第14章流体力学

Water pressure in homes is sometimes lower than normal during times of heavy use, such as hot summer days. The drop in pressure occurs in the water main before it reaches individual homes. Let us consider flow through the water main as illustrated in Figure 14.39. We can understand why the pressure $p_1$ to the home drops during times of heavy use by rearranging the equation for flow rate:

In this case, $p_2$ is the pressure at the water works and R is the resistance of the water main. During times of heavy use, the flow rate Q is large. This means that $p_2-p_1$ must also be large. Thus $p_1$ must decrease. It is correct to think of flow and resistance as causing the pressure to drop from $p_2$ to $p_1$. The equation $p_2-p_1=RQ$ is valid for both laminar and turbulent flows.

We can also use $p_2-p_1=RQ$ to analyze pressure drops occurring in more complex systems in which the tube radius is not the same everywhere. Resistance is much greater in narrow places, such as in an obstructed coronary artery. For a given flow rate Q, the pressure drop is greatest where the tube is most narrow. This is how water faucets control flow. Additionally, R is greatly increased by turbulence, and a constriction that creates turbulence greatly reduces the pressure downstream. Plaque in an artery reduces pressure and hence flow, both by its resistance and by the turbulence it creates.

An indicator called the Reynolds number $N_{\text{R}}$ can reveal whether flow is laminar or turbulent. For flow in a tube of uniform diameter, the Reynolds number is defined as

where $\rho$ is the fluid density, v its speed, $\eta$ its viscosity, and r the tube radius. The Reynolds number is a dimensionless quantity. Experiments have revealed that $N_{\text{R}}$ is related to the onset of turbulence. For $N_{\text{R}}$ below about 2000, flow is laminar. For $N_{\text{R}}$ above about 3000, flow is turbulent.

For values of $N_{\text{R}}$ between about 2000 and 3000, flow is unstable—that is, it can be laminar, but small obstructions and surface roughness can make it turbulent, and it may oscillate randomly between being laminar and turbulent. In fact, the flow of a fluid with a Reynolds number between 2000 and 3000 is a good example of chaotic behavior. A system is defined to be chaotic when its behavior is so sensitive to some factor that it is extremely difficult to predict. It is difficult, but not impossible, to predict whether flow is turbulent or not when a fluid’s Reynold’s number falls in this range due to extremely sensitive dependence on factors like roughness and obstructions on the nature of the flow. A tiny variation in one factor has an exaggerated (or nonlinear) effect on the flow.