CIVL2611 Chap.5 Kinematics, Continuity and Momentum
Kinematics, Continuity and Momentum
This chapter first describes fluid motion through velocity fields, streamlines and steady-flow language, then chooses control volumes for conservation balances. Continuity connects inlet and outlet flow rates, while linear momentum turns a vector velocity change into an external-force requirement. Every result names whether the force acts on the fluid or the hardware.
A steady field can still accelerate spatially through a nozzle, while uniformity describes variation across space rather than time. Streamlines show the instantaneous direction field; particle histories need not coincide with them in unsteady motion. A control volume is an accounting region whose boundary cuts the flow at useful sections.
Mass conservation is solved before momentum so areas and known velocities determine missing section speeds. Linear momentum is then written component by component, with pressure, weight and support actions drawn as external forces on the fluid. Negative values correct assumed directions instead of being discarded.
Original nozzle, junction, jet and right-angle bend problems show how equal speed can still produce momentum change when direction rotates. Newton's third law is stated explicitly when converting a force on the fluid into the opposite force on a plate, bend or nozzle. Scaling checks also show why a stopping force can grow with speed squared when fixed jet area makes both mass flow and velocity change proportional to speed.
What this chapter covers
- 01
Velocity fields and streamlines
- 02
Steady and uniform flow
- 03
Control volumes and control surfaces
- 04
Volumetric and mass flow rates
- 05
Continuity for incompressible flow
- 06
Linear momentum and reaction forces
Worked example · free
Nozzle velocity and axial force
- +1Calculate flow rate as 0.036 cubic metres per second.
- +1Use continuity to find exit speed 6.0 metres per second.
- +1Find mass flow rate 36 kilograms per second.
- +2Calculate 144 newtons on the fluid in the flow direction.
- +1State the equal opposite force exerted on the nozzle.
Key terms
- Velocity field
- A vector assignment giving fluid velocity at each position and time.
- Control surface
- The boundary of a chosen control volume through which fluxes and forces are evaluated.
- Momentum flux
- The rate at which linear momentum crosses a control surface with flowing mass.
Kinematics, Continuity and Momentum FAQ
Can a steady flow still accelerate?
Yes. Steady means conditions at a fixed location do not change with time. Fluid can still accelerate spatially through a nozzle because velocity differs from one position to another even though the field remains time independent.
Why is continuity usually solved first?
Continuity often converts areas and one known velocity into another section velocity. That reduces the unknowns before momentum or energy equations are applied and makes the later squared or vector velocity terms easier to audit.
How do I report a momentum reaction?
First state the net external force acting on the fluid control volume. Then use Newton's third law to give the equal and opposite force exerted by the fluid on the nozzle, bend, plate or support.
Exam move
Practise drawing control volumes around nozzles, bends, junctions and jets. For each sketch, shade the selected region, mark every place mass crosses its surface and draw an outward normal. State whether the description is steady or unsteady and uniform or nonuniform before writing a balance.
Review streamlines, pathlines and streaklines by explaining when they coincide and when a particle history differs from the instantaneous field. Use continuity first to reduce unknown section velocities. Begin with one incompressible liquid, then add a problem in which densities differ so mass flow rather than volume flow must balance.
Treat a negative computed branch flow as information about the actual direction and revise the arrow on the diagram. For momentum, choose axes and resolve each velocity into signed components before writing equations. Keep the x and y balances on separate lines, especially for bends. List every external pressure, weight and support force, including any term deliberately neglected.
Draw pressure forces inward on cut faces and distinguish them from wall reactions. After solving for the net force on the fluid, write a separate Newton's-third-law sentence for the force exerted by the fluid on the hardware. This object naming is as important as the sign. Create scaling checks for jet problems: at fixed area and density, mass flow is proportional to speed and stopping force is proportional to speed squared.
Use that trend to test revised numerical cases. For a junction, verify that total mass inflow equals total mass outflow before attempting momentum. During timed practice, reserve one line for continuity, one for mass flow, one for vector momentum and one for the reaction statement. If an answer is wrong, classify the cause as an area conversion, flow direction, missing pressure force, omitted component or object reversal.
Repeat a fresh geometry that targets only the failed stage rather than copying the full previous solution. Close the session by reversing one outlet direction and predicting which momentum component must change sign before repeating the balance. Mark all vector directions explicitly.