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Steady, Laminar Flow in Circular Tubes01:23

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Hagen-Poiseuille flow describes a viscous fluid's steady, incompressible flow through a cylindrical tube with a constant radius R. This flow profile is often applied to understand fluid transport in narrow channels, such as capillaries. It serves as a foundational example of laminar flow. In this model, cylindrical coordinates (r,θ,z) are used to describe the radial (r), angular (θ), and axial (z) dimensions within the tube. For Hagen-Poiseuille flow, the velocity profile is purely axial,...
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When a body is submerged in water, it experiences fluid pressure acting normal on its surface and distributed over its area. For better design structures, it is crucial to determine the magnitude and location of the resultant force acting on the surface. In the case of a rectangular plate of constant width submerged in water, the pressure increases with depth, resulting in a linearly varying trapezoidal pressure distribution from the upper to the lower edge of the plate.
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Fluid Pressure over Flat Plate of Variable Width01:02

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When a flat plate is submerged in a fluid, the fluid exerts pressure on the plate. This pressure can lead to many different phenomena, including drag and buoyancy. To understand the behavior of the fluid over a flat plate of variable width, it is essential to analyze the distribution of the pressure exerted.
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When a curved plate of constant width is submerged in a liquid, the pressure acting normal to the plate varies continuously both in magnitude and direction. Calculating the magnitude and location of the resultant force at a point is often challenging for such cases. One of the methods to determine the resultant force and its location involves separately calculating the horizontal and vertical components of the resultant force. This complex calculation can be simplified by representing the...
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Understanding steady, laminar flow between parallel plates is essential for analyzing and designing flow in narrow rectangular channels, commonly found in various water conveyance and drainage systems. The Navier-Stokes equations govern fluid motion and are generally challenging to solve due to their nonlinearity. However, simplifications are possible in certain cases, like the steady laminar flow between parallel plates. For this scenario, we assume steady, incompressible, laminar flow.
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First Law: Particles in Two-dimensional Equilibrium01:18

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Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic equilibrium involves objects in motion without acceleration; but it is important to remember that these conditions are relative. For instance, an object may be at rest when viewed from one frame of reference, but that same object would appear to be in motion when viewed by someone moving at a constant velocity.
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Related Experiment Video

Updated: Mar 15, 2026

Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures
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Simulating confined particles with a flat density profile.

Airidas Korolkovas1

  • 1Institut Laue-Langevin, 71 rue des Martyrs, 38000 Grenoble, France and Université Grenoble Alpes, Liphy, 140 Rue de la Physique, 38402 Saint-Martin-d'Hères, France.

Physical Review. E
|September 15, 2016
PubMed
Summary

Simulations of soft matter liquids using new mirror-and-shift boundary conditions create realistic, monotonic density profiles. This method accurately models polymer brushes, aligning simulation results with experimental data.

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Area of Science:

  • Soft matter physics
  • Computational materials science
  • Polymer physics

Background:

  • Particle simulations with sharp walls often yield unrealistic oscillatory density profiles.
  • This behavior is particularly problematic for modeling soft matter liquids and polymer systems.
  • Existing simulation methods struggle to accurately represent density changes near interfaces.

Purpose of the Study:

  • To develop a novel simulation method that produces monotonic density profiles near interfaces.
  • To reconcile discrepancies between particle simulations and experimental observations in soft matter systems.
  • To accurately model the density profile of a polymer brush grafted on a substrate.

Main Methods:

  • Introduction of mirror-and-shift boundary conditions to particle simulations.
  • Mapping simulation interfaces to distant parts of themselves to avoid artificial oscillations.
  • Application of the method to simulate a polymer brush in explicit solvent on a silicon substrate.

Main Results:

  • The proposed boundary conditions lead to an almost monotonic density increase from zero to bulk density.
  • The density profile develops over a short distance, approximately one particle diameter.
  • Simulations of a polymer brush show excellent agreement with neutron reflectometry and self-consistent field theory.

Conclusions:

  • Mirror-and-shift boundary conditions are effective in generating realistic density profiles in particle simulations.
  • This method provides a more accurate way to model soft matter systems, particularly polymer brushes.
  • The approach bridges the gap between computational simulations and experimental measurements.