In a steady or streamline flow of a fluid, which quantity at a fixed point in the fluid remains constant with time?

Introduction / Context:

Fluid dynamics distinguishes between different types of flow, such as steady (streamline) flow and unsteady (turbulent or time varying) flow. Understanding what remains constant in a steady flow at a particular point is important for applying Bernoulli equation and continuity equation. This question asks you to identify which physical quantity stays the same at a given point in a streamline flow as time passes.

Given Data / Assumptions:

• We are dealing with a streamline or steady flow of a fluid.
• At a fixed point in space, the properties of the fluid may or may not change with time.
• Steady flow means that certain properties are time independent at each point.
• We consider speed, pressure, force, temperature and density as possible candidates.

Concept / Approach:

In fluid mechanics, steady flow (often used interchangeably with streamline flow in basic texts) is defined as a flow in which the fluid properties at any given point do not change with time. The most important property in this definition is the velocity of the fluid. In a steady flow, the velocity vector at each fixed point in space remains constant in magnitude and direction over time, although it may vary from point to point in the flow field. Pressure, density and temperature may also be constant in some special flows, but in general they can vary with position and sometimes along streamlines even in steady conditions. The key defining feature is the constancy of velocity (and thus speed) at each point with respect to time.

Step-by-Step Solution:

Verification / Alternative check:

Visualise water flowing smoothly through a pipe at a constant rate. At any fixed point inside the pipe, the speed and direction of flow remain the same with time; if you drop dye into the flow, it moves in smooth paths and the pattern stays steady. This is a steady or streamline flow. However, pressure can change from one point to another along the pipe due to height differences or constrictions. Density in incompressible flows (like liquids) may remain constant overall, but in compressible flows (like gases) it can vary with pressure, even under steady conditions. Thus, the only universally correct answer that fits the definition at a given point is the constancy of velocity or speed with respect to time.

Why Other Options Are Wrong:

Option A (force): Force on fluid elements can vary with position and may not remain constant at a point even in steady flow, especially if external fields or pressure gradients are present.

Option B (pressure): Pressure often changes along the direction of flow, for example in a Venturi tube, even when the flow is steady.

Option D (temperature): Temperature may change along a streamline in flows involving heat transfer, even if the velocity field is steady.

Option E (density): In compressible flows, density can change as pressure and temperature change, despite the flow being steady in terms of velocity.

Common Pitfalls:

Students sometimes think that steady flow means all properties are constant everywhere, which is not correct. Steady means no change with time at a fixed point, but properties can still vary from point to point in space. The velocity field is the primary focus in the definition of steady or streamline flow: at each fixed point, the velocity does not change with time. Remember that when you see the phrase steady flow, you should immediately think of a velocity field that is time independent, even though it may vary spatially.

Final Answer:

In a streamline (steady) flow, the speed (magnitude of velocity) at each fixed point remains constant with time.