# 2.1: Introduction

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Steady flow past a solid sphere is important in many situations, both in the natural environment and in the world of technology, and it serves as a good reference case for extension to more complicated situations, involving unsteady flows and/or nonuniform flows and/or nonspherical bodies. It is also an excellent starting point for development of a number of important principles and techniques that are essential for later development in these notes. In particular, I hope to be able to convince you of the importance and utility of careful dimensional reasoning about flows of fluids.

You can think in terms of fluid flowing past a stationary sphere, or of a sphere moving through stationary fluid. The two cases are almost, but not quite, equivalent. And in the latter case you could imagine the sphere being moved through the fluid in three different ways: fastened to a rigid strut, or towed with a flexible line, or pulled downward through the fluid under its own weight. For now, do not worry about these distinctions; just view the fluid from the standpoint of the sphere. I will return to the differences briefly later. For the sake of definiteness, assume here that the sphere is towed or pushed through still fluid. All that is said here about the flow is then with reference to a point fixed relative to the moving sphere.

Just from considerations of space and motion, it is clear that the approaching fluid must both move faster and be displaced laterally as it flows past the sphere. On the other hand, the no-slip condition requires that the fluid velocity be zero everywhere at the surface of the sphere; this implies the existence of gradients (that is, spatial rates of change) of velocity, very sharp under some conditions, at and near the surface of the sphere. These velocity gradients produce a shear stress on the surface of the sphere; see Equation 1.3.6. When summed over the surface, the shear stress exerted by the fluid on the sphere represents the part of the total drag force on the sphere called the **viscous drag**. Your intuition probably tells you (correctly in this case) that the pressure of the fluid, the normal force per unit area, is greater on the front of the sphere than on the back. The sum of the pressure forces over the entire surface of the sphere represents the other part of the drag force, called the **pressure drag** or **form drag**. You will see later that the relative importance of viscous drag and pressure drag, as well as the qualitative flow patterns and the distance out into the fluid the sphere makes its presence felt, are greatly different in different ranges of flow.

You can see now that even in such a seemingly simple flow as the passage of a steady and uniform approach flow around a smooth sphere there is a great variation in flow phenomena. Complexity of this kind in deceptively simple flows is common in fluid dynamics; you need to be on your guard against theorizing about phenomena of fluid flow without the ground truth of experiment and observation.