An ideal gas passes isothermally through a long horizontal uniform cross-section pipe under steady flow. Consider that the pressure gradient in the pipe is sufficient for a finite change in the density of the gas. If the flow of the gas is purely pressure driven and subsonic througout, then the average flow velocity

The correct option is: increases along the flow.

To understand why this is correct, let us analyze the physical conditions of the gas flow in the pipe:

1. Steady Flow: For a steady flow in a horizontal pipe of uniform cross-sectional area ($A$), the mass flow rate ($\dot{m}$) must be constant at every cross-section along the length of the pipe. The mass flow rate is given by the continuity equation:

$\dot{m}=\rho ·A·v=\text{constant}$

where $\rho$ is the density of the gas and $v$ is the average flow velocity.

2. Uniform Cross-section: Since the cross-sectional area $A$ is uniform and constant throughout the pipe, the product of density and velocity must remain constant along the direction of flow ($x$):

$\rho ·v=\text{constant}$

3. Pressure-Driven Flow: The gas flows from a region of higher pressure to a region of lower pressure. This means the pressure $P$ decreases along the direction of flow:

$\frac{dP}{dx}<0$

4. Isothermal Process: The gas is ideal and the flow is isothermal, which means the temperature $T$ remains constant. For an ideal gas, the equation of state is:

$P=\rho ·R·T$

where $R$ is the specific gas constant. Since $R$ and $T$ are constant, the pressure is directly proportional to the density:

$P\propto \rho$

Since the pressure $P$ decreases along the flow direction, the density $\rho$ must also decrease along the flow direction to satisfy the isothermal condition.

5. Velocity Behavior: Re-examining the continuity relation $\rho ·v=\text{constant}$, we see that velocity is inversely proportional to density:

$v\propto \frac{1}{\rho }$

Because the density $\rho$ decreases along the flow, the average flow velocity $v$ must increase along the flow to maintain a constant mass flow rate.