WHEN I WATER THE LAWN with a garden hose, I sometimes partially block the end of the hose with a finger in order to make the water go farther. The action increases the speed of the water by forcing it to pass through a smaller opening.

Does the partial blockage increase the volume of water issuing from the hose per unit of time? No, the flow rate (as it is called) actually decreases, because my finger creates turbulence that removes some of the energy in the flow. Decreasing the size of the opening even more might further increase the speed of the water, but it would also decrease the flow rate. This experience is just as commonplace in a laboratory. When I want to decrease the flow of water or any other liquid