Fluid flows steadily through a converging nozzle (the cross-section decreases in the flow direction). Fluid particles accelerate as they move through the nozzle. Which term in the material derivative accounts for this acceleration?
ALocal acceleration ∂V/∂t, because the flow is changing at each point
BConvective acceleration (V·∇)V, because velocity changes with position along the flow
CBoth terms contribute equally in a nozzle
DNeither term — steady flow means zero acceleration everywhere
In steady flow, ∂V/∂t = 0 at every fixed point because the velocity field does not change over time. However, as a fluid parcel moves to a narrower section, it occupies a region where the velocity is higher. The convective term (V·∇)V captures this: the parcel is carried into a different location where the velocity field has a different value. Steady flow does not mean zero acceleration — it means the velocity pattern is frozen in space, not that individual parcels are stationary.
Question 2 True / False
In steady flow, streamlines, pathlines, and streaklines are all identical.
TTrue
FFalse
Answer: True
In steady flow the velocity field does not change with time, so a streamline drawn now will still be valid later. A fluid particle follows the instantaneous streamline because the streamline never changes — the pathline coincides with the streamline. Dye continuously injected at a point also follows the unchanging streamline, so the streakline does too. In unsteady flow, the velocity field evolves, so a particle no longer follows the current streamline — it follows the sequence of streamlines that existed at each moment of its journey, which diverges from both the current streamline and the dye injection pattern.
Question 3 Short Answer
Explain in physical terms what the material derivative D/Dt measures, and how it differs from the partial derivative ∂/∂t.
Think about your answer, then reveal below.
Model answer: The partial derivative ∂/∂t measures how a quantity changes over time at a fixed point in space (Eulerian perspective). The material derivative D/Dt = ∂/∂t + (V·∇) measures the total rate of change experienced by a fluid parcel as it moves with the flow (Lagrangian perspective). The extra term (V·∇) — the convective derivative — accounts for the change that occurs simply because the parcel has moved to a new location where the field has a different value.
A weather analogy: a thermometer fixed at an airport measures ∂T/∂t — how the local temperature changes over time. A balloon drifting with the wind experiences DT/Dt — temperature changes both because the local weather evolves AND because the balloon moves into warmer or cooler air masses. The convective term captures the second effect: it is the dot product of the velocity (how fast the parcel moves) with the spatial gradient of the field (how much the field varies with position).