An enclosed square duct of side \(s\) has a horizontal axis and vertical sides. It runs full of water and at one position there is a curved right-angled bend where the axis of the duct has radius \(r\). If the flow in the bend is assumed frictionless so that the velocity distribution is that of a free vortex, show that the volume rate of flow is related to \(\Delta h\), the difference of static head between the inner and outer sides of the duct, by the expression $$ Q=\left(r^{2}-\frac{s^{2}}{4}\right)(s g \Delta h / r)^{1 / 2} \ln \left(\frac{2 r+s}{2 r-s}\right) $$

In fluid dynamics, a free vortex is a type of flow where the tangential velocity changes inversely with the radius. This means that as you move closer to the center of the vortex (smaller radius), the velocity increases, and as you move farther away (larger radius), the velocity decreases. The formula for tangential velocity in a free vortex is given by:

\(v_{\theta} = \frac{C}{r}\),
where \(C\) is a constant that depends on the specific conditions of the flow, and \(r\) is the radius at which you measure the velocity.

The Bernoulli equation is a fundamental principle in fluid mechanics that relates the pressure, velocity, and height in a flowing fluid. For a streamline flow, the Bernoulli equation is expressed as:

\(P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}\)
Here, \(P\) represents the pressure, \(\rho\) is the fluid density, \(v\) is the flow velocity, and \(h\) is the height relative to a reference point. When applied to the given problem, the equation helps to relate the static head difference between the inner and outer sides of the duct to the velocities at these points. Using the Bernoulli equation to connect the inner and outer velocities, it transitions into:

\( \Delta h = \frac{1}{2g}(v_{outer}^2 - v_{inner}^2)\)

The static head difference \(\Delta h\) is the difference in height of the liquid columns at two points in the flow. This difference is caused by variations in pressure and velocity between these points. In the case of the duct with a bend, the static head difference originates from the curvature of the duct and the centrifugal forces acting on the fluid.

Since the fluid undergoes a change in velocity due to the shape of the duct, the static head difference directly connects to these velocity changes. The relation is derived from Bernoulli’s equation, where the term \( \Delta h \) links the velocities \(v_{outer}\) and \(v_{inner}\) through the expression:

\( \Delta h = \frac{1}{2g}(v_{outer}^2 - v_{inner}^2)\)

The volume flow rate \(Q\) in a fluid system represents the quantity of fluid passing a point per unit time. It is typically measured in cubic meters per second (m³/s) or liters per second (L/s). For ducts, volume flow rate is calculated using the cross-sectional area \(A\) and the average velocity \(\bar{v}\):

\(Q = A \bar{v}\)
In this exercise, the side length of the square duct is \(s\), so the area \(A = s^2\). The average velocity \(\bar{v}\) is determined through the flow characteristics within a free vortex, integrating the speed variations across the radius.

Eventually, due to the logarithmic relationship established through integration and balancing pressure differences, we get:

\[ Q = \left(r^{2}-\frac{s^{2}}{4}\right)(sg\Delta h / r)^{1 / 2} \ln\left(\frac{2r + s}{2r - s}\right) \]