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The first characteristic of fluids that must be appreciated for an understanding of viscosity is the “no‐slip condition” with respect to solids. That is, at the interface between a solid and a fluid flowing over it, the velocity of the fluid is zero. A zero‐velocity boundary layer is created, whose thickness depends on the velocity of the fluid flow. At the solid–fluid boundary, fluids stick to solids absolutely. Any object in a flow thus creates a shear, as the fluid particles at the no‐slip boundary must be moving at a different velocity than those at a distance from the body in the flow.

Figure 1.2 Dynamic viscosity ( μ ). A fluid’s “stickiness” or resistance to shear. Expressed mathematically as μ = Fl/US.

The resistance to shear is the dynamic viscosity of the fluid, and it is best described using the classical “flat plate” analogy. Imagine two plates of negligible thickness oriented parallel to one another and with a fluid between them (Figure 1.2). The bottom plate is stationary, and the movable plate is pushed forward with a force that results in a constant velocity. Since the fluid at the boundary of each plate has zero velocity due to the no‐slip rule, the fluid between them is deformed or sheared, and a uniform velocity gradient develops between them in response to the constant pushing. The force needed to maintain the constant velocity depends upon the velocity itself, upon the area of the plate because the amount of fluid that needs to be moved is a function of the size of plate, and upon the viscosity of the fluid, or how easily the fluid is deformed. The resulting equation is:

(1.1)

where U is velocity, S is the area of the plate, μ is viscosity, and l is the distance between the plates. Think of the flat plates as the bottom and top card in a deck of playing cards. The fluid is all the cards between them, and the stickiness between the cards determines how easy it is to deform the deck. Viscosity is that stickiness. The viscosity (dynamic viscosity) is an important property of fluids because it determines how easy it is to move them and to move through them.

A second type of viscosity is quite important in understanding flow around and through objects: the kinematic viscosity or υ. It is the ratio of dynamic viscosity ( μ ) to density ( ρ ):

(1.2)

Kinematic viscosity is considerably less easy to grasp on an intuitive level, but it relates two important properties of a fluid that will be significant to us in examining the locomotion of open‐ocean fauna. Viscosity and density have much to do with patterns of flow around an organism. On the one hand, viscosity measures how adjacent particles retard a fellow fluid particle’s movement when it encounters a body in a flow. On the other, density is a measure of how likely it is that a fluid particle will keep moving. The ratio of the two forces, inertial and viscous, is the subject of our next topic, the Reynolds number.