Fluid Mechanics

Fluid Mechanics: Lecture Notes

Laminar and Turbulent Flow

  • Laminar Flow: Characterized by smooth and orderly motion of the fluid.
  • More internal friction leads to increased resistance to mixing.
    • Example: Water in rivers seldom exhibits laminar flow due to turbulence and other influences.
    • For laminar flow, thickness of flow layers is minimal to ensure smooth transitions.
  • Turbulent Flow: Typically appears in larger, more chaotic conditions such as rivers.

Shear Stress in Fluids

  • Shear stress (τ\tau) is defined as the force per unit area exerted by one layer of fluid on another.
    • SI units: Newtons per square meter (N/m²).
    • Calculation: Shear Force ($F_s$) = Shear Stress (τ\tau) × Area ($A$).
  • A moving plate will affect the fluid under it, exerting forces and creating shear stress in the fluid and vice versa.

Boundary Conditions

  • No-slip Boundary Condition: Fluid in contact with a surface has zero velocity relative to the surface; thus, the velocity at the bottom layer is at rest while the top layer moves at a velocity $u$.
  • The velocity profile between layers is often assumed to be linear, unless otherwise specified.
    • It is important for performing calculations concerning fluid flow and shear stresses.

Example Problem with Slopes and Fluids

  • Setup involves a block on a slope with a thin layer of fluid.
    • Given parameters: weight of block, fluid properties, contact area, and slope.
  • Forces acting on the block include:
    • Downhill force due to gravity: $F_g = m imes g$ (component along slope).
    • Shear force due to viscosity of the fluid reducing friction.
    • Need to establish a force balance:
    • ΣF=F<em>g−F</em>shear=0\Sigma F = F<em>g - F</em>{shear} = 0 when terminal velocity is reached.

Shear Force Calculation

  • Shear stress at the slope can be expressed as:
    • τs=μdudy\tau_s = \mu \frac{du}{dy}, where $du/dy$ is the velocity gradient.
    • A linear profile yields: aus=μuhau_s = \mu \frac{u}{h}, where $h$ is the fluid layer thickness.

Viscosity Measurement Devices

  • Various setups use a cylinder within a fluid-filled container to measure viscosity.
  • Resistance to spinning of the cylinder relates back to the fluid’s viscosity being tested.

Forces in Fluid Mechanics

  • Shearing Forces: Caused by the motion of fluids and affect surfaces in contact.
  • Normal Forces: Act perpendicular to the surfaces and described as pressure forces.
    • Similar in action to shearing forces but operate statically in fluids at rest, characterized mainly by pressure.

Pressure Fundamentals

  • Pressure (P):
    • Defined as force per unit area, thus SI units: N/m², or Pascals (Pa).
    • Variations in pressure depend on gravitational forces acting on fluid columns above any point.
  • Types of Pressure:
    • Absolute Pressure: Measured from a vacuum (zero pressure reference).
    • Gauge Pressure: Measured relative to atmospheric pressure (can be negative).

Pressure Derivations and Calculations

  • Using small fluid prisms to balance forces based on pressure and gravity.
  • Balancing pressure forces and weight leads to the expression:
    • dPdz=−γ\frac{dP}{dz} = -\gamma (where γ\gamma is specific weight).
  • Integration of the equation provides relationships for pressure differences in static fluids:
    • P<em>2=P</em>1+γ(z<em>1−z</em>2)P<em>2 = P</em>1 + \gamma (z<em>1 - z</em>2).

Hydrostatic Pressure Distribution

  • Pressure increases linearly with depth in a fluid at rest:
    • The change in pressure is directly proportional to the fluid’s weight above that point.
  • Integration allows determination of pressures at different depths, which can be applied in various real-world situations.

Manometers

  • Manometers are used for measuring fluid pressures by readjusting pressures through a gauge fluid.
  • The difference in heights of fluids indicates pressure changes in relation to known pressures in setup conditions.

Pascal's Law

  • Explains that pressure applied at any point in a closed fluid system is transmitted undiminished and isotropically throughout the fluid.
  • Hydraulic systems utilize this principle to amplify forces: small force applied on small piston translates to much larger force exerted on a larger piston or surface.

Summary

  • Understanding laminar vs. turbulent flows, shear forces, normal forces, pressure definitions, and how to balance forces through integration is foundational in fluid mechanics, crucial for applications in engineering and environmental sciences.