Name: irrotational flow near boundaries
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 28 s
Channel: JoVE

Irrotational flow refers to the movement of fluid particles that do not rotate along the flow path, characterized by a velocity field with zero curl and no vorticity.
A uniform flow with only one non-zero component of velocity, a constant u=U, is an example of irrotational flow because it lacks velocity gradients.
When a solid object is introduced into a uniform flow, the flow remains irrotational except near the boundaries. Near its surface, the velocity sharply increases from zero, creating steep gradients and significant shear stress.
In the case of a truly inviscid flow, the flow would glide past the boundary, preserving irrotational motion. However, in real fluids, viscosity forms a boundary layer near fixed surfaces, producing rotational effects and sometimes causing downstream separation.
In pipe flow, the entrance region initially exhibits an irrotational core with uniform flow, but as a boundary layer forms along the pipe wall and thickens, it transitions into a fully developed region dominated by viscous forces.
Assuming the flow is irrotational, incompressible, steady, and inviscid, Bernoulli&#39;s equation can be applied in the flow field.

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:
<img alt="Fluid dynamics equation &#8706;v/&#8706;x = &#8706;u/&#8706;y diagram, illustrating incompressible flow concept." src="/files/ftp_upload/18059/18059_Equation_1.svg" style="text-align: center; height: 27px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
must be satisfied, along with similar conditions for other axes:
<img alt="Partial derivatives formula, fluid dynamics equations, illustrating continuity in vector fields." src="/files/ftp_upload/18059/18059_Equation_2.svg" style="text-align: center; height: 27px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
A uniform flow, where u is constant and v and w are both zero, satisfies these conditions and serves as a simple example of irrotational flow. However, when a solid object is introduced, irrotationality only persists far from the object. Near the boundary, the velocity increases sharply from zero (due to the no-slip condition) to the freestream value, creating significant shear stress and a boundary layer in viscous fluids.
When fluid enters a pipe from a large reservoir, the initial flow is nearly uniform, resulting in an irrotational core where the velocity distribution is flat. As the fluid moves downstream, a boundary layer forms along the pipe walls due to viscosity, causing the velocity profile to develop gradients that ultimately lead to a fully developed parabolic profile. In this fully developed region, rotational effects dominate near the walls, while a central core may remain approximately irrotational.
In the initial irrotational region of pipe flow, Bernoulli's equation
<img alt="Bernoulli's equation: \( \frac{p}{\gamma} + \frac{V^2}{2g} + z = \text{constant} \), fluid dynamics formula." src="/files/ftp_upload/18059/18059_Equation_3.svg" style="text-align: center; height: 27px;" data-altupdatedat="1762173717000" data-alt="Equation 3" />
can be applied between points along a streamline as the conditions of incompressibility, inviscid flow, and irrotationality hold. However, as the boundary layer grows and rotational effects become significant near the walls, the assumptions of Bernoulli's equation break down. This means that Bernoulli's equation is only applicable in the entrance region and the irrotational core, but not in the fully developed, rotational region where viscosity effects dominate.

Irrotational flow refers to the movement of fluid particles that do not rotate along the flow path, characterized by a velocity field with zero curl and ...

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JoVE Core

Civil Engineering

Unit 1

Differential Analysis of Fluid Flow

KEY TERMS AND CONCEPTS

irrotational flow near boundaries

Irrotational flow is fluid motion in which particles do not spin around their own axes. It has zero vorticity, which means the curl of the velocity field is zero. That condition places limits on the velocity gradients in the flow.

A simple example is uniform flow. In this case, the velocity u stays constant, while v and w are both zero. This satisfies the irrotational conditions and gives a basic model of non-rotating flow.

When a solid object is added, irrotational flow usually remains only far from the object. Near the surface, the fluid speed rises sharply from zero because of the no-slip condition. This creates strong shear stress and a boundary layer in viscous fluids.

A similar change happens when fluid enters a pipe from a large reservoir. At first, the flow is nearly uniform, so the center region forms an irrotational core with a flat velocity profile. Farther downstream, viscosity causes boundary layers to grow along the pipe walls.

As the flow continues, the velocity profile develops more gradient and becomes fully developed and parabolic. In that region, rotational effects are strongest near the walls, while the center may still be approximately irrotational. Bernoulli's equation can apply in the entrance region and in the irrotational core, where the flow is incompressible, inviscid, and irrotational. It no longer applies well in the fully developed region, where viscosity and rotation dominate.

Irrotational Flow Near Boundaries

euler&#39;s equations of motion

Pressure and Velocity in Inviscid Flow

Pressure and velocity in inviscid flow are linked by Euler's equations of motion. In fluid mechanics, viscosity creates shear stress, which is a force that acts along a surface. For low-viscosity fluids like water, these shear stresses are very small. That lets the fluid be treated as inviscid, or frictionless. In an inviscid fluid, shear stresses are absent. Only normal stresses remain, and these act perpendicular to a fluid element. Pressure is the negative of the normal stress. It stays the...

stream function

Stream Function for Incompressible Flow

The stream function helps describe two-dimensional incompressible flow. In this type of flow, the continuity equation keeps mass conserved. That means fluid entering a region must be balanced by fluid leaving it. For incompressible flow, the density stays constant. The continuity requirement then becomes a zero-divergence condition for the velocity field. In two dimensions, the horizontal and vertical velocity components are usually written as u and v. A stream function, written as ψ, is used...

