Understanding Diverging Sections and Diffusers in Fluid Flow

In fluid dynamics, the shape of a duct or channel significantly affects the flow properties such as velocity and pressure. A diverging section is a part of a duct where the cross-sectional area increases in the direction of flow.

A diffuser is a component designed to slow down the fluid flow and increase its static pressure. This is the opposite function of a nozzle, which accelerates flow and decreases pressure.

Diverging Section Behavior in Incompressible Flow

The question specifically mentions incompressible flow. Incompressible flow is a type of flow where the density of the fluid remains constant, regardless of changes in pressure. This is often a good approximation for liquids and for gases flowing at low speeds (much less than the speed of sound, i.e., subsonic speeds).

For incompressible flow, the principle of conservation of mass (continuity equation) simplifies greatly:

$\rho_1 A_1 V_1 = \rho_2 A_2 V_2$

Since density $\rho$ is constant in incompressible flow ($\rho_1 = \rho_2$), the equation becomes:

$A_1 V_1 = A_2 V_2$

or simply,

$A V = \text{constant}$

This equation tells us that for incompressible flow, the product of the cross-sectional area ($A$) and the flow velocity ($V$) is constant along the duct. Therefore, if the area increases in a diverging section ($A_2 > A_1$), the velocity must decrease ($V_2 < V_1$) to keep the product constant.

According to Bernoulli's principle for incompressible, steady, inviscid flow, an increase in pressure is associated with a decrease in velocity (assuming negligible changes in elevation). Since a diverging section causes the velocity to decrease in incompressible flow, the pressure must increase.

Thus, in incompressible flow, a diverging section slows down the fluid and increases its pressure. This is precisely the definition of a diffuser.

Comparison with Compressible Flow (Subsonic vs. Supersonic)

While the question specifies incompressible flow, it's useful to understand why the answer options mention subsonic and supersonic flows. Incompressible flow is a valid approximation for subsonic compressible flow (typically Mach number < 0.3). The behavior of a diverging section changes dramatically in supersonic flow.

• Subsonic Flow (M < 1): The area-velocity relationship for compressible flow is $\frac{dV}{V} = - \frac{dA}{A} \frac{1}{1-M^2}$. For subsonic flow, $M^2 < 1$, so $1-M^2 > 0$. Thus, `dV/V` and `dA/A` have opposite signs. A diverging section (`dA > 0`) causes velocity to decrease (`dV < 0`). This matches the incompressible/subsonic behavior. Pressure increases, acting as a diffuser.
• Supersonic Flow (M > 1): For supersonic flow, $M^2 > 1$, so $1-M^2 < 0$. Thus, `dV/V` and `dA/A` have the same sign. A diverging section (`dA > 0`) causes velocity to increase (`dV > 0`). Pressure decreases, acting as a nozzle.

Based on this, a diverging section acts as a diffuser specifically for subsonic flow (which includes incompressible flow as a low-speed approximation).

Analyzing the Options

• sonic state only: Incorrect. Sonic flow (M=1) typically occurs at the throat of a nozzle or diffuser. A simple diverging section does not act as a diffuser for sonic flow throughout.
• Both subsonic and supersonic flows: Incorrect. It acts as a diffuser only for subsonic flow; for supersonic flow, it acts as a nozzle.
• supersonic flows only: Incorrect. It acts as a nozzle for supersonic flow, not a diffuser.
• subsonic flows only: Correct. As explained above, for incompressible flow (which is a type of subsonic flow), a diverging section reduces velocity and increases pressure, functioning as a diffuser. This behavior holds true for compressible subsonic flow as well.

Therefore, for incompressible flow, a diverging section acts as a diffuser in the downstream for subsonic flows only.