Fluid Mechanics: Kinematics and Streamlines

Kinematics

• Kinematics studies the motion of particles without considering the forces causing it.
• Velocity determines pressure distribution and forces on a fluid.

Streamlines

• A streamline is an imaginary line in a flow field where the tangent represents velocity direction.
• No flow occurs across a streamline.
• The differential equation for a streamline is: dx/u = dy/v = ds/w, where u, v, w are velocity components and ds is a short distance along the streamline.

Path Lines

• A path line is the trajectory of a specific fluid particle.
• It's the curve traced by a single particle as it moves through the flow.
• Path lines can be different from streamlines in unsteady flows.

Streak Lines

• A streak line shows the path of fluid particles passing through a fixed point (marked with dye or smoke).
• It connects all particles passing through that point
• In steady flow, streamlines, path lines, and streak lines coincide.

Instantaneous Streamlines

• Instantaneous streamlines represent the location of streamlines at a particular instant in time.
• In time-dependent flows, streamlines change with time.

Stream Tubes

• Streamlines forming a closed curve create a stream tube.
• Fluid cannot cross the stream tube's boundary.
• Fluid entering and leaving a stream tube must be equal.
• Streamtubes are assumed to have a uniform velocity.

Flow Classifications

• Steady/Unsteady:
  • Steady flow: fluid properties at a point don't change with time (dv/dt = 0, dp/dt = 0, dρ/dt = 0).
  • Unsteady flow: fluid properties change with time (dv/dt ≠ 0, dp/dt ≠ 0, dρ/dt ≠ 0).
• Uniform/Non-uniform:
  • Uniform flow: velocity at a given time doesn't change with space.
• Laminar/Turbulent:
  • Laminar flow: smooth, layer-like movement along streamlines.
  • Turbulent flow: zigzag movement, eddies, and high energy loss. Reynolds number (VD/v) determines type, for pipe flow.
• Compressible/Incompressible:
  • Compressible flow: fluid density changes (ρ ≠ Constant).
  • Incompressible flow: fluid density is constant (ρ = Constant). Liquids are generally incompressible; gases are compressible.
• Rotational/Irrotational:
  • Rotational flow: fluid particles rotate.
  • Irrotational flow: no particle rotation.
• One, Two, Three-Dimensional:
  • One-dimensional: flow parameters are functions of time and one spatial coordinate.
  • Two-dimensional: flow parameters are functions of time and two spatial coordinates.
  • Three-dimensional: flow parameters are functions of time and three spatial coordinates.

Discharge (Rate of Flow)

• Discharge (Q) is the fluid quantity flowing per second.
• For liquids, units are m³/s or L/s.
• For gases, units are kgf/s or N/s.