1. For an incompressible fluid, the flow rate is:
Verification Matrix:
For an incompressible fluid undergoing steady flow, the volume flow rate ($Q = A \cdot v$) must remain constant at every cross-section throughout the pipe network. Whatever volume enters one end must simultaneously exit the other end.
2. Bernoulli’s principle states that for horizontal flow of a fluid through a tube, the sum of the pressure and energy of motion per unit volume is:
Verification Matrix:
For horizontal streamlines, potential energy changes are zero ($h_1 = h_2$). Bernoulli's equation simplifies to $P + \frac{1}{2}\rho v^2 = \text{constant}$. This states that static pressure ($P$) plus dynamic pressure (kinetic energy per unit volume, $\frac{1}{2}\rho v^2$) remains uniform along a streamline.
3. Which of the following is associated with the law of conservation of energy in fluids?
Verification Matrix:
Bernoulli's principle is directly derived by applying the Work-Energy Theorem to an ideal, moving fluid segment. It acts as a mathematical statement of the law of conservation of energy for fluid mechanics.
4. As the speed of a moving fluid increases, the pressure in the fluid:
Verification Matrix:
From the horizontal expression $P + \frac{1}{2}\rho v^2 = \text{constant}$, an escalation in speed ($v$) increases the dynamic pressure term ($\frac{1}{2}\rho v^2$). To keep the total sum constant, the local static pressure ($P$) must decrease.
5. If the cross-sectional area of a pipe decreases, what happens to the fluid velocity?
Verification Matrix:
According to the equation of continuity ($A_1 v_1 = A_2 v_2$), cross-sectional area and fluid velocity share an inverse functional relationship. Therefore, a reduction in channel width/area forces the speed of the streaming fluid to accelerate.
6. A sky diver falls through the air at terminal velocity. The force of air resistance on him is:
Verification Matrix:
Terminal velocity represents a steady-state condition where acceleration drops to zero ($\vec{a} = 0$). By Newton's Second Law ($\Sigma \vec{F} = 0$), the upward drag force vector exerted by air resistance must perfectly match the magnitude of the downward gravitational force vector (the skydiver's weight).
7. Wind speeding up as it blows over the top of a hill:
Verification Matrix:
As streamlines crowd together over the top of a hill, wind velocity increases. By Bernoulli's Principle, regions featuring higher localized stream speeds exhibit lower static pressures compared to surrounding slower zones.
8. A fluid is undergoing “incompressible” flow. This means that:
Verification Matrix:
Incompressibility is a material field property where mass density remains uniform under stress conditions. Mathematically, the material derivative of density is zero ($\frac{D\rho}{Dt} = 0$), meaning density ($\rho$) stays constant across all spatial points and time intervals.
9. A fluid is undergoing steady flow. Therefore:
Verification Matrix:
By Eulerian definition, "steady flow" indicates that fluid properties (such as velocity vectors, pressure, or temperature fields) at any fixed spatial coordinates are constant with respect to time ($\frac{\partial \vec{v}}{\partial t} = 0$). Individual molecules can still change velocity as they move to different spots.
10. The equation of continuity for fluid flow can be derived from the conservation of:
Verification Matrix:
The equation of continuity ($\Delta m_{\text{in}} = \Delta m_{\text{out}}$) is a direct accounting expression tracking mass conservation. It states that for a bounded transport network with no internal sources or sinks, the total mass flowing in per unit time must balance the total mass flowing out.
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