Pressure varies depending on the depth. Because of this, the fluid force acting on the object differs depending on the depth; hence, the computation for the total force will require integration.
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In this post, we'll explore how to find the total force acting on an object in a fluid experiencing varying forces.

Varying Fluid Force

Let's say we have a submerged plate oriented vertically in a fluid with uniform unit weight. In addition, let's also consider its pressure solid. From this graph, pressure increases from the top of the object to its bottom. It implies that pressure varies depending on the depth.

To find the total force \(F\) acting on the body, it is \(F=\gamma h A\); however, because pressure varies, the fluid force on the bottom of the plate is not the same as that of the top.

Approximation

Approximating the fluid force exerted on an object

To compute the total force, we can approximate it. We divide the pressure solid into multiple solid sections. This would result to submerged cross-sectional areas \(A_1\), \(A_2\), \(A_n\). From this, we establish the heights \(h_1\), \(h_2\), \(h_n\) of these areas from a reference. We can compute the fluid force for each division as follows:

\(F_n = \gamma h_n A_n\)

The total force is then the sum of these individual fluid forces. If we want a better approximation, we increase the number of solid sections.

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Exact Solution

Finding the exact fluid force acting on an object

Borrowing this idea from approximation, we can determine the pressure solid's total fluid force using integration. If we consider multiple differential fluid force \(dF\) and sum all of these, we have the total fluid force \(F\) as:

\(F = \int dF\)

  • \(F\) is the total fluid force
  • \(dF\) is the differential fluid force
The differential fluid force

Let's expand on \(dF\), it is equal to:

\(dF = \gamma h dA\)

As a result, \(F\) is equal to:

\(F = \int_{a}^{b} \gamma h dA\)

  • \(F\) is the total fluid force
  • \(a\) is the lower limit (origin - fluid surface)
  • \(b\) is the upper limit (total depth of surface)
  • \(\gamma\) is the unit weight of the fluid; If the unit weight varies per height, it must be a math function of \(y\)
  • \(h\) is the height from the fluid surface. Usually, a math function of \(y\)
  • \(dA\) is the differential area.

The differential area \(dA\) will depend on the surface we want to investigate. Generally, it is equal to:

\(dA = L dy\)

  • \(dA\) is the differential area of the submerged surface
  • \(L\) is the length of the surface. Must be a function of \(y\)
  • \(dy\) is the height dependent variable

These equations are the general formula for the total fluid force acting on the submerged project. It is the equation we use to solve for the fluid force of any pressure solid.

Summary

It implies that pressure varies depending on the depth. Because of this, the fluid force acting on the object differs depending on the depth.
The general expression for finding such an object is \(F = \int_{a}^{b} \gamma h dA\). \(F\) is the total fluid force, \(a\) is the lower limit (origin - fluid surface), \(b\) is the upper limit (total depth of surface), \(\gamma\) is the unit weight of the fluid; If the unit weight varies per height, it must be a math function of \(y\), \(h\) is the height from the fluid surface - usually, a math function of \(y\), and \(dA\) is the differential area.
The differential area is \(dA = L dy\). \(dA\) is the differential area of the submerged surface, \(L\) is the length of the surface - it must be a function of \(y\), and \(dy\) is the height dependent variable

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Created On
June 5, 2023
Updated On
February 23, 2024
Contributors
Edgar Christian Dirige
Founder
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