A pipe carrying water tapers from a cross-section of \(0.3 \mathrm{~m}^{2}\) at \(A\) to \(0.15 \mathrm{~m}^{2}\) at \(B\) which is \(6 \mathrm{~m}\) above the level of \(A .\) At \(A\) the velocity, assumed uniform, is \(1.8 \mathrm{~m} \cdot \mathrm{s}^{-1}\) and the pressure \(117 \mathrm{kPa}\) gauge. If frictional effects are negligible, determine the pressure at \(B\).

The Continuity Equation is a fundamental concept in fluid mechanics that expresses the conservation of mass for an incompressible fluid. It states that the product of the cross-sectional area (A) and the fluid velocity (v) at any two points along a streamline must be constant. This can be mathematically represented as: \[ A_A \times v_A = A_B \times v_B \] where \( A_A \) and \( A_B \) are the cross-sectional areas at points A and B, respectively, and \( v_A \) and \( v_B \) are the velocities at these points. This relationship ensures that the same volume of fluid flows through any cross-section of the pipe per unit time. In our exercise, the area at A is 0.3 m² and the velocity at A is 1.8 m/s, while the area at B is 0.15 m². Using the Continuity Equation, we can solve for the velocity at B ( \( v_B \)). Hence: \[ 0.3 \times 1.8 = 0.15 \times v_B \] Solving for \( v_B \): \[ v_B = \frac{0.3 \times 1.8}{0.15} = 3.6 \text{ m/s} \] This shows that as the cross-sectional area decreases, the velocity of the fluid increases.

Bernoulli's Equation is another cornerstone of fluid mechanics that illustrates the conservation of energy in a flowing fluid. For a fluid moving along a streamline, Bernoulli’s Equation states that the sum of the fluid's potential energy, kinetic energy, and pressure energy remains constant. This can be expressed mathematically as: \[ P_A + \frac{1}{2} \rho v_A^2 + \rho g h_A = P_B + \frac{1}{2} \rho v_B^2 + \rho g h_B \] Here, \( P_A \) and \( P_B \) are the pressures, \( v_A \) and \( v_B \) are the velocities, \( h_A \) and \( h_B \) are the heights, and \( \rho \) is the fluid density at points A and B, respectively. Bernoulli’s Equation allows us to calculate the pressure at one point (B) if the pressures, velocities, and heights are known at another point (A). In our example, frictional effects are negligible, simplifying the calculation.

An incompressible fluid is one whose density does not change significantly with variations in pressure. This assumption significantly simplifies fluid mechanics calculations, as it implies that the fluid's volume remains constant. In our exercise, water is treated as an incompressible fluid. The density ( \(\rho \)) of water is approximately \( 1000 \text{ kg/m}^3 \), and it does not alter appreciably with changes in pressure and temperature under normal conditions. This consistent density allows us to apply the Continuity Equation and Bernoulli’s Equation directly, ensuring straightforward calculations for velocity and pressure.

Pressure calculation is an essential part of applying Bernoulli’s Equation. By knowing the pressures, velocities, and heights at points A and B, as well as the density and gravity, we can solve for the unknown pressure. Substituting the known values into Bernoulli’s Equation, we have: \[ 117 \text{ kPa} + \frac{1}{2} \times 1000 \frac{\text{kg}}{\text{m}^3} \times (1.8 \text{ m/s})^2 = P_B + \frac{1}{2} \times 1000 \frac{\text{kg}}{\text{m}^3} \times (3.6 \text{ m/s})^2 + 1000 \frac{\text{kg}}{\text{m}^3} \times 9.81 \frac{\text{m}}{\text{s}^2} \times 6 \text{ m} \] This equation accounts for the pressure energy, kinetic energy, and potential energy at both points. Simplifying and solving for \( P_B \) provides the pressure at point B: \[ P_B = 53.28 \text{ kPa} \]

The velocity of the fluid in a pipe depends largely on the pipe's cross-sectional area. From the Continuity Equation, we see that if the area is halved, the velocity must double to maintain the same flow rate. In our example, the velocity at the wider section A is 1.8 m/s, whereas at the narrower section B, the velocity increases to 3.6 m/s. This concept is key in understanding how fluid dynamics work in varying pipe dimensions and is crucial for designing efficient fluid transportation systems. Increased velocity in narrower sections leads to changes in pressure distribution along the pipe, as described by Bernoulli’s Principle.