The cross-section of a river \(30 \mathrm{~m}\) wide is rectangular. At a point where the bed is approximately horizontal the width is restricted to \(25 \mathrm{~m}\) by the piers of a bridge. If a flood of \(450 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\) is to pass the bridge with the minimum upstream depth, describe the flow past the piers and calculate the upstream depth.

The Continuity Equation is essential in fluid mechanics for understanding how the flow of a fluid behaves as it moves through varying cross-sectional areas. For incompressible fluid like water, the Continuity Equation states that the mass flow rate must remain constant at all points along the streamtube. This can be mathematically written as

\[ Q = A_1V_1 = A_2V_2 \] where:

\t
• \(Q\) = flow rate

\t
• \(A_1\) = upstream cross-sectional area

\t
• \(V_1\) = upstream velocity

\t
• \(A_2\) = downstream cross-sectional area

\t
• \(V_2\) = downstream velocity

By applying this equation to the river flow problem, we can figure out how to relate the areas and velocities upstream and downstream of the bridge that narrows the river. This reasoning ensures that the volume of water remains constant as it flows through different sections of the river.

Bernoulli's Equation connects the pressure, velocity, and height at different points in a flowing fluid. It derives from the principle of conservation of energy, asserting that the total energy along a streamline is constant. Mathematically, Bernoulli’s equation is written as:

\[ P + \frac{1}{2} \rho V^2 + \rho gh = \text{constant} \] For this problem, since the elevation of the riverbed is consistent, the terms involving height (\(h\)) cancel out. This simplifies to:

\t
• \(P_1 + \frac{1}{2} \rho V_1^2 = P_2 + \frac{1}{2} \rho V_2^2\)

Assuming the atmospheric pressure is the same at both points, the equation further simplifies to:

\[ \frac{1}{2} V_1^2 = \frac{1}{2} V_2^2 \] This relationship helps us understand how velocities change with varying cross-sectional areas. Notably, if the cross-section narrows (width at the piers), the velocity must increase to conserve energy.

In fluid mechanics, flow rate calculations are critical for determining how much fluid moves through a particular cross-section per unit of time. For our river problem, given the flow rate (\(Q\)) as 450 cubic meters per second (\(m^3/s\)), we can apply the Continuity Equation to understand velocities at various points in the river. The upstream flow rate formula is:

\[ Q = A_u \cdot V_u \] where

\t
• \(A_u\) = upstream area (width \(W_u\) \times height \(h_u\))

\t
• \(V_u\) = upstream velocity

By plugging in the known values and manipulating the equation, we can find unknown variables. Similarly, for the downstream section, the relationship ensures conservation of flow:

\[ Q = A_d \cdot V_d \] Flow rate calculations guide us to examine how the variable (\(V\)) and area (\(A\)) are interconnected.

Determining upstream depth is the ultimate goal in our river problem. By using the Continuity and Bernoulli's equations, we can calculate how deep the water will be upstream of the bridge. We start with calculating the downstream velocity:

\[ V_d = \frac{Q}{A_d} = \frac{450}{25h} \] where

\t
• \(V_d\) = downstream velocity

\t
• \(A_d\) = downstream area

\t
• \(h\) = downstream depth

We then use Bernoulli’s equation to relate this velocity to the velocities at other points. Finally, solve for \(h\) yielding the upstream depth needed to ensure the flow rate remains constant through the restricted section near the bridge. These principles help us understand and predict fluid behaviors crucial for civil engineering and flood management.