# Fluid flow

#### Prerequisites

Flow — the physical motion of a fluid from one place to another — plays a crucial role in biological organisms larger than a few cells in a number of ways. Living organisms are dynamic systems, extracting energy and organization from chemical reactions and dumping heat and waste products. They continually need to bring in new chemicals and remove used ones. Without flow, organisms are limited to using diffusion to move things around. Diffusion is inherently slow, and the time for something to diffuse increases with the square of the distance. We can calculate typical diffusion times, $t_{diff}$, from the equation (see Fick's law):

$$t_{diff} = \frac{x^2}{2D}$$

where $x$ is the diffusion distance and $D$ is the diffusion coefficient.

Oxygen has a diffusion coefficient of 1.8 x 10-5 cm2/s in water. Based on this, it will take just 2.5 ms for oxygen to diffuse 3 μm (from a red blood cell to the capillary wall), 280 s for it to diffuse 1 mm, and 2.8 x 108 s (8.8 years) for it to diffuse 1 m between the lungs and a hand.

While diffusion across 3 μm is a key process for life, diffusion as far as 1 m takes an unacceptably long time. Therefore, as organisms become bigger, they need to have an active means for moving nutrients and waste products around the body. Similarly, plants need a way to provide water to leaves for photosynthesis and to move the resulting products to their roots for storage.

Organisms are constantly managing and interacting with fluids. There are two key geometries of fluid flow. For internal flow, the fluid moves through a small opening or tube, often inside of an organism. This includes when air is inhaled into a set of lungs, blood flows through a circulatory system, or sap rises up a tree. We refer to this as internal flow.

But some organisms also move relative to an external fluid. This might occur when an organism walks, flies, or swims through a fluid. However, it might also occur when an organism is fixed and the fluid moves over it, such as when the wind blows past a plant wanting to disperse its seeds or an ocean current flows past a barnacle wanting to gather food.  We refer to such a situation as external flow. Because the dimensions of internal flow tend to be smaller than those for external flow, we will consider these two cases separately.

## Patterns of flow

Fluid flow can occur in many different ways. Sometimes, flows occur slowly over a smooth surface.  In this case the flow is quite regular and orderly, with all the fluid moving in smooth sheets. We call this laminar flow (from the Latin lamina meaning a thin layer). However, sometimes the flow is quite fast or occurs over uneven surfaces.  In this case, the flow is unorganized or turbulent, and there can be a lot of mixing, with whirlpools and vortices as shown in the figure below.

Another important aspect of fluids is that when a fluid moves across a solid surface, the fluid right next to the surface doesn’t move. This is called the no-slip condition and results from the molecular attraction (adhesion) between the molecules of the fluid and the solid surface.  Further from that surface, the velocity increases until it reaches the velocity of the free fluid, unperturbed by the surface. As a result of the no-slip condition, the flow has a velocity gradient, with zero velocity right next to the surface and an increasing velocity the further away from the surface you get as shown in the figure at the right. The region where the velocity is lower is often called the boundary layer, as it is the fluid that is under the influence of the boundary or surface.

In the two follow-ons to this section, we quantify the flow and discuss some scenarios and considerations that organisms take into account in order to take advantage of different aspects of flow, in air and water.

Resources:

Steven Vogel, Comparative Biomechanics: Life's Physical World (Princeton U. Press, 2013) chapters 6, 7, 9, 10.

Steven Vogel, Life in Moving Fluids: The Physical Biology of Flow (Princeton U. Press, 1996) chapters 9, 13.

Karen Carleton and Joe Redish 10/26/11

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