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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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Bernoulli's Equation for Flow Normal to a Streamline01:16

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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.
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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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Uniform Depth Channel Flow01:27

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Uniform depth channel flow keeps fluid depth consistent along channels such as irrigation canals. In natural channels, such as rivers, approximate uniform flow is often assumed. This condition occurs when the channel’s bottom slope matches the energy slope, balancing potential energy lost from gravity with head loss due to shear stress. This balance prevents depth changes along the channel length, resulting in a steady, uniform flow.Uniform flow in open channels with a constant cross-section...
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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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Related Experiment Video

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Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
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Variational approach for Stokes flow through a two-dimensional non-uniform channel.

Abhishek Banerjee1,2,3, Alexander Oron4, Yehuda Agnon5

  • 1Department of Mathematics, SRM Institute of Science and Technology Kattankulathur, Chennai, 603203, India. abhishek.rajnagr@gmail.com.

Scientific Reports
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PubMed
Summary

This study introduces a variational method to calculate pressure drop in non-uniform channels, offering accurate estimates using channel shape functions.

Keywords:
Euler-Lagrange equationFinite volume methodStokes flowVariational calculus

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Area of Science:

  • Fluid dynamics
  • Applied mathematics

Background:

  • Stokes flow analysis is crucial for microfluidics and biomechanics.
  • Understanding pressure drop in non-uniform channels is complex.

Purpose of the Study:

  • To develop a variational approach for Stokes flow in 2D non-uniform channels.
  • To derive an explicit formula for average pressure drop based on channel geometry.

Main Methods:

  • Utilizing the stationarity of the Lagrangian to derive Euler-Lagrange equations.
  • Formulating a set of ordinary differential equations for pressure drop estimation.
  • Comparing results with second-order extended lubrication theory.

Main Results:

  • The variational approach accurately estimates average pressure drop.
  • Results show excellent agreement with established lubrication theory.
  • Higher-order formulations enhance the precision of pressure drop calculations.

Conclusions:

  • The proposed variational method provides an efficient and accurate tool for analyzing Stokes flow.
  • The derived pressure drop formula offers direct correlation with channel geometry.
  • This work contributes to the precise modeling of fluid behavior in complex geometries.