Fluid Statics and Dynamics: Pressure, Buoyancy, and Flow

Fluid Statics

Force and Pressure

Pressure is a fundamental concept in fluid mechanics, describing the force exerted per unit area. In everyday life, atmospheric pressure acts on every surface, including the human body.

• Pressure (P): Defined as force per unit area, $P = \frac{F}{A}$.

• Force due to Pressure: $F = PA$

• Force due to Pressure Difference: $F = (\Delta P)A$

• Example: Every square meter of your body experiences a force of approximately 101,300 N due to atmospheric pressure. Your cells maintain a similar internal pressure to prevent being crushed.

Measuring Pressure: Tire Gauge and Manometer

Pressure can be measured using various devices, such as tire gauges and manometers.

• Tire Gauge: Measures the gauge pressure inside a tire, which is the difference between the internal and atmospheric pressure.

• Pressure Relationships:

  • $P = P_G + P_{atm}$

  • $P_G = P - P_{atm}$

• Manometer: A device that measures the pressure of a gas by comparing it to atmospheric pressure using a column of liquid. $P = P_{atm} + \rho gh$

Barometer

A barometer is used to measure atmospheric pressure, typically using a column of mercury.

• Principle: The height of the mercury column is proportional to atmospheric pressure.

• Standard Atmospheric Pressure: 760 mm Hg

Units of Pressure

Pressure can be expressed in several units, depending on context and region.

• Common Units:

  Unit	Value in Pa	Equivalent
  atm	1.013 × 105 Pa	1 atm = 1.013 bar
  bar	1.000 × 105 Pa	1 bar = 0.987 atm
  psi	6.90 × 103 Pa	1 atm = 14.7 psi
  mm Hg	1.33 × 102 Pa	1 atm = 760 mm Hg
  torr	1.33 × 102 Pa	1 atm = 760 torr

• Example: Atmospheric pressure is often quoted as 760 mm Hg or 1 atm.

Pressure Difference and Height

Pressure in a fluid varies with height due to the weight of the fluid above.

• Pressure Difference: $\Delta P = \rho_{air} g h$

• Example Calculation: If a barometer reads 760.0 torr on the first floor, the pressure on higher floors (each 4 m higher) decreases by $\Delta P = \rho_{air} g h$.

• Conversion: $1 \text{ torr} = 133 \text{ Pa}$

Pascal's Principle and Hydraulic Systems

Pascal's Principle

Pascal's Principle states that an external pressure applied to a confined fluid is transmitted undiminished throughout the fluid.

• Equation: $P = \rho g h + P_{atm}$

• Application: Used in hydraulic systems to multiply force.

Hydraulic Systems

Hydraulic systems use fluids to transmit force and multiply mechanical advantage.

• Pressure Equality: $P_1 = P_2$

• Force and Area Relationship: $\frac{F_1}{A_1} = \frac{F_2}{A_2}$

• Force Multiplication: $F_2 = F_1 \frac{A_2}{A_1}$

• Example: A small input force applied to a small piston can lift a heavy object using a larger piston.

Buoyancy and Archimedes' Principle

Buoyancy

Buoyancy is the upward force exerted by a fluid on an object immersed in it. This force is responsible for objects floating or sinking.

• Buoyant Force: $F_B = \rho_F g V$

• Pressure at Top and Bottom:

  • $P_{top} = \rho_F g h_1$

  • $P_{bottom} = \rho_F g h_2$

• Force from Pressure Difference: $F = (\Delta P)A = (\rho_F g h)A$

• Equivalence: $F_B = m_F g$ (where $m_F$ is the mass of fluid displaced)

• Archimedes' Principle: The buoyant force on an object is equal to the weight of the fluid displaced by the object.

• Example: A block submerged in water experiences an upward force equal to the weight of the water displaced.

Additional info:

• Some slides reference test and re-test information, which is administrative and not relevant to physics concepts.

• Images illustrate pressure measurement devices and hydraulic systems, supporting the explanations above.