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Related Concept Videos

Couette Flow01:22

Couette Flow

244
Couette flow represents the flow of fluid between two parallel plates, with one plate fixed and the other moving with a constant velocity. This configuration allows for a simplified analysis using the Navier-Stokes equations, which govern fluid motion under conditions of viscosity and incompressibility. For Couette flow, the assumptions include a steady, laminar, incompressible flow with a zero-pressure gradient in the flow direction. This flow type is beneficial for understanding shear-driven...
244
Bernoulli's Equation for Flow Along a Streamline01:30

Bernoulli's Equation for Flow Along a Streamline

956
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:
956
Steady, Laminar Flow Between Parallel Plates01:17

Steady, Laminar Flow Between Parallel Plates

172
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.
172
Velocity Potential01:20

Velocity Potential

366
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:
366
Navier–Stokes Equations01:28

Navier–Stokes Equations

471
For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...
471
Bernoulli's Equation for Flow Normal to a Streamline01:16

Bernoulli's Equation for Flow Normal to a Streamline

841
Bernoulli's equation for flow normal to a streamline explains how pressure varies across curved streamlines due to the outward centrifugal forces induced by the fluid's curvature. The pressure is higher on the inner side of the curve, near the center of curvature, and decreases outward to balance these centrifugal forces.
The pressure difference depends on the fluid's velocity and radius of curvature. The pressure variation is minimal in flows with nearly straight streamlines.
841

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Related Experiment Video

Updated: Jun 23, 2025

Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions
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Variational solution to the lattice Boltzmann method for Couette flow.

Joseph T Johnson1, Mahyar Madadi2, Daniel R Ladiges3

  • 1School of Mathematics and Statistics, <a href="https://ror.org/01ej9dk98">University of Melbourne</a>, Victoria 3010, Australia.

Physical Review. E
|June 22, 2024
PubMed
Summary

This study presents a new variational approach for lattice Boltzmann method (LBM) simulations, offering exact analytical solutions for Couette flow. This method is computationally more efficient than traditional moment methods for fluid dynamics beyond the continuum limit.

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Area of Science:

  • Computational fluid dynamics
  • Non-equilibrium hydrodynamics
  • Lattice Boltzmann methods

Background:

  • Lattice Boltzmann methods (LBM) are increasingly used to study hydrodynamics beyond the continuum limit.
  • Existing methods, like moment equations, face challenges in computational efficiency for complex flow problems.
  • Analytical solutions for fluid dynamics problems, especially those involving specific boundary conditions, are crucial for validation and understanding.

Purpose of the Study:

  • To derive exact analytical solutions for the bulk velocity and shear stress in Couette flow using the lattice Boltzmann method.
  • To develop a novel variational approach for solving LBM equations with Maxwell-type boundary conditions.
  • To demonstrate the computational advantages of the new variational method over existing moment-based approaches.

Main Methods:

  • Solving equivalent moment equations to obtain analytical solutions for Couette flow under diffuse reflection.
  • Formulating a systematic variational approach based on the linearity of bulk velocity and shear stress with respect to Mach number.
  • Comparing the growth rate of partial differential equations (PDEs) in the variational method versus the moment method.

Main Results:

  • Exact analytical solutions for bulk velocity and shear stress in Couette flow with Maxwell-type boundary conditions were obtained.
  • The bulk velocity and shear stress were proven to be inherently linear in Mach number for 2D isothermal LBM.
  • The variational method exhibits superior computational efficiency, with a linear increase in PDEs versus a quadratic increase in the moment method with quadrature order.

Conclusions:

  • A novel variational approach provides exact analytical solutions for lattice Boltzmann models of Couette flow with Maxwell-type boundary conditions.
  • This method offers significant computational advantages over traditional moment methods, particularly in terms of the number of PDEs.
  • The developed variational approach is expected to be valuable for solving analytical solutions in new LBM quadrature schemes and other flow regimes.