Chapter 6: Fluid Dynamics

• Fluid

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  • Substance that can flow (liquids and gases).
  • No definite shape.

• Density ($\rho$)

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  • Mass per unit volume.
  • $\rho = \frac{m}{V}$.
  • Unit: $kg/m^3$.
  • Dimensions: $[ML^{-3}]$.
  • Scalar quantity.

• Pressure (P)

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  • Force per unit area perpendicular to the surface.
  • $P = \frac{F}{A}$.
  • Unit: Pascal (Pa) = $N/m^2$.
  • Dimensions: $[ML^{-1}T^{-2}]$.
  • Scalar quantity.

  • Pressure in Fluids

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    • Pressure at depth $h$: $P = \rho gh$.
    • Total pressure at depth $h$: $P_{total} = P_{atm} + \rho gh$.
    • Pressure is same at same horizontal level in a static fluid.

  • Atmospheric Pressure

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    • Pressure exerted by the weight of the atmosphere.
    • Standard atmospheric pressure: $1~atm = 1.013 \times 10^5~Pa = 760~torr = 760~mmHg$.
    • Measured by barometer.

  • Gauge Pressure

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    • Difference between absolute pressure and atmospheric pressure.
    • $P_{gauge} = P_{abs} - P_{atm}$.

• Pascal's Principle

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  • Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel.
  • Hydraulic systems (e.g., hydraulic lift, brakes) work on this principle.
  • $\frac{F_1}{A_1} = \frac{F_2}{A_2}$.
  • Mechanical Advantage of hydraulic lift: $MA = \frac{F_2}{F_1} = \frac{A_2}{A_1}$.

• Archimedes' Principle (Buoyancy)

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  • An object wholly or partially immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced.
  • Buoyant Force ($F_b$) = Weight of displaced fluid = $\rho_f V_f g$.
  • $\rho_f$: density of fluid.
  • $V_f$: volume of displaced fluid.

  • Floating and Sinking

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    • Float: $F_b \ge W_{object}$ or $\rho_{object} \le \rho_{fluid}$.
    • Sink: $F_b < W_{object}$ or $\rho_{object} > \rho_{fluid}$.

  • Apparent Weight

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    • Apparent Weight = Actual Weight - Buoyant Force.
    • $W_{apparent} = W_{actual} - F_b$.

• Fluid Flow

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  • Ideal Fluid Characteristics

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    • Non-viscous (no internal friction).
    • Incompressible (density constant).
    • Steady flow (velocity at any point constant over time).
    • Irrotational flow (no turbulence, no swirling).

  • Streamline Flow (Laminar Flow)

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    • Smooth, orderly flow.
    • Fluid particles follow paths that do not cross.

  • Turbulent Flow

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    • Irregular, chaotic flow with eddies and swirls.

• Equation of Continuity

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  • For an incompressible, steady flow, the mass flow rate is constant.
  • $A_1v_1 = A_2v_2 = \text{constant}$ (Volume flow rate).
  • $Av = \text{constant}$.
  • $A$: cross-sectional area.
  • $v$: fluid speed.
  • Implies: where area is smaller, speed is greater.

• Bernoulli's Equation

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  • For an ideal fluid in steady flow, the sum of pressure, kinetic energy per unit volume, and potential energy per unit volume is constant along a streamline.
  • $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$.
  • $P$: static pressure.
  • $\frac{1}{2}\rho v^2$: dynamic pressure.
  • $\rho gh$: hydrostatic pressure.

  • Applications of Bernoulli's Principle

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    • Venturi Effect: Fluid speed increases as cross-sectional area decreases, leading to a drop in pressure.
    • Venturi Relation (from Bernoulli's): $P_1 - P_2 = \frac{1}{2}\rho (v_2^2 - v_1^2)$ (for horizontal flow).
    • Aerodynamics (Lift): Airfoil shape causes faster air flow over top surface, lower pressure, resulting in upward lift.
    • Blood Flow: Narrowed arteries increase blood velocity, decreasing pressure and potentially causing collapse.
    • Spray Guns/Carburetors: High-speed air creates low pressure, drawing liquid into the air stream.

• Viscosity ($\eta$)

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  • Measure of a fluid's resistance to flow (internal friction).
  • Unit: Pascal-second (Pa s) or Poise (P). $1~Pa~s = 10~Poise$.
  • For liquids, viscosity decreases with temperature.
  • For gases, viscosity increases with temperature.

  • Stokes' Law

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    • Drag force on a spherical object moving through a viscous fluid.
    • $F_d = 6\pi\eta rv$.
    • $r$: radius of sphere.
    • $v$: speed of sphere.

  • Terminal Velocity ($v_t$)

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    • Constant velocity achieved by an object falling through a fluid when drag force equals gravitational force (and buoyant force).
    • $v_t = \frac{2r^2g(\rho_s - \rho_f)}{9\eta}$.
    • $\rho_s$: density of sphere.
    • $\rho_f$: density of fluid.

• Fluid Dynamics in Real World

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  • Applications in engineering (e.g., pipe design, aerodynamics).
  • Biological systems (e.g., blood circulation).
  • Meteorology (e.g., weather patterns).

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