbulent Flow: Problem Solving

177
Carbonation is a process used to dissolve carbon dioxide gas in a liquid, commonly used in the production of carbonated beverages. Achieving efficient carbonation requires careful control of temperature, pressure, and flow conditions. By adjusting these parameters, carbonation efficiency can be maximized, producing a higher concentration of CO2 in the liquid.
Temperature is a key factor in CO2 solubility. In this case, the CO2 gas and the liquid are cooled to 20°C. Lower temperatures...
177
Bernoulli's Equation for Flow Along a Streamline01:30

Bernoulli's Equation for Flow Along a Streamline

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Bernoulli's equation relates the energy conservation in a fluid moving along a streamline. The equation applies to incompressible and inviscid fluids under steady flow. For such a flow, Newton's second law is applied to a small fluid element, which experiences forces due to pressure differences, gravity, and velocity variations. The force balance leads to the following form of Bernoulli's equation:
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Optical Coherence Tomography Based Biomechanical Fluid-Structure Interaction Analysis of Coronary Atherosclerosis Progression
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Penalty Virtual Element Method for the 3D Incompressible Flow on Polyhedron Mesh.

Lulu Li1, Haiyan Su1, Yinnian He1

  • 1College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China.

Entropy (Basel, Switzerland)
|August 26, 2022
PubMed
Summary

A new penalty virtual element method (VEM) effectively solves 3D incompressible flow on polyhedral meshes. This method bypasses complex computations, offering a robust solution for fluid dynamics simulations.

Keywords:
3D incompressible flowerror analysispenalty methodpolyhedron meshvirtual element method

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Area of Science:

  • Computational Fluid Dynamics
  • Numerical Analysis
  • Applied Mathematics

Background:

  • Standard finite element methods face challenges with arbitrary meshes.
  • Virtual Element Method (VEM) offers a mesh-independent approach.
  • Solving 3D incompressible flow requires robust numerical techniques.

Purpose of the Study:

  • To propose and analyze a penalty virtual element method (VEM) for 3D incompressible flow.
  • To address limitations of standard finite element discretizations on complex geometries.
  • To provide a rigorous error analysis for the proposed method.

Main Methods:

  • Implementation of a penalty virtual element method (VEM) on polyhedral meshes.
  • Approximation of velocity and pressure using lowest equal-order virtual element spaces (Xh, Qh).
  • Application of a penalty method to handle the inf-sup condition deficiency.

Main Results:

  • Rigorous error estimation was proven for the VEM approach.
  • The method successfully handles 3D incompressible flow problems.
  • Numerical results validated the theoretical findings and method's effectiveness.

Conclusions:

  • The penalty VEM is a viable and effective method for 3D incompressible flow.
  • VEM simplifies computations by avoiding explicit trial and test space calculations.
  • The method demonstrates strong performance on 3D polygonal meshes.