Question about Kelvin’s circulation theorem

In summary, Kelvin's circulation theorem states that the circulation around a closed curve moving with a fluid is constant if the fluid is inviscid and the flow is steady. This principle helps to analyze fluid motion and is essential in understanding the behavior of rotating fluids and the conservation of angular momentum in fluid dynamics.
  • #1
Seyn
6
1
In Currie’s fluid mechanics textbook, there is a statement “the vorticity of each fluid particle will be preserved.” as the result of Kelvin’s circulation theorem.
Kelvin’s circulation theorem claims that
For inviscid flow, constant density or barotropic fluid, conservative body force,
the circulation around an arbitrary contour is conserved following same fluid particle.
Does Kelvin’s theorem also guarantee the vorticity on each fluid particles? Why?
 
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  • #2
Take a look at the vorticity equation under the same conditions as Kelvin's circulation theorem. Does it reduce to [itex]\frac{D\omega}{Dt} = 0[/itex]?
 

FAQ: Question about Kelvin’s circulation theorem

What is Kelvin’s circulation theorem?

Kelvin’s circulation theorem states that the circulation around a closed curve moving with the fluid is constant over time, provided that the flow is inviscid (non-viscous) and the fluid is barotropic (density only depends on pressure). This theorem is fundamental in understanding the behavior of ideal fluid flows and is applicable in various fields of fluid dynamics.

What are the assumptions made in Kelvin’s circulation theorem?

The primary assumptions of Kelvin’s circulation theorem are that the fluid is inviscid, meaning there is no viscosity acting on the fluid, and barotropic, indicating that the fluid's density is a function of pressure alone. Additionally, the flow must be steady and the closed curve must move with the fluid.

How is circulation defined in the context of Kelvin’s theorem?

Circulation is defined as the line integral of the velocity field around a closed curve. Mathematically, it is expressed as Γ = ∮ V · dl, where Γ is the circulation, V is the velocity vector, and dl is the differential length element along the closed curve. In the context of Kelvin’s theorem, this circulation remains constant as the curve moves with the fluid.

What are some applications of Kelvin’s circulation theorem?

Kelvin’s circulation theorem has several applications, including in meteorology for understanding large-scale atmospheric circulation patterns, in oceanography for studying ocean currents, and in aerospace engineering for analyzing the flow around aircraft and other vehicles. It is also used in theoretical studies of vortex dynamics and turbulence.

What is the significance of Kelvin’s circulation theorem in fluid dynamics?

The significance of Kelvin’s circulation theorem lies in its ability to provide insights into the conservation of angular momentum in fluid flows. It helps in predicting the behavior of ideal fluids and understanding the stability of vortices, which is crucial for both theoretical studies and practical applications in engineering and natural sciences.

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