Fluid Dynamics Summary

Fluid Dynamics

  • Study of fluids in motion.

Conservation Laws

  • Mass: Governed by the Continuity Equation.

  • Energy: Governed by Bernoulli's Principle.

Conservation of Mass and Fluid Flow Rate

  • Mass is conserved in a fluid system.

  • Liquid (e.g., water) is treated as incompressible (ideal fluid).

  • Volume is conserved, so fluid flow rate is constant:

    • Fluid Flow Rate: Q = \text{volume/time}

    • Units: \text{m}^3/\text{s}

Continuity Equation Derivation

  • Fluid Flow Rate is constant: Q = Q1 = Q2.

  • A1ν1 = A2ν2 (where A is area and ν is speed).

Example Problem

  • If blood enters a vessel segment with diameter 1.6 mm and speed u, find exit speed from 0.4 mm diameter segments:

    • Choices: 2v, 4v, 8v, 16v, where v is the inlet speed.

Terms to Remember

  • Conservation of Energy: Total mechanical energy remains constant unless affected by external forces.

  • Kinetic Energy: Energy due to motion.

  • Gravitational Potential Energy: Energy due to an object's position in a gravitational field.

  • Work: Energy transfer by external forces.

  • Pressure defined as Force/Area.

Bernoulli's Equation

  • Represents conservation of energy as a principle in fluid dynamics:

    • P1 + \rho g y1 + \frac{1}{2} \rho v1^2 = P2 + \rho g y2 + \frac{1}{2} \rho v2^2 (where P is pressure, (\rho) is density, g is acceleration due to gravity, v is velocity, y is height).

Toricelli's Theorem states that the speed of efflux of a fluid under the force of gravity through an orifice is given by the equation:
v = ext{sqrt}(2g h)
where:

  • v is the speed of the fluid as it exits the orifice,

  • g is the acceleration due to gravity,

  • h is the height of the fluid above the opening.
    The theorem highlights that the speed of fluid exiting is determined by the height of the fluid column above the opening, not by the volume of the fluid or the size of the orifice. It is a practical application of Bernoulli's principle and is useful in calculating fluid flows in various engineering applications, including tanks and pipes.