ME 310 Chap.6 Fluid Kinematics and Control Volume Analysis
Fluid Kinematics and Control Volume Analysis
Fields, not particles
The kinematics lecture, titled on the calendar as fields not particles, changes how flow is described. Instead of tracking individual particles, the velocity is written as a field that gives the value at every fixed point and time.
Newton's law still needs the acceleration of a particle, so the material derivative rebuilds it from the field: a local part, the change at a fixed point, and a convective part, the change a particle sees by moving to where the velocity differs.
A steady flow through a nozzle has no local acceleration, yet every particle speeds up.
Flow visualisation
Streamlines, pathlines and streaklines are three different lines.
In steady flow they coincide; in unsteady flow a dye streak can look nothing like the path any single particle followed.
The Reynolds transport theorem
Conservation laws belong to a fixed mass of fluid, but devices are easier to analyse as control volumes that fluid passes through.
The transport theorem connects the two: the rate of change for the system equals storage inside the control volume plus net outflow through its surface.
Choosing the transported property gives mass, momentum or energy conservation in control-volume form.
Finite control volume examples
The calendar then works finite control-volume examples for mass and linear momentum, a lecture on moving control volumes, and energy for one inlet and one outlet. Mass conservation balances inflow, outflow and storage.
The momentum equation turns the change in momentum flux into the force on a nozzle, bend, vane or plate. A moving control volume uses velocities relative to it. The energy equation restores shaft work and losses to Bernoulli and is the tool for pumps and turbines.
Exam II covers textbook chapters 4 to 7, so this material appears there and on the final.
Sign conventions that save points
Pick positive directions once, draw them on the sketch, and give every velocity and every force a sign from that drawing. Outflow through a control surface is positive and inflow negative in the flux terms.
What this chapter covers
- 01
Eulerian velocity fields
- 02
Local and convective acceleration
- 03
Streamlines, pathlines and streaklines
- 04
System versus control volume
- 05
Reynolds transport theorem
- 06
Mass conservation with storage
- 07
Linear momentum on fixed and moving control volumes
- 08
Energy equation with pump head and losses
Worked example · free
Force on a moving vane
- 2Attach the control volume to the vane; the relative speed is W = 25 − 10 = 15 m/s in and out.
- 2Mass flow through the moving control volume: ρ A W = 1000 × 5.0 × 10⁻⁴ × 15 = 7.5 kg/s.
- 2Momentum in the jet direction: F = ṁ W (1 − cos 120°) = 7.5 × 15 × 1.5.
- 1F = 169 N on the vane, in the direction of the jet.
Key terms
- Material Derivative
- The material derivative is the rate of change of a quantity following a fluid particle, combining local and convective change.
- Control Volume
- A control volume is a chosen region of space through whose surface fluid flows, used to apply conservation laws to devices.
- Reynolds Transport Theorem
- The Reynolds transport theorem relates the rate of change of a property for a system to storage and net outflow for a control volume.
- Momentum Flux
- Momentum flux is the rate at which momentum is carried across a surface by the flow, mass flow rate times velocity for uniform flow.
Fluid Kinematics and Control Volume Analysis FAQ
Why can a steady flow still have acceleration?
Steady means the velocity at each fixed point does not change with time. A particle travelling through a converging section still moves into regions of higher speed, so it accelerates convectively even though the field is frozen.
How should I choose a control volume?
Cut the flow where velocities and pressures are known or wanted, cut through any solid whose force you need, and keep surfaces perpendicular to the flow where possible so the flux terms are simple products.
When do I use the energy equation instead of Bernoulli?
Whenever a pump or turbine adds or removes shaft work, or when friction losses between the two points are not negligible. The energy equation adds a pump or turbine head and a head-loss term to the Bernoulli terms.
How do I find the force a fluid exerts on a pipe bend?
Draw a control volume around the fluid in the bend, apply the momentum equation in each direction including pressure forces on the cut sections, solve for the force the bend exerts on the fluid, and reverse its sign for the force on the bend.
Exam move
Sketch the control volume first, list every force acting on its contents, then write momentum separately for each direction with signs for velocity and for outflow.
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