Name: parabolic flow in circular tubes
Uploaded: 2025-02-13T15:43:26.000Z
Duration: 1 min 23 s
Channel: JoVE

Consider steady, incompressible flow through a straight cylindrical tube of constant radius R, known as Hagen&#8211;Poiseuille flow.
The cylindrical geometry allows the use of coordinates r, &#952;, and z for radial, angular, and axial measurements.
The fluid flowing parallel to the tube&#39;s axis has a purely axial velocity component and varies only with the radial distance.
These conditions simplify the Navier&#8211;Stokes equations, revealing a parabolic velocity profile showing that the velocity is highest at the center and decreases towards the walls, reaching zero at the boundary.
This demonstrates the laminar flow, where fluid particles move smoothly in organized layers.
Integrating this profile over the tube&#39;s cross-section gives Poiseuille&#39;s law, which expresses the flow rate Q as dependent on the fourth power of the tube&#39;s radius, fluid viscosity, and pressure gradient.
The mean velocity V, calculated from Q, is half the maximum velocity observed at the tube&#39;s center.
This velocity distribution confirms that flow is highest at the center and decreases parabolically to zero at the wall.

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,&#952;,z) are used to describe the radial (r), angular (&#952;), and axial (z) dimensions within the tube. For Hagen-Poiseuille flow, the velocity profile is purely axial, meaning the velocity vector points only along the z-axis and varies solely with the radial distance r from the center of the tube.
The governing equations for fluid motion, the Navier-Stokes equations, simplify under these conditions. Without angular or radial velocity components, the axial velocity vz(r) emerges as a function of r alone. Solving the reduced Navier-Stokes equations yields a parabolic velocity profile:
<img alt="Fluid dynamics equation: \( v_z/v_{max} = 1 - (r/R)^2 \); velocity profile analysis." src="/files/ftp_upload/18065/18065_Equation%201.svg" style="text-align: center; height: 27px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
This profile demonstrates that viscous drag decreases the fluid velocity from the maximum at the center to zero at the tube wall. This zero velocity at the boundary, known as the no-slip condition, results from the friction between the fluid and the tube wall. Consequently, the flow is laminar, with fluid particles moving in parallel layers without lateral mixing.
Integrating the velocity profile over the cross-sectional area of the tube yields the volumetric flow rate Q, known as Poiseuille's law:
<img alt="Poiseuille's law formula, static equilibrium, fluid dynamics equation for flow rate analysis." src="/files/ftp_upload/18065/18065_Equation%202.svg" style="text-align: center; height: 27px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
Where &#916;P is the pressure difference across the length L of the tube, and &#956; is the fluid's dynamic viscosity. This relation shows that Q is highly sensitive to the tube's radius, increasing with the fourth power of R. It follows that even a slight increase in radius significantly boosts the flow rate, a principle critical in fluid transport applications.
From Poiseuille's law, the mean velocity V of the fluid can be expressed as:
<img alt="Poiseuille's law formula diagram, fluid flow through a pipe equation V=R&#178;&#916;p/8&#956;&#8467;." src="/files/ftp_upload/18065/18065_Equation%203.svg" style="text-align: center; height: 27px;" data-altupdatedat="1762173717000" data-alt="Equation 3" />
This average velocity is precisely half of the maximum velocity at the tube's center, confirming the parabolic velocity distribution and demonstrating the predictable, layered nature of laminar flow in a cylindrical tube.

Consider steady, incompressible flow through a straight cylindrical tube of constant radius R, known as Hagen&#8211;Poiseuille flow.
The cylindrical ...

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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 only along the z-axis and changes with distance r from the center of the tube.

Under these conditions, the Navier-Stokes equations become simpler. With no angular or radial velocity components, the axial velocity vz(r) depends only on r. Solving the reduced equations gives a parabolic velocity profile. The fluid moves fastest at the center and slows toward the wall because viscous drag reduces the speed.

At the tube wall, the velocity is zero. This is the no-slip condition, which comes from friction between the fluid and the wall. The result is laminar flow, where fluid particles travel in parallel layers with no lateral mixing.

Integrating the velocity profile over the tube cross section gives the volumetric flow rate Q. This is Poiseuille’s law. In this relation, ΔP is the pressure difference across tube length L, and μ is the fluid’s dynamic viscosity. The flow rate depends strongly on the tube radius and increases with the fourth power of R, so even a small increase in radius can greatly raise Q.

Poiseuille’s law also gives the mean velocity V of the fluid. The average velocity is exactly half of the maximum velocity at the tube center. This matches the parabolic profile and shows the predictable, layered nature of flow in a circular tube.

Parabolic Flow in Circular Tubes

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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...

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key terms and concepts

Explore how Hagen-Poiseuille flow creates a parabolic velocity profile in circular tubes, with no-slip at the wall and Poiseuille’s law for flow rate.

Parabolic Flow in Circular Tubes | JoVE K12

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Parabolic Flow in Circular Tubes

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