Hydrodynamics Overview

Hydrodynamics

• Fluid flow can be characterized as streamline (laminar) or turbulent.
• Streamline flow: Each particle follows the same smooth path as previous particles.
• Streamlines do not cross each other.
• Streamline direction coincides with the fluid velocity at that point.
• Turbulent flow: Irregular or chaotic flow, above a certain velocity or under conditions causing abrupt changes in velocity.
• Eddy currents are characteristic of turbulent flow.
• Viscosity: The degree of internal friction in a fluid.
• Viscosity is the resistance between adjacent fluid layers moving relative to each other.
• Fluids with lower viscosity flow more easily (e.g., kerosene).
• Ideal fluid conditions:
  • Non-viscous: No internal friction between adjacent layers.
  • Incompressible: Constant density.
  • Steady flow: Velocity, density, and pressure at any point do not change with time.
  • No turbulence: No eddy currents.

Steady Flow of Liquids

• Volume flow rate through different cross-sections of a pipe is constant if the fluid is incompressible.
• A1v1 = A2v2
• The volume of fluid entering one end of a tube equals the volume leaving in the same time interval (no leaks).
• The equation is called the equation of continuity.

Kinetic Energy of a Liquid

• A moving fluid possesses kinetic energy (1/2mv2).
• Work done to move fluid to a pipe = difference of pressure × volume of fluid.
• This work gives the fluid its kinetic energy.
• Fluid velocity coming out of a tank is the same as if the fluid fell through the same height.
  • v² = 2gh

Bernoulli's Equation

• Relates pressure, speed, and elevation in a flowing fluid.
• A consequence of energy conservation (ideal fluids).
• The sum of pressure, kinetic energy per unit volume, and potential energy per unit volume is constant along a streamline.
  • (1/2)pv² + pgh + p = constant
• Assumes steady, incompressible, and inviscid flow.

Applications of Bernoulli's Equation

• Vascular flutter: In arteriosclerosis, plaque constricts arteries, forcing higher blood speeds, potentially lowering pressure, and collapsing the artery.

Viscosity

• Viscosity is the internal friction of a fluid, resisting the movement of one layer over another.
• Non-viscous fluids: All fluid particles move at the same velocity through a pipe.
• Viscous fluids: The fluid's velocity is maximum at the centre of the pipe and zero at the pipe walls; layers move at different velocities.

Flow of Fluids through Narrow Tubes

• Laminar flow: Velocity profile depends on the radius of the tube.
  • Velocity is greatest at the center and zero at the pipe wall.
• Poiseuille's Law: Flow rate is inversely proportional to fluid viscosity but directly proportional to pressure difference and the fourth power of pipe radius.
  • V = (π/8)(P₁ - P₂)(r⁴)/ηL

Motion Through a Viscous Medium (Stokes' Law)

• Stokes' Law describes the resistive force on a sphere moving slowly in a viscous fluid.
• The resistive force = 6πηrv.
• Terminal velocity: Speed when the resistive force equals the weight and buoyant force.
• Terminal velocity depends on size, density of the object and the surrounding fluid (viscosity and density).

Kinematic Viscosity

• Ratio of dynamic viscosity to density.
• V = µ/ρ

Reynolds Number

• Indicates the flow type based on factors like fluid velocity, viscosity, density, and pipe radius.
• Re = (ρVD)/µ.
• Low Re: Laminar flow.
• High Re: Turbulent flow.