10. Mechanical Properties of Fluids 1

Mechanical Properties of Fluids

Fluid

  • Definition: A fluid is a substance that flows under the action of an applied force and lacks a definite shape.

  • Examples: Liquids and gases.

Fluid Statics

  • Characteristics:

    • Fluid is either at rest or moving uniformly without relative motion between adjacent particles.

    • No shearing stress exists in the fluid; only pressure acts.

    • Pressure is the force that develops on the surfaces of particles.

Pressure

  • Definition: Pressure is the normal force exerted by a fluid per unit area.

  • Units of Measurement:

    • Pascal (Pa) = N/m².

    • Commonly used bigger units: Kilopascal (1 kPa = 10³ Pa) and Megapascal (1 MPa = 10⁶ Pa).

    • Other units: bar, atm, kgf/cm².

  • Formula: P = F/A

Pressure at a Point

  • Properties:

    • The pressure at any point in a fluid is uniform in all directions.

    • Pressure is a scalar quantity (it has magnitude but no specific direction).

Pressure Variation with Depth

  • Concept:

    • A layer of fluid exerts pressure on the layer beneath it due to the weight of the fluid above.

    • Reason for Increased Pressure with Depth: The pressure increases with depth because of the weight of the fluid above.

Blood Pressure

  • Variation: Blood pressure varies with vertical position; for instance, it can be higher in the feet than in the head when standing due to gravitational effects.

Pressure Exerted by a Liquid Column

  • Formula for Pressure:

    • F = mass of liquid in the column of depth h × g = Volume × Density × g

    • F = A × h × ρ × g

  • Pressure Measurement at Depth h:

    • P = F/A = A × h × ρ × g / A = h × ρ × g

    • Conclusively, Pressure ∝ height of fluid column & Density of fluid.

Pressure Difference in Fluid

  • Variation of Liquid Pressure with Depth:

    • Forces: a) Top force (F1) = P1A (downward)b) Bottom force (F2) = P2A (upward)c) Weight of the cylinder (W) = Ahρg (downward)

    • Equilibrium Condition: F1 + W = F2

    • Rearranging gives: P2 - P1 = hρg

Pascal’s Law

  • Statement: A change in pressure applied to an enclosed incompressible fluid is transmitted undiminished to every point of the fluid and its container's walls.

  • Equilibrium for fluid elements: Fa/Aa = Fb/Ab = Fc/Ac or Pa = Pb = Pc.

  • Conclusion: The pressure is exerted equally in all directions.

Applications of Pascal’s Law

  • Hydraulic Lift: Used for lifting heavy objects, where pressure on smaller pistons exerts larger force on bigger pistons.

  • Comparison: Force on larger piston: F = P × A, leading to F > f when A > a.

Hydrostatic Paradox

  • Pressure is independent of container shape when the fluid is at rest.

  • Horizontal pressure in a fluid is the same at all points at the same level.

Viscosity

  • Definition: Viscosity is the measure of a fluid's resistance to flow, indicating internal friction when the fluid is in motion.

  • Viscosity characterizes fluids under motion and varies with temperature.

Coefficient of Viscosity

  • Formula:

    • According to Newton: F = -η A (dv/dx),

    • η = Coefficient of viscosity.

    • SI Units: N.s/m² or Kg/m.s.

    • CGS Units: dyne.s/cm² or poise.

  • Effects of Temperature:

    • With heating, kinetic energy increases, leading to reduced viscosity in liquids.

    • In gases, viscosity rises with temperature due to increased molecular diffusion.

Poiseuille’s Formula

  • Flow Through a Capillary Tube: Q = (πpr⁴)/(8ηℓ), where Q = volume of liquid flowing per second.

Stokes’ Law

  • The viscous force F opposing the motion of a sphere in a viscous fluid is given as: F = 6πηrv, where r is the radius and v is the velocity.

Terminal Velocity

  • When a body is dropped in a viscous fluid, it first accelerates, then reaches constant velocity (terminal velocity).

  • Forces at Terminal Velocity:

    • Weight upward thrust from fluid (U) and backward viscous force (F) must balance at this velocity.

Flow Types: Laminar versus Turbulent

  • Laminar Flow: Characterized by orderly layers, occurs at lower velocities typically in high-viscosity fluids.

  • Turbulent Flow: Irregular motion at high velocities with fluctuating forces, causing vortex formations.

  • Transitional Flow: Alternates between laminar and turbulent states.

Streamline Flow

  • Defined as the flow where fluid particles follow path lines; characterized by smooth conditions with no intersection of streamlines.

  • Properties:

    • Fluid velocity can be constant along streamlines.

    • More streamlines indicate higher fluid velocity.

Critical Velocity

  • The limiting velocity that differentiates between laminar flow and turbulent flow.

  • Influencing factors:

    • Directly proportional to fluid's viscosity,

    • Inversely proportional to fluid density and tube diameter.

Bernoulli’s Principle

  • States that the total pressure, kinetic energy, and potential energy per unit volume remains constant in streamlined flow.

  • Relates to variations in area and flow velocity across a streamline.

  • Applications: Used in devices to measure fluid flow like Venturimeter and in aerodynamics with aerofoils.