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Bernoulli's Equation for Flow Along a Streamline

Bernoulli's Equation for Flow Along a Streamline

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:

Steady, Laminar Flow in Circular Tubes

Steady, Laminar Flow in Circular Tubes

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...

Steady, Laminar Flow Between Parallel Plates

Steady, Laminar Flow Between Parallel Plates

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.

Irrotational Flow

Irrotational Flow

Irrotational flow is characterized by fluid motion where particles do not rotate around their axes, resulting in zero vorticity. For a flow to be irrotational, the curl of the velocity field must be zero. This imposes specific conditions on velocity gradients. For instance, to maintain zero rotation about the z-axis, the gradient condition:

Bernoulli's Equation for Flow Normal to a Streamline

Bernoulli's Equation for Flow Normal to a Streamline

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.

Couette Flow

Couette Flow

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...

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Updated: Oct 23, 2025

Magnetically Induced Rotating Rayleigh-Taylor Instability

Magnetically Induced Rotating Rayleigh-Taylor Instability

Published on: March 3, 2017

Elastically driven Kelvin-Helmholtz-like instability in straight channel flow.

Narsing K Jha^1,2, Victor Steinberg^3,4

¹Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel.

Proceedings of the National Academy of Sciences of the United States of America

|August 19, 2021

Summary

Kelvin-Helmholtz instability (KHI) is observed in viscoelastic channel flow, challenging prior beliefs about fluid stability. This elastic KHI drives coherent structures and is synchronized by elastic waves.

Keywords:

Kelvin–Helmholtz instability elastic turbulence elastically driven instability

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Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions

Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions

Published on: February 22, 2018

Area of Science:

Fluid Dynamics
Rheology
Non-Newtonian Flows

Background:

Kelvin-Helmholtz instability (KHI) typically describes fluid interfaces with velocity and density differences.
KHI is a fundamental concept in fluid dynamics, observed across various natural and industrial scales.
Elastic turbulence (ET) is a phenomenon in viscoelastic flows characterized by chaotic behavior without significant inertia.

Purpose of the Study:

To report the observation of an elastically driven KH-like instability in straight viscoelastic channel flow.
To challenge the established view that interface perturbations are stable in low-inertia flows.
To elucidate the novel mechanism behind this instability in elastic turbulence.

Main Methods:

Experimental observation of viscoelastic channel flow exhibiting elastic turbulence.
Analysis of coherent structures (CSs) and velocity fluctuations.
Investigation of the role of elastic waves and vorticity dynamics.

Main Results:

Observation of KH-like instability in viscoelastic channel flow, contradicting stability assumptions at low inertia.
Identification of self-organized, cycling coherent structures (streaks) synchronized by elastic waves.
Demonstration that elastic waves interact with wall-normal vorticity to amplify perturbations, driving the instability.

Conclusions:

Elastic turbulence exhibits a novel KH-like instability distinct from Newtonian KHI.
The instability is driven by the interaction of elastic waves with vorticity, overcoming stabilizing hoop stress.
This finding expands the understanding of instabilities in viscoelastic flows and challenges existing paradigms.