Steady flow in fluid mechanics In continuum fluid mechanics (ignoring compressibility effects unless stated), a flow is called steady when which condition holds true at a fixed point in space?

Introduction / Context:
Steady flow is a foundational concept in fluid mechanics used when applying Bernoulli’s equation, continuity, and momentum analyses. Correctly distinguishing “steady” from “uniform” helps avoid common modeling errors in pipelines, channels, and turbomachinery.

Given Data / Assumptions:

• The fluid is treated as a continuum.
• Any property may be considered: velocity, pressure, density, temperature, etc.
• We assess behavior at a fixed spatial point (Eulerian viewpoint).

Concept / Approach:

In the Eulerian description, steadiness means the partial derivative with respect to time of any property at a fixed location is zero. That is, for velocity components u, v, w at a point, ∂u/∂t = ∂v/∂t = ∂w/∂t = 0, and similarly for pressure and other scalars. This does not require spatial uniformity—only temporal invariance at each point.

Step-by-Step Solution:

Verification / Alternative check:

Plot any property (say, pressure) vs. time at a sensor fixed in space. If the value is constant (ignoring noise), the flow is steady at that point regardless of spatial gradients.

Why Other Options Are Wrong:

(a) “Change steadily” still means unsteady. (c) Same at adjacent points defines spatial uniformity, not steadiness. (d) Restricting to streamlines is unnecessary; steadiness applies to all properties at the point. (e) Uniformity along the pipe may occur without steadiness, and vice versa.

Common Pitfalls:

Confusing steady with uniform; believing a converging nozzle cannot be steady because velocity varies spatially—it can be steady if the flow rate and local properties are time-invariant.

Final Answer:

conditions of flow do not change with time at a point