Fluids Summary

• Fluids are states of matter characterized by a lack of long range order. They are characterized by their density $\rho$ and their compressibility. Liquids such as water are (typically) relatively incompressible; gases can be significantly compressed. Fluids have other characteristics, for example viscosity (how ``sticky'' the fluid is). We will ignore these in this course.

• Pressure is the force per unit area exerted by a fluid on its surroundings:
  

  $$P = F/A$$ (8.1)

  
  
  Its SI units are pascals where 1 pascal = 1 newton/meter squared. Pressure is also measured in ``atmospheres'' (the pressure of air at or near sea level) where 1 atmosphere $\approx 10^5$ pascals. The pressure in an incompressible fluid varies with depth according to:
  

  $$P = P_0 + \rho g D$$ (8.2)

  
  
  where $P_0$ is the pressure at the top and $D$ is the depth.

• Pascal's Principle Pressure applied to a fluid is transmitted undiminished to all points of the fluid.

• Conservation of Flow We will study only steady/laminar flow in the absence of turbulence and viscosity.
  

  $$A_1 v_1 = A_2 v_2$$ (8.3)

  
  

• Bernoulli's Equation
  

  $$P + \frac{1}{2}\rho v^2 + \rho g h = {\rm constant}$$ (8.4)

  
  

• Toricelli's Rule: If a fluid is flowing through a very small hole (for example at the bottom of a large tank) then the velocity of the fluid at the large end can be neglected in Bernoulli's Equation.

• Archimedes' Principle The buoyant force on an object
  

  $$F_b = \rho g V_{\rm disp}$$ (8.5)

  
  
  where frequency $V_{\rm disp}$ is the volume of fluid displaced by an object.

• Venturi Effect The pressure in a fluid increases as the velocity of the fluid decreases. This is responsible for e.g. the lift of an airplane wing.