Question

STATEMENT-1 The stream of water flowing at high speed from a garden hose pipe tends to spread like a fountain when held vertically up, but tends to narrow down when held vertically down. and STATEMENT-2 In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant. (A) STATEMENT-1 is True, STATEMENT- 2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1 (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 (C) STATEMENT-1 is True, STATEMENT-2 is False (D) STATEMENT-1 is False, STATEMENT-2 is True

Short answer

(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

Step-by-step solution

Analyzing Statement-1

Examine the behavior of a stream of water from a garden hose pipe when held vertically up and down. When held vertically up, the stream spreads due to gravity affecting the parabolic trajectory of water particles, causing them to disperse. When held vertically down, the stream narrows as all the water is flowing in the same direction aided by gravity.

Analyzing Statement-2

Understand the principle of conservation of mass applied to incompressible fluids. This principle states that in steady flow, the mass flow rate, and consequently the volume flow rate, must remain constant. This is also known as the equation of continuity for incompressible fluids.

Examining the Link Between Statement-1 and Statement-2

Determine whether the constancy of volume flow rate (Statement-2) explains the behavior of the water stream from the hose (Statement-1). The constancy of volume flow rate doesn't directly explain the spreading and narrowing of the stream; it is explained by the added effect of gravity and inertia acting on the water particles.

Key concepts

Conservation of Mass

In fluid mechanics, the conservation of mass is a fundamental principle stating that mass cannot be created or destroyed within a closed system. During fluid flow, this principle implies that the mass of fluid entering a system must equal the mass exiting it, assuming no accumulation of mass within the system. For incompressible fluids - those whose density remains constant - this translates into the volume flow rates being consistent, provided that the flow is steady; there are no variations in flow characteristics with time.

When water flows out of a garden hose, conservation of mass is at play. Regardless of whether the hose is pointed upwards or downwards, the same amount of water must exit the hose over a given time period. The difference in spread or narrowing of the water stream is due to gravity and inertial effects instead of a change in the amount of water being conserved.

Incompressible Fluid Flow

In the realm of fluid mechanics, an incompressible fluid is characterized by a constant density. Water, at typical conditions encountered in everyday experiences, is often treated as an incompressible fluid. This simplifies many problems and is a reasonable assumption for most practical purposes, especially at low velocities and standard pressures.

In the case of the water streaming from a hose, the behavior observed when the hose is held vertically up or down is a result of fluid dynamics involving incompressible flow. For an incompressible fluid, factors such as pressure changes, velocity, and geometry of space through which it flows will affect its behavior visibly, but not its density. The narrow stream when the hose is down and the fanned-out spray when it's up are due to physics - fluid trajectories and forces - all while treating water as incompressible.

Equation of Continuity

The equation of continuity is a mathematical expression derived from the conservation of mass specifically for fluid flow. It essentially states that for an incompressible, steady-flow condition, the product of cross-sectional area and flow velocity must remain constant along the streamline. Formally, for any two points along a streamline, the equation is expressed as:

\( A_1 v_1 = A_2 v_2 \)
where \( A \) is the cross-sectional area and \( v \) is the velocity of the fluid at points 1 and 2 respectively.

This equation illuminates why, for a constant volume flow rate, if the area of the hose's exit were to be reduced, the velocity of water exiting would increase to maintain the flow rate - which is why firefighters' nozzles are designed to narrow for a high-velocity jet. However, it doesn't directly explain the spreading or narrowing of the stream as that involves gravitational forces not accounted for in the equation of continuity.