9.7. SUMMARY

1. The basic property of a fluid is that it can flow. The fluid does not have any resistance to change of its shape. Thus, the shape of a fluid is governed by the shape of its container.

Fluids, unlike solids, can change shape easily. This means they take the shape of the container they're in. For example, if you pour water into a glass, the water changes its shape to match the glass.

Think of a cloud or smoke; they continuously change shape depending on the wind and surrounding conditions, demonstrating the property of fluidity.

2. A liquid is incompressible and has a free surface of its own. A gas is compressible and it expands to occupy all the space available to it.

Liquids do not change their volume easily when pressure is applied, which is why they are considered incompressible. Gases, however, can compress and expand, meaning they can fill any container completely.

Consider filling a balloon with air. As you blow more air into the balloon, it expands, demonstrating the compressibility of gases. On the other hand, if you try to compress a bottle full of water, you will find it nearly impossible, showcasing how liquids remain the same volume.

3. If F is the normal force exerted by a fluid on an area A then the average pressure Pav is defined as the ratio of the force to area Pav = F/A.

Pressure is defined as the force applied per unit area. For instance, if you push your hand against a wall, the pressure exerted is calculated by dividing the force of your hand by the area of your hand that contacts the wall.

Imagine standing on a soft surface like sand. If you stand still, you create pressure with your weight. But if you lie down, you spread out your weight over a larger area, reducing the pressure on the sand so you won’t sink.

4. The unit of the pressure is the pascal (Pa). It is the same as N m-2. Other common units of pressure are 1 atm = 1.01 × 10^5 Pa, 1 bar = 10^5 Pa, 1 torr = 133 Pa = 0.133 kPa, 1 mm of Hg = 1 torr = 133 Pa.

Pressure can be measured using various units, with the pascal being the SI unit. To put it simply, 1 Pascal is defined as the pressure exerted by a force of one newton acting over an area of one square meter.

When measuring air pressure in weather forecasts, you’ll often see it given in millibars or atmospheres. These units help us understand not just pressure but also its effects, like how the weather develops based on atmospheric pressure.

5. Pascal’s law states that: Pressure in a fluid at rest is the same at all points which are at the same height. A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the walls of the containing vessel.

Pascal's Law indicates that in a closed system, if you apply pressure at any point, that pressure is transmitted equally throughout the entire fluid. This is important for understanding how hydraulic systems work.

Think of squeezing a tube of toothpaste. When you apply pressure to the tube, toothpaste comes out evenly at the opening. This is similar to how pressure is transmitted in hydraulics.

6. The pressure in a fluid varies with depth h according to the expression P = Pa + ρgh, where ρ is the density of the fluid, assumed uniform.

This equation shows that the pressure increases with depth due to the weight of the fluid above. As you go deeper into a pond or the ocean, pressure increases because of the additional water above you.

When you swim, you might feel pressure building on your body as you dive deeper. This pressure is due to the weight of water above you, demonstrating how depth affects pressure in fluids.

7. The volume of an incompressible fluid passing any point every second in a pipe of non-uniform cross-section is the same in the steady flow. vA = constant (v is the velocity and A is the area of cross-section).

This principle helps us understand how fluids behave in pipes and tubes. If a pipe narrows at one point, the fluid must flow faster to keep the volume constant, showing an inverse relationship between area and velocity.

Imagine a garden hose. When you place your thumb over the end, the opening becomes smaller, making the water flow out faster. This is a practical demonstration of how a change in cross-sectional area affects fluid velocity.

8. Bernoulli’s principle states that as we move along a streamline, the sum of the pressure (P), the kinetic energy per unit volume (ρv2/2), and the potential energy per unit volume (ρgy) remains a constant.

Bernoulli's equation is a statement of the conservation of energy for flowing fluids. It implies that when a fluid speeds up, its pressure decreases, and when it slows down, its pressure increases.

When you blow air across the top of an empty paper cup, the cup might be pulled upward. This happens because fast-moving air creates a low-pressure area above the cup, demonstrating Bernoulli's principle.

9. Though shear strain in a fluid does not require shear stress, when a shear stress is applied to a fluid, the motion is generated which causes a shear strain growing with time. The ratio of the shear stress to the time rate of shearing strain is known as coefficient of viscosity, η.

Viscosity is a measure of a fluid's resistance to flow. When a force is applied to a fluid, it can change shape (strain), and the coefficient of viscosity quantifies this relationship.

Think of honey compared to water. Honey is thick and flows slowly, while water is thin and flows easily. This difference in flow ability is due to varying viscosities.

10. Stokes’ law states that the viscous drag force F on a sphere of radius a moving with velocity v through a fluid of viscosity is, F = 6 πηav.

This law provides a way to calculate the drag force experienced by objects moving through a fluid. The force is dependent on the speed of the object, the viscosity of the fluid, and the size of the object.

Picture a ball falling through water. A larger or faster ball will experience more drag compared to a smaller, slower ball due to Stokes’ law.

11. Surface tension is a force per unit length (or surface energy per unit area) acting in the plane of the interface between the liquid and the bounding surface. It is the extra energy that the molecules at the interface have as compared to the interior.

Surface tension causes the surface of a liquid to behave like a stretched elastic membrane. It is why small objects can float on the surface of water even if they are denser than water.

Think of a needle that can float on the surface of water despite being metal. This is due to surface tension, which acts to hold the liquid together at the surface.

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