5.2 Fluid statics

Fluid statics is all about understanding how liquids and gases behave when they're not moving. It's crucial for figuring out things like water pressure in pipes or how ships float. This topic covers key concepts like hydrostatic pressure, buoyancy, and pressure measurement.

In this part of fluid mechanics, we'll learn about Pascal's law, Archimedes' principle, and how to use manometers. These ideas are super important for designing everything from dams to submarines, and they'll help us solve real-world engineering problems.

Hydrostatic Pressure and Depth

Hydrostatic Pressure Principles

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• Hydrostatic pressure is the pressure exerted by a fluid at rest due to the weight of the fluid above it
• Hydrostatic pressure acts equally in all directions at a given depth in a fluid (isotropic)
• The hydrostatic pressure at a given depth is independent of the shape or size of the container holding the fluid (Pascal's law)
• Hydrostatic pressure is a scalar quantity, having magnitude but no specific direction

Relationship Between Hydrostatic Pressure and Depth

• Hydrostatic pressure increases linearly with depth in a fluid
• The relationship is given by the equation $P = \rho gh$, where:
  • $P$ is the hydrostatic pressure
  • $\rho$ is the fluid density
  • $g$ is the acceleration due to gravity
  • $h$ is the depth below the surface
• The hydrostatic pressure difference between two points in a fluid is proportional to the vertical distance between the points and the density of the fluid
• Example: The pressure at the bottom of a swimming pool (3 m deep) is greater than the pressure at a depth of 1 m in the same pool

Force and Center of Pressure

Hydrostatic Force on Submerged Surfaces

• The hydrostatic force on a submerged surface is the resultant force exerted by the fluid pressure acting on the surface
• The magnitude of the hydrostatic force on a submerged surface is equal to the product of the fluid pressure at the centroid of the surface and the area of the surface, $F = PA$, where:
  • $F$ is the hydrostatic force
  • $P$ is the fluid pressure at the centroid
  • $A$ is the area of the surface
• The direction of the hydrostatic force on a submerged surface is always perpendicular to the surface, pointing from the high-pressure side to the low-pressure side
• Example: The force on a submerged dam wall is perpendicular to the wall surface and increases with depth

Center of Pressure

• The center of pressure is the point on a submerged surface where the resultant hydrostatic force acts
• The location of the center of pressure depends on the shape and orientation of the submerged surface and the pressure distribution acting on it
• For a vertically oriented rectangular surface, the center of pressure is located below the centroid of the surface
• The vertical distance of the center of pressure from the centroid is given by the equation $y_p = (I_c / Ah_c) + h_c$, where:
  • $y_p$ is the distance of the center of pressure from the surface
  • $I_c$ is the second moment of area about the centroidal axis
  • $A$ is the area of the surface
  • $h_c$ is the depth of the centroid below the surface
• Example: The center of pressure on a vertically submerged rectangular gate is always below the centroid of the gate

Buoyancy and Archimedes' Principle

Buoyancy Concept

• Buoyancy is the upward force exerted by a fluid on an object immersed in it, which opposes the weight of the object
• The magnitude of the buoyant force is given by the equation $F_b = \rho gV$, where:
  • $F_b$ is the buoyant force
  • $\rho$ is the density of the fluid
  • $g$ is the acceleration due to gravity
  • $V$ is the volume of the fluid displaced by the object
• An object will float in a fluid if the buoyant force is equal to the weight of the object, sink if the weight is greater than the buoyant force, and remain neutrally buoyant if the weight and buoyant force are equal
• The stability of a floating object depends on the relative positions of its center of gravity and center of buoyancy

Archimedes' Principle and Applications

• Archimedes' principle states that the buoyant force acting on an object immersed in a fluid is equal to the weight of the fluid displaced by the object
• Archimedes' principle has various applications, such as in the design of ships, submarines, and hot air balloons
• It is also used in determining the density of objects and fluids (hydrostatic weighing)
• Example: A ship floats because the weight of the water it displaces is equal to the ship's weight
• Example: A helium balloon rises because the buoyant force exerted by the displaced air is greater than the balloon's weight

Manometers and Pressure Measurement

Manometers

• Manometers are devices used to measure the pressure difference between two points in a fluid system
• A simple U-tube manometer consists of a U-shaped tube filled with a liquid (mercury or water), with one end connected to the point of unknown pressure and the other end open to the atmosphere or a reference pressure
• The pressure difference between the two points is determined by the height difference of the liquid columns in the manometer and the density of the manometer fluid, using the equation $P_1 - P_2 = \rho gh$, where:
  • $P_1$ and $P_2$ are the pressures at the two points
  • $\rho$ is the density of the manometer fluid
  • $g$ is the acceleration due to gravity
  • $h$ is the height difference between the liquid levels
• Other types of manometers include inclined manometers (for measuring smaller pressure differences) and differential manometers (for directly measuring the pressure difference between two points)

Pressure Measurement Devices

• Pressure gauges, such as Bourdon tubes and diaphragm gauges, are commonly used to measure pressure in fluid systems
• When solving problems involving manometers and pressure gauges, consider the reference pressure (usually atmospheric pressure), the density of the manometer fluid, and the height differences between the liquid levels or the deflection of the pressure gauge
• Example: A U-tube manometer connected to a gas pipeline can measure the pressure difference between the pipeline and the atmosphere
• Example: A Bourdon tube pressure gauge can measure the absolute pressure in a compressed air tank