A tall cylindrical body having an oval cross-section with major and minor dimensions \(2 X\) and \(2 Y\) respectively is to be placed in an otherwise uniform, infinite, two-dimensional air stream of velocity \(U\) parallel to the major axis. Assuming irrotational flow and a constant density, show that an appropriate flow pattern round the body may be deduced by postulating a source and sink each of strength \(|m|\) given by the simultaneous solution of the equations $$ m / \pi U=\left(X^{2}-b^{2}\right) / b \quad \text { and } \quad b / Y=\tan (\pi U Y / m) $$ Determine the maximum difference of pressure between points on the surface.

Potential flow theory helps us to study irrotational flows which means that the flow does not have any rotation or vorticity. This theory assumes an ideal fluid with no viscosity and incompressibility. Understanding potential flow theory is essential when dealing with fluid flow around objects like the oval cylinder in this exercise.

Potential flows can be considered as combinations of simpler flow patterns like sources, sinks, and vortices. Here, we model the flow around the oval cylinder using a source and a sink to simulate the actual flow pattern.

In this problem, we set up and solved equations using potential flow theory to understand how the fluid flows around the cylinder. We began by postulating a source and sink, each with a strength denoted by \(|m|\). This involves breaking down complex flows into simple elements and adding their effects together, making potential flow a powerful method for analyzing fluid mechanics problems.

Bernoulli's equation is a fundamental principle in fluid dynamics, describing the relationship between the pressure, velocity, and height in a moving fluid. For irrotational flow, like the one we have, Bernoulli's equation helps determine pressure differences on the surface of objects in the flow.

Bernoulli’s equation in its simplest form is: \[ p + 0.5 \rho U^{2} + \rho g h = \text{constant} \] In our scenario, we neglect the height term as we consider a two-dimensional flow, focusing instead on pressure and velocity. The pressure difference \(\triangle p \) between two points on the surface can be determined using: \[ \triangle p = \rho U^{2} \]
This equation helps us find the maximum pressure difference around the oval cylinder, influenced primarily by the flow velocity. Higher flow velocity leads to a larger pressure difference.

In many fluid dynamics problems, exact analytical solutions are challenging or impossible to obtain. In such cases, numerical methods become essential tools. They allow us to approximate solutions iteratively, making complex problems more manageable.

In this problem, numerical methods play a crucial role, especially for solving the second equation: \[ b = Y \tan \left( \frac{ \pi U Y b }{ \pi U (X^{2} - b^{2}) / b }\right) \]
Given its complexity, solving it analytically is impractical. Therefore, we use numerical techniques like iterative methods or computer simulations to approximate the value of \(|b|\). These methods involve initially guessing a value and refining it through multiple iterations until convergence is achieved.
Understanding basic numerical methods, such as iteration and root-finding algorithms, can significantly enhance one's capability to tackle complex fluid dynamics problems efficiently.