Notes on Fluid Mechanics

Chapter 12: Fluid Mechanics

1. Introduction to Fluid Mechanics

• Fluid Mechanics is the study of fluids (liquids and gases) and the forces acting on them.

  • Fluids can flow and change the shape of the volume they occupy.

2. Properties of Fluids

• Density: An important property of any material, fluid, or solid, defined as mass per unit volume.

  • Density formula: $<br>ho = rac{m}{V}$

  • SI Unit of density: kilograms per cubic meter (kg/m³).

  • Homogeneous materials (e.g., ice, iron) maintain the same density throughout.

  • Float, submerged, sink scenarios based on density:

  • Example: Different masses and volumes can have the same density (e.g., steel wrench and nail).

Densities of Common Substances (Table 11.1)

3. Pressure in Fluids

• Pressure at depth $h$:

  • Formula: $P = <br>ho g h$

• Pascal's Principle: Pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid. This leads to the equation:
  $P_2 = P_1 + <br>ho gh$

• Pressure Measurements:

  • Absolute pressure vs. gauge pressure.

  • Atmospheric pressure at sea level: $P_{atm} = 1.013 imes 10^5 ext{ Pa}$.

Example Problem: Pressure on a Swimmer’s Hand

• Given pressure $P = 1.2 imes 10^5 ext{ Pa}$ and surface area $A = 8.4 imes 10^{-3} m^2$,

  • Calculate the force acting on the hand: $F = PA = (1.2 imes 10^5)(8.4 imes 10^{-3}) = 1.0 imes 10^{3} ext{ N}$.

4. Fluid Flow

4.1 Types of Fluid Flow

• Laminar Flow: Smooth and orderly; adjacent layers slide past one another.

• Turbulent Flow: Chaotic; characterized by eddies and vortices.

• Steady vs. Unsteady Flow: In steady flow, conditions at a point do not change over time, while in unsteady flow, conditions vary.

• Compressible vs. Incompressible Flow: Most liquids are nearly incompressible, whereas gases are compressible.

4.2 Continuity Equation

• For an incompressible fluid, the continuity equation is: $A_1 v_1 = A_2 v_2$ Where:

  • $A$ is the cross-sectional area,

  • $v$ is the flow velocity.

5. Bernoulli's Equation

• Derivation reflecting the conservation of energy in a flowing fluid:
  $P_1 + rac{1}{2} <br>ho v_1^2 + <br>ho gy_1 = P_2 + rac{1}{2} <br>ho v_2^2 + <br>ho gy_2$

• Significance: Relates pressure, velocity, and height at two points along a streamline, assuming incompressible, nonviscous flow.

6. Buoyancy and Archimedes' Principle

• Buoyant Force: An upward force exerted by a fluid opposing the weight of an object submerged in it.

• Archimedes' Principle: An object immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces.

  • Formula: $F_B = <br>ho g V_{displaced}$

7. Viscosity

• Viscosity: A measure of a fluid's resistance to flow.

  • Examples of viscous fluids include honey and motor oil.

• Laminar vs. Turbulent Flow: Laminar flows are characterized by lower viscosity, while turbulent flows involve higher viscosity and chaotic movement.

8. Applications of Fluid Mechanics

8.1 Real-world Implications

• Understanding pressure in deep water scenarios: As depth increases, pressure increases due to the weight of the overlying fluid.

• Applications in aviation: Bernoulli's principle explains how lift is generated on airplane wings.

  • Fast-moving air over the wing creates lower pressure, allowing the airplane to rise.

8.2 Pressure Gauges and Measurements

• Used to measure fluid pressure in various applications, including blood pressure in medicine and tire pressure in vehicles.

• Gauge pressure = absolute pressure - atmospheric pressure.

9. Conclusion

• Fluid mechanics is fundamental in various fields, including engineering, meteorology, and medicine. Mastery of concepts like density, pressure, buoyancy, and flow dynamics is critical for practical applications in science and technology.