## Ideal Fluid

Cardiologists employ fluid dynamic principles on a daily basis in evaluating patients with heart disease. One of the most frequently invoked is a conservation of energy principle, the simplified Bernoulli equation (SBE). In its most frequently utilized clinical form, the SBE relates the inter-conversion of pressure and kinetic energy along a streamline. As usually employed, the SBE contains no provisions for energy losses due to fluid friction, time-varying (pulsatile) aspects of blood flow, or other forms of energy such as gravitational potential. Cardiologists must understand certain aspects of fluid dynamics, at least unto limitations of application for the SBE.

The SBE treats energy losses due to friction as negligible. I prefer to call this an approximation as opposed to an assumption -- it is an approximation that can be made when certain conditions about the flow are present. One such circumstance exists when fluid flow occurs in such a way that the friction produced is negligible; fluid layers do not shear against each other and frictional forces are largely avoided. An example of this flow situation occurs when the fluid moves as a unified mass although that is not the only way that this can occur. While it may seem unlikely, this situation is commonplace in many natural settings including, seemingly, flow in the vicinity of the heart. Another applicable situation is when the fluid itself has very low viscosity or, more specifically, when the Reynolds number is high.

The Reynolds number is literally a ratio of inertial (acceleration) to viscous forces associated with a flow. For many important applications, a fluid can be approximated as having viscosity equal to zero. This is said to be an ** ideal fluid** and generates no friction, even if the fluid layers were to slide against each other. It must be appreciated, however, that all physiologic fluids possess viscosity and that viscous forces are never negligible in the vicinity of a solid boundary. Near a solid boundary, and often times not so near one, a real fluid moves at a velocity that is nearly the same as the boundary and the fluid elements immediately adjacent to the boundary are stopped! (More specifically, they move at the same velocity as the boundary.) Whether we consider blood flow through an artery or airflow over a wing, this means that there will be a region of great variation in velocity between the boundary and much higher velocity in the free stream (there will be a large

**). This region is termed a**

*velocity gradient***where layers of a real fluid slide against each other and generate significant friction.**

*boundary layer*

In generating models for flow over the cylinders and airfoils shown here, the fluid has been approximated as an ideal one with no viscosity whatsoever. All the models depict flow as originating from the left side of the image and traveling to the right. It may seem odd or inappropriate to consider the airfoil as stationary while flow (e.g. air) moves past it. However, this is a matter of perspective only, one of ** frame of reference**. From your travels through the air, you may be accustomed to considering the airfoil as the object in motion in this problem, but it makes no difference whatsoever to the physical principles to consider the airfoil as fixed in space and the air as the movable component.

Models here are also 2 dimensional; the flow does not depend on the coordinate perpendicular to the image plane. It's as if the cylinder or airfoil extends for an infinite distance both in to and out of the plane of the image. (Note that other 2 dimensional flows do not necessarily have this property. For example, an axis-symmetric flow in a tube with changing radius is also 2 dimensional.) Far away from the airfoil the flow is ** uniform**. Fluid velocity is directed left to right with all fluid elements moving at the same speed; this is called the

**. The fluid approaching the airfoil must, of course, flow around it. The airfoil**

*free stream velocity**perturbs*the free stream in such a way that fluid elements must move

*tangentially*to the surface of the airfoil. If they did not, they would be moving through the solid surface!