## Mathematical Models

Flow and pressure depictions shown in this section were derived using mathematical methods; no experiments were done to derive them. The notion that it is possible to predict the trajectories of all the fluid elements moving past a wing may be foreign to you. You may be skeptical of the results -- and you should be. The solutions shown here are created from a relatively limited mathematical model of fluid flow, one in which the fluid possesses no viscosity and the flow exhibits no turbulence. However, those limitations were chosen so as to allow rapid computations and to make the explorations more interesting, partly by examining the departures from reality. Much more complex mathematical models of fluid flow than this one are applied on a daily basis in our culture today. It is commonplace to use these techniques in the design of everything from rockets and airplanes to medical pumps and interventional devices. This is rocket science.

The terms model and theory may suggest a lack of reliability to you. Mathematical models in particular are often discounted by those not familiar with the processes involved in their development. With a moment's pause, however, you should recognize that human beings conceptualize (model) virtually everything about the universe (even my dog conceptualizes the trajectory of a tennis ball and alters his own for an efficient interception). Everything that you *understand* about cardiology, medicine, and the world around you is based on models -- your *expectation* of what will happen under a given set of circumstances. That you don't normally perform any computations before formulating a therapy for your patient may be a limitation of our current culture -- a limitation that may not be supportable in the future.

Mathematical models have the distinction of being unambiguous. Equations are supremely precise in their description of how outcomes depend on specific variables. This doesn't necessarily mean that they're *correct*. A mathematical model can be based on observations and applied in a descriptive sense, or it can be based on physical law; it's clear from the cardiology literature that many do not appreciate the distinction. The models shown in this section are derived from extremely robust physical principles -- conservation laws -- that will be discussed in other sections. These laws include the principles of conservation of mass, momentum, and energy. Suffice it to say that these principles are thought to be inviolate throughout the universe -- more consistent than death or taxes. Nevertheless, our ability to apply or understand these laws is limited so that the predictions of mathematical models may not agree with observations. In fact we can assume a priori that no model will predict an outcome *exactly; all models are approximations*. By evaluating the departure of a model from observations, we can continue to refine the model (and measurements) to improve understanding.

The models of fluid flow shown in this section using the Joukowski Active Figure have a limited range of applicability. For our purposes, the limitations and departures from reality are as valuable as the solutions that are realistic. They allow for an educational opportunity so that readers can better understand the physical laws being described. Limitations of the potential flow model will be discussed as they are encountered.

The transcendent power of mathematics is that once a physical relationship has been expressed as a set of equations, manipulation of the equations within the applicable rules continues to yield true relationships. This is how it is possible for a young Albert Einstein, seated at his desk, to make fantastical predictions about the universe that continue to be supported by a century of observations and research.