9. Fluid Mechanics

Welcome to this lecture on fluid mechanics. As we discussed in the last class fluid at rest and fluid statics we have derived basic equations of fluid statics. That is what the pressure equations with related to gravity field.

In this introduction, we delve into the fundamental aspects of fluid mechanics, focusing on fluids at rest, also known as fluid statics. The key equations related to the behavior of fluids under the influence of gravity are derived. Understanding these basics is crucial as they form the foundation for more complex concepts in fluid dynamics.

Think of a calm, glassy lake. The surface of the lake represents a fluid at rest, and the principles of fluid statics govern how pressure is distributed within this body of water.

Now, just we will have two applications of the fluid at rest or hydrostatic pressure distributions. One is manometer. Another is for an inclined surface, a somewhat inclined surface what could be the pressure distributions, what could be the total pressure force acting on that plane, also the center of pressure.

This section discusses two primary applications of fluid statics: manometers and analyzing pressure distributions on inclined surfaces. A manometer is a device used to measure fluid pressure based on hydrostatic principles, while the analysis of inclined surfaces helps determine how pressure acts on sloped planes.

Consider a simple manometer like a glass tube filled with water. When one end of the tube is open to a fluid, the height of the water column will change based on the pressure of that fluid, illustrating how hydrostatic pressure measurements work in practice.

Very simple devices are used to measure the pressure like you have a conduit, the pipes carrying any liquids and you want to measure it what could be the pressure on that pipe.

Manometers are practical instruments employed to gauge the pressure of fluids in conduits. They typically work by observing the height of a liquid column that corresponds to the pressure being exerted by the fluid in the pipe. This simple yet effective measurement technique is a staple in fluid mechanics.

Imagine a straw dipped in a glass of juice. When you cover the top of the straw and lift it out of the glass, some juice stays in the straw due to atmospheric pressure, similar to how a manometer uses liquid columns to measure pressure differences.

You can have a U-shaped manometer; we make it U shape. And then if this is the conditions if you look it that I am to find out what is a pressure is acting on this surface which is a pressure at the centroid of the pipe.

U-shaped manometers are a specific design used for measuring pressure differences between two points. In a U-tube, one side is connected to the fluid in the pipe and the other is open to the atmosphere or another reference pressure. The difference in liquid height between the two sides indicates the pressure differential.

Visualize a seesaw with two children at different heights. The height difference between the two sides tells you how weighted the children are on each side, just like the height in a U-tube indicates pressure differences.

Another case let we have an inclined manometer which has quite an advantage in the sense that it amplifies the pressure reading as compared when you are talking about low pressure ranges.

Inclined manometers allow for more precise measurements of low pressures due to their design, which amplifies the reading. This is particularly beneficial when small differences in pressure must be accurately detected, as the inclination increases the height of the liquid column for the same pressure change.

Think of an inclined slide at a playground. If children slide down at a gentle slope, they take longer to reach the ground. Similarly, an inclined manometer 'stretches out' the height difference for easier reading, perfect for detecting small pressures.

Here, we do not measure a particular point pressure measurements where you consider two points. Because many of our applications, we are not worried about absolute pressure.

Differential manometers focus on measuring the pressure difference between two points rather than measuring the absolute pressure at each point. This is especially useful in cases such as fluid flow in pipes, where knowing the difference between pressures can tell us about the flow dynamics and energy losses.

Picture two balloons of unequal sizes. Instead of looking at the actual size of each balloon, you measure the difference in air pressure between them to understand how much air is needed to balance them—this is similar to how differential manometers work.

If I take the surface of A, I can equate the pressure. Because pressure along the surface will be the same.

In a U-tube manometer, pressure is equated along the same horizontal level because fluids at rest exhibit equal pressure at the same depth. By measuring the height difference of liquids in the two arms of the tube, one can determine the pressure differential between the two points connected to the manometer.

Imagine two friends on a flat table sliding weighted boxes back and forth. If they push with equal force, the boxes will slide together; similarly, the pressure balance in the U-tube ensures measurements are consistent across its surface.

Let us come to a very interesting topic on pressure acting on a submerged plane surface.

Micro manometers represent a refined approach to measuring pressure differences in various fluids, including oils and slurries. They utilize a reservoir system to enhance accuracy in low-pressure conditions, which is particularly useful in industrial applications.

Think of a precise kitchen scale used for measuring small quantities of ingredients. Just as a micro scale helps in getting exact measurements, a micro manometer is engineered to capture minute pressure differences accurately.

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