Lagrange-Young equation in Fluid Mechanics

In summary, the Young-Laplace equation in Fluid Mechanics describes the relationship between surface tension, pressure, and the curvature of a fluid interface. The equation states that the change in pressure multiplied by the area of the interface is equal to the surface tension multiplied by the circumference of the interface. This can be derived from energy minimization principles and is related to Young's equation for three-phase lines. The pressure acting on a fluid interface is always perpendicular to the surface due to the inability of fluids to support shear stresses. However, fluids can exhibit shear stresses when subjected to viscous deformation. This does not apply to non-Newtonian fluids with yield stresses.
  • #1
Nikitin
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Young-Laplace equation in Fluid Mechanics

EDIT: I meant the Young-Laplace equation, of course, not the Young-Lagrange.. sorry!

Heya!

According to said equation, ΔP*area = γ*circumference, for an interface of spherical fluid-element.

Can I pls get some explanations?

1) Why is tension, or γ, purely a result of chemical forces? Isn't there a pressure parallel to the surface, just like there is a pressure inside the fluid-element? Because if there were, the equation should be ΔP*area + P*area = γ*circumference.

2) Is there a physical explanation for the formula? Why does γ=ΔP*area/circumference?
 
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  • #2
I am not familiar with the name Lagrange-Young equation, but it seems to describe how surface tension is related to the pressure difference over the surface.

Then it makes perfect sense. The force of tension at the boundary of some surface segment must stretch it, and the total "stretching" force over the entire boundary must be balanced by the force due to the pressure difference over the area of the segment.

Think about a balloon made of rubber. Because the rubber is stretched, it presses against that content of the balloon, thus the internal pressure is higher than the external. Now cut a piece of the rubber from the balloon. It will be acted upon by the pressure difference, tending to separate it from the rest of the balloon, so in order to keep it in place, you must apply force to its edges.
 
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  • #3
You're right to not be familiar with it, because it's called the Laplace-Young equation.. Sorry, I keep mixing lagrange and laplace. :P here's what I was talking about: http://en.wikipedia.org/wiki/Young–Laplace_equation

Anyway: I read the explanation offered by Cimbala & Cengel's fluid mechanics, and I think I got it now. Though I am unsure on what delta-P actually is: is it the difference between the pressure inside the fluid and outside, or is it the pressure-change inside the fluid?
 
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  • #4
The referenced article explains what ## \Delta p ## is.
 
  • #5
Nikitin said:
EDIT: I meant the Young-Laplace equation, of course, not the Young-Lagrange.. sorry!

Heya!

Isn't there a pressure parallel to the surface, just like there is a pressure inside the fluid-element?

Pressure only acts normal to surfaces. There is no shear component of pressure. Any way you orient an area element within a fluid, the pressure only acts perpendicular to the area element. In a fluid, the viscous portion of the stress tensor has shear components on arbitrarily oriented area elements, but not the pressure portion.

Chet
 
  • #6
Nikitin said:
1) Why is tension, or γ, purely a result of chemical forces? Isn't there a pressure parallel to the surface, just like there is a pressure inside the fluid-element? Because if there were, the equation should be ΔP*area + P*area = γ*circumference.

2) Is there a physical explanation for the formula? Why does γ=ΔP*area/circumference?

The second question is easier to answer: it arises from energy minimization of a dividing surface. The detailed calculation can be difficult to follow, but a good reference is here:

http://www.sciencedirect.com/science/article/pii/0001868695002820#

Similarly, Young's equation is derived from conservation of momentum at the three-phase line.

I may not understand what you are asking in question (1): the interfacial energy is just that- the energy associated with an interface. Since fluids can't support a shear stress, there is no 'parallel' component. Solids can support a shear, so the interfacial energy of a solid-vacuum interface is not easy to define (Wulff constructions are commonly used). A good starting point is here:

http://www.virginia.edu/ep/SurfaceScience/Thermodynamics.html
 
  • #7
Andy Resnick said:
Since fluids can't support a shear stress, there is no 'parallel' component.
I think that this needs to be qualified a little. Fluids can't support a shear stress statically, but can exhibit shear stresses by deforming viscously with time (i.e., with time in a Lagrangian sense). For example, fluid being sheared continuously between parallel plates causes a shear stress on the plates.

Chet
 
  • #8
Chestermiller said:
I think that this needs to be qualified a little. Fluids can't support a shear stress statically, but can exhibit shear stresses by deforming viscously with time (i.e., with time in a Lagrangian sense). For example, fluid being sheared continuously between parallel plates causes a shear stress on the plates.

Chet

Yes, fluids flow in response to shear. Sometimes the flow is reversible, most often it is not. In any case, a fluid's inability to 'store' elastic energy is why there is no parallel component in the OP.

Edit: my comments above are restricted to Newtonian fluids only- viscoelastic fluids, bingham fluids, and other fluid phases possessing a yield stress violate what I said above.
 

Related to Lagrange-Young equation in Fluid Mechanics

1. What is the Lagrange-Young equation in fluid mechanics?

The Lagrange-Young equation, also known as the Euler-Lagrange equation, is a fundamental equation in fluid mechanics that describes the motion of a fluid in terms of its velocity and pressure. It is derived from the Navier-Stokes equations and is used to study complex fluid flows, such as turbulence and vortices.

2. How is the Lagrange-Young equation derived?

The Lagrange-Young equation is derived by applying the principle of least action to the Navier-Stokes equations, which states that the actual path of a fluid particle is the one that minimizes the action, or the integral of the Lagrangian, along its trajectory. This results in a set of partial differential equations that govern the motion of the fluid.

3. What are the applications of the Lagrange-Young equation?

The Lagrange-Young equation has a wide range of applications in fluid mechanics, including the study of aerodynamics, oceanography, and weather patterns. It is also used in engineering and design to optimize the performance of fluid systems, such as pumps, turbines, and pipes.

4. How does the Lagrange-Young equation relate to other equations in fluid mechanics?

The Lagrange-Young equation is closely related to other fundamental equations in fluid mechanics, such as the Bernoulli equation and the continuity equation. It is also used in combination with the Navier-Stokes equations to model more complex fluid flows, such as those found in turbulent or multiphase systems.

5. Are there any limitations to the Lagrange-Young equation?

While the Lagrange-Young equation is a powerful tool for studying fluid dynamics, it does have some limitations. It assumes that the fluid is incompressible, inviscid, and irrotational, which may not always be the case in real-world scenarios. Additionally, it does not account for external forces, such as gravity or electromagnetic fields, which may have an impact on the fluid's motion.

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