Fluid Flow 1

Fluid Flow 1

1. Units & Dimensions

   • Dimensions
     • Length
     • Time
     • Mass
     • Temperature
   • All quantities can be derived from the "basic" dimensions
     • Dimensions ARE NOT the same as units!

2. Forces in Fluids

   • Pressure Forces
     • Pressure = Force / Area [Nm^-2]
     • Pascal's Law
       • In the absence of shear stresses, the pressure at any point in a fluid is independent of direction
   • Shear Forces
   • Overall Forces

3. Static Fluids

   • Pressure varies with depth
     • Not along horizontal axis UNLESS container is accelerating
     • Shape of vessel is unimportant
     • P = rho*g*h
       • From dp/dz = -rho*g   When density is constant 
       • Overall Pressure Force on a Surface
         • dF = rho * g * b * y dy
           • Integrate between 0 and (h)eight of wall
           • Moment of a Force
             • dM = rhp * g * b * y^2 dy
               • Result is the height the force acts FROM the surface
   • Vertical Pressure Forces
     • Force due to pressure of water above a horizontal surface
       • The vertical force on a horizontal component equals the weight of fluid above it
     • Buoyancy Forces

4. Pressure Measurement

   • Absolute & Gauge Pressure
     • Absolute Pressure - pressure measured against a vacuum (for example, in an open container filled with static fluid, absolute pressure is the atmospheric pressure + pressure at that depth as result of fluid
     • Gauge pressure - pressure measured using local atmospheric pressure as a reference
     • Pressure is often expressed as a "head" of fluid
   • Barometers
     • Are used to measure local atmospheric pressure By equating the pressures at the level of the free surface... P(atm) = P(vap) + rho*g*h
   • Manometers
     • Allows comparison of pressures e.g calculates pressure DIFFERENCES
     • U-Tube Manometers
       • Use 2 fluids - the pressure difference is the height difference between the two surfaces of the gauge fluid
       • To solve, equate the pressures at two level points on the manometer
         • Pressure (C) = Pressure (D)
           • Hence, P(A) + rho(m)*g*h = P(B) + rho(f)*g*h Where m is density of manometer fluid and f is density of other fluid
           • Tubes on each side may not be the same diameter
             • Equate pressures at a level as normal P(X) = P(X') (1) Then equate volume displaced on LHS with RHS y*pi*D^2 /4 = z*pi*d^2 /4  Hence y = z(d/D)^2 (2) Then substitute (2) back into (1)
             • The advantage of this is that if D is very large compared to d, then the height y moved by the large leg is negligible.  Hence we only have to measure the height z, which is easier to do than measuring both!
           • Inclined Manometers
             • Used because they are more accurate
             • h = x*sin(i)
               • Where h is the pressure head caused by pressure difference, x is the distance moved by the fluid and i is the inclination of the manometer

5. Shear forces

   • Result of tangentially applied forces
     • Fluid: substance which deforms continuously under shear stress
   • No Slip Condition
     • For any fluid flowing past a solid boundary, the fluid next to the boundary moves at the same speed as the boundary
     • Newtonian Fluids
       • As shear stress (tau) increases, the velocity gradient increases. 
       • Tau = mu * d(u) / dy
         • where mu = dynamic viscosity (Ns/m^-2) u = velocity of fluid (average?) y = width of fluid flow
         • V = mu / rho
           • V = kinematic viscosity mu = dynamic viscosity rho = density 
   • Shear Stress = Shear Force / Area

6. Drag Forces

   • Act on objects moving through fluids
     • Act to retard an object's motion
     • Complex combination of pressure forces and shear forces
     • Archimede's Principal
       • Upthrust acting upon an object is equal to the weight of the FLUID displaced by the object
       • U = V*rho*g
   • D = 0.5*Cd*rho*A*u^2
     • where  D = drag force Cd = drag coefficient rho = density  A = area  u = velocity
     • Flow Types
       • Laminar Flow has a higher Cd
       • Turbulent Flow has a much lower Cd