The Pressure Excess Notion: Why is the Formula Multiplied by 2?

In summary, the pressure excess notion is a concept used in fluid mechanics to describe the pressure exerted by a fluid on a surface. This notion takes into account both static and dynamic pressures, with the formula multiplied by 2 for a more accurate calculation. Static pressure is the force due to weight, while dynamic pressure is the force due to motion. The pressure excess notion has many real-world applications, but it also has limitations such as assuming incompressibility and not accounting for turbulence or boundary effects.
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kthouz
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in the surface tension about the pressure excess notion i have problem: why for a full dropplet of water the formula is p=2T/r whereas for a bubble p=4T/r. Why does it be multiplied by 2.
 
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In a filled drop there is only a surface. In a bubble there is also the internal surface. When you increase the radius, in the first cas,e only a surface increases. In the second, two surfaces increase.
 
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The reason for the difference in the formula for the pressure excess between a full droplet of water and a bubble is due to the different shapes and surface areas of these two objects. The formula for pressure excess, p=2T/r, is derived from the Young-Laplace equation which relates the pressure difference across a curved interface to the surface tension, T, and the radius of curvature, r.

In the case of a full droplet of water, the surface tension acts on the entire surface area of the droplet, which is equal to 4πr^2. Therefore, when we plug in this value for surface area into the Young-Laplace equation, we get p=2T/r. This is because the surface tension is acting on both the inside and outside surfaces of the droplet, resulting in a pressure difference twice that of a single surface.

On the other hand, for a bubble, the surface tension only acts on the inner surface of the bubble, which has a smaller surface area compared to the full droplet. The outer surface of the bubble is exposed to the surrounding air, so it does not contribute to the pressure difference. Thus, when we plug in the surface area of the inner surface, which is equal to 2πr^2, we get p=4T/r. This is because the surface tension is only acting on one surface, resulting in a pressure difference four times that of a single surface.

In summary, the difference in the formula for the pressure excess between a full droplet of water and a bubble is due to the different surface areas and the number of surfaces that the surface tension is acting on. This is why the formula is multiplied by 2 in the case of a full droplet and by 4 in the case of a bubble.
 

Related to The Pressure Excess Notion: Why is the Formula Multiplied by 2?

1. What is the "pressure excess notion"?

The pressure excess notion is a concept used in fluid mechanics to describe the pressure exerted by a fluid on a surface. It is based on the idea that the pressure at any given point is equal to the sum of the static pressure and the dynamic pressure.

2. Why is the formula multiplied by 2?

The formula for pressure excess is multiplied by 2 because it takes into account both the static and dynamic pressures. By multiplying the formula by 2, it allows for a more accurate calculation of the total pressure exerted by a fluid on a surface.

3. Can you explain the difference between static and dynamic pressure?

Static pressure is the force exerted by a fluid on a surface due to the weight of the fluid itself. Dynamic pressure, on the other hand, is the force exerted by a fluid on a surface due to its motion or velocity. Both static and dynamic pressures are important in calculating the total pressure exerted by a fluid on a surface.

4. How is the pressure excess notion used in real-world applications?

The pressure excess notion is used in many real-world applications, such as in the design of aircraft wings and car aerodynamics. It is also used in the design of pipes and pumps in plumbing systems, as well as in the study of ocean currents and weather patterns.

5. Are there any limitations to the pressure excess notion?

While the pressure excess notion is a useful concept in fluid mechanics, it does have some limitations. It assumes that the fluid is incompressible and inviscid, which may not always be the case in real-world scenarios. Additionally, it does not take into account factors such as turbulence or boundary effects, which can affect the accuracy of calculations.

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