Irrotational flow is fluid motion in which particles do not spin around their own axes. It has zero vorticity, which means the curl of the velocity field is zero. That condition places limits on the velocity gradients in the flow. A simple example is uniform flow. In this case, the velocity u stays constant, while v and w are both zero. This satisfies the irrotational conditions and gives a basic model of non-rotating flow. When a solid object is added, irrotational flow usually remains only...

velocity potential

Velocity Potential in Pipe Flow

Velocity potential in pipe flow describes the motion of fluid in the central region of a long, straight pipe with a uniform cross-section. In this region, the flow is steady, incompressible, and irrotational. Irrotational means the fluid particles do not spin around their own axes. A scalar function called the velocity potential, written as φ, is used to describe this motion. The velocity field V is defined as the gradient of φ. Because the curl of a gradient is always zero, this form of V...

plane potential flows

Plane Potential Flow Patterns and Models

Plane potential flows describe fluid motion using simple flow patterns and models. They assume the fluid is irrotational and incompressible. These assumptions let the flow be written with a velocity potential function, φ, which gives the flow speed in a direction, and a stream function, ψ, which shows the flow path. Both are governed by Laplace's equation and are useful for estimating flow patterns, velocity distributions, and pressure fields around hydraulic structures. Uniform flow is the...

navier&#8211;stokes equations

Fluid Stress and Flow Equations

Fluid stress and flow equations describe how incompressible Newtonian fluids move under pressure, shear, and gravity. In these fluids, density stays constant, and stress changes linearly with the rate of deformation. The main stress types are normal stress and shear stress. Normal stress depends on pressure acting on the fluid and on deformation rate in specific directions. Shear stress acts along the surface of a fluid layer. It shows how neighboring layers slide past one another, so velocity...

steady, laminar flow between parallel plates

steady-laminar-flow-between-parallel-plates

Laminar Flow Profile Between Parallel Plates

Laminar flow between parallel plates shows how fluid moves through a narrow, flat channel. This model helps describe flow in water conveyance and drainage systems, where the space is limited and the motion stays orderly. It is also useful for analyzing flow in rectangular channels used in many engineering settings. The flow is studied with the Navier-Stokes equations, which govern fluid motion. These equations are often difficult to solve because they are nonlinear. In this case, the problem...

couette flow

Linear Shear Between Moving Plates

Linear shear between moving plates describes fluid motion in a simple channel with two parallel plates. One plate stays fixed, and the other moves at a constant speed. This setup helps show how viscosity and incompressible flow shape fluid motion under the Navier-Stokes equations. The model uses a steady, laminar, incompressible flow with no pressure gradient in the flow direction. The x-axis points along the moving direction of the upper plate, and the y-axis is perpendicular to both plates.

steady, laminar flow in circular tubes

Parabolic Flow in Circular Tubes

Parabolic flow in circular tubes describes the steady, incompressible motion of a viscous fluid through a tube with a constant radius. This Hagen-Poiseuille flow is a classic example of laminar flow. It is often used to study fluid transport in narrow channels, including capillaries. Cylindrical coordinates are used to describe the tube. The radial direction is r, the angular direction is θ, and the axial direction is z. In this flow, the velocity is purely axial. That means the fluid moves...

design example: flow of oil through circular pipes

design-example-flow-of-oil-through-circular-pipes

Pressure Drop in Oil Pipe Design

Pressure drop in oil pipe design depends on pipe size, oil properties, and the flow rate needed through a circular pipe. For steady, incompressible, laminar flow, Hagen-Poiseuille's law gives an exact solution from the Navier-Stokes equations. It is used to find the pressure difference needed to keep oil moving through a pipeline section. The pressure drop equation uses the oil's dynamic viscosity, the pipeline length, the desired volumetric flow rate, and the pipe's inner radius. This...

key terms and concepts

Learn how irrotational flow behaves near solids and in pipe entrance flow, including boundary layers, rotational effects, and when Bernoulli's equation applies.

Irrotational Flow Near Boundaries | JoVE K12

JoVE Core Civil Engineering offers a foundational exploration of essential construction materials such as bricks, stones, steel, wood, and concrete, along with fluid mechanics topics like flow over immersed bodies, open channel flow, and dimensional modeling. It also covers key surveying concepts, including leveling, traverse adjustments, curve design, topographic mapping, and geospatial technologies (GPS & GIS). Science-in-action videos showcase laboratory research on materials, fluids, and surveying practices in real-world engineering systems.

civil engineering

Irrotational Flow Near Boundaries

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