ENGG2500 Chap.3 Kinematics, Visualisation and Continuity
Kinematics, Visualisation and Continuity
Fluid kinematics describes motion without yet solving for the forces that cause it. The Eulerian view uses fields at fixed positions, while the Lagrangian view follows particles. Streamlines, pathlines and streaklines have different meanings in unsteady flow. Continuity then balances mass accumulation and flux across a control surface, with area times average velocity as a useful one-dimensional reduction.
The chapter distinguishes local and convective acceleration, explains why a streamtube has no normal side flow and treats average velocity as a flux-preserving definition rather than a claim of uniform velocity.
What this chapter covers
- 01
Eulerian and Lagrangian descriptions
- 02
Velocity fields and acceleration
- 03
Streamlines, pathlines and streaklines
- 04
Streamtubes and control surfaces
- 05
Volume and mass flow
- 06
Steady incompressible continuity
Worked example · free
Flow through a reducer
- 1Convert both diameters to areas. The marks shown in this solution are not a university mark allocation.
- 1Apply A1 V1 = A2 V2 to obtain 5.00 m/s.
- 1Compute Q = 0.006283 m3/s.
- 1Compute mass flow as 6.27 kg/s.
Key terms
- Velocity field
- A velocity field assigns a velocity vector to each position and time.
- Streamline
- A streamline is tangent to the instantaneous velocity everywhere.
- Streamtube
- A streamtube is bounded by streamlines and has no normal side flow.
- Mass flow
- Mass flow is the rate at which mass crosses a surface.
Kinematics, Visualisation and Continuity FAQ
Can steady flow still accelerate?
Yes. Steady means the field at a fixed point does not change with time, so local acceleration is zero. A particle can still experience convective acceleration when it moves through a spatially varying velocity field, such as a reducer or curved streamline.
When do streamlines and pathlines coincide?
They coincide in steady flow because the velocity field does not change while particles move through it. In unsteady flow, a streamline is an instantaneous geometric construction, while a pathline records the trajectory of one particle over time, so the curves can differ.
Why use average velocity in continuity?
Actual velocity varies over most pipe cross-sections. Average velocity is defined so that its product with area equals the true integrated volume flow. That preserves total flux in a one-dimensional model without pretending that the local profile is uniform.
What is the difference between volume flow and mass flow?
Volume flow measures volume crossing a surface per unit time, while mass flow weights that flux by density. For uniform density, mass flow equals density times volume flow. In compressible flow or a section with density variation, the density must remain inside the surface integral or be represented by an appropriate average.
Why is there no flow across a streamtube side?
A streamtube side is constructed from streamlines, and velocity is tangent to a streamline at every point. Its normal component on that side is therefore zero. Fluid can enter or leave through the end sections, which makes the tube a convenient control volume for continuity and energy analysis.
How should branch flows be signed?
Choose outward normals or a clear inflow-positive convention and keep it through the full balance. Under the usual outward-normal convention, outflow flux is positive and inflow flux is negative. A negative solved branch value means the actual direction is opposite the assumed arrow, not that flow magnitude is impossible.
What does a flow visualisation line actually show?
It depends on how the line was created. A pathline follows one particle, a streakline joins particles that passed through one release point, and a streamline is tangent to the instantaneous field. They coincide in steady flow, but identifying an unsteady image requires knowing the release and observation procedure.
What does the accumulation term mean physically?
It measures the rate at which mass stored inside the chosen control volume changes. A filling tank has positive accumulation even if its liquid density is constant. Steady flow sets this term to zero, but steady does not mean that every inlet and outlet flow is individually equal when several openings exist.
Exam move
Start by translating field notation into a picture: mark fixed observation points for an Eulerian description and particle trajectories for a Lagrangian one. Practise identifying whether acceleration is local, convective or both. For continuity, draw the control surface and outward normals, label every opening with area, average normal velocity and density, then write accumulation plus net outward mass flux equals zero.
Only after that line should steady flow remove accumulation or constant density cancel from all terms. Convert diameters to areas explicitly and keep volume flow separate from mass flow. In branch problems, solve signed flows and reverse arrows only after the algebra identifies direction. Use a reducer to connect continuity with convective acceleration, and use a storage tank to keep the unsteady accumulation term active.
Finish each problem by checking that mass units close and that the computed speed change agrees with the area change for incompressible flow.