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Navier–Stokes Equations01:28

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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...
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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...
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Updated: Aug 16, 2025

Evolution of Staircase Structures in Diffusive Convection
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Layer resolving numerical scheme for singularly perturbed parabolic convection-diffusion problem with an interior

Gemadi Roba Kusi1, Aknaw Hailemariam Habte1, Tesfaye Aga Bullo1

  • 1Department of Mathematics, College of Natural Science, Jimma University, Jimma, Ethiopia.

Methodsx
|December 22, 2022
PubMed
Summary

This study presents a new numerical scheme to accurately solve singularly perturbed parabolic convection-diffusion problems with interior layers. The developed method offers stability, consistency, and improved accuracy over existing techniques.

Keywords:
Accurate solutionInterior layerLayer resolvingLayer resolving numerical schemeParabolic problemsSingularly perturbed

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Area of Science:

  • Numerical Analysis
  • Partial Differential Equations
  • Computational Mathematics

Background:

  • Singularly perturbed parabolic convection-diffusion problems present challenges due to interior layers caused by sign-changing convection coefficients.
  • Accurate numerical solutions are crucial for understanding phenomena governed by these equations.

Purpose of the Study:

  • To introduce a novel layer-resolving numerical scheme for singularly perturbed parabolic convection-diffusion problems with interior layers.
  • To analyze the stability, consistency, and accuracy of the proposed numerical method.

Main Methods:

  • The numerical scheme discretizes the temporal variable using a uniform mesh.
  • Spatial discretization is performed on a piecewise uniform Shishkin mesh to effectively capture interior layers.
  • Theoretical analysis is complemented by numerical experiments to validate the scheme's performance.

Main Results:

  • The proposed scheme is proven to be stable and consistent.
  • The numerical solution achieves almost first-order convergence.
  • The method demonstrates superior accuracy compared to existing numerical techniques in the literature.

Conclusions:

  • The developed numerical scheme provides an effective and accurate approach for solving singularly perturbed parabolic convection-diffusion problems with interior layers.
  • The scheme's stability, consistency, and enhanced accuracy make it a valuable tool for researchers in computational mathematics and related fields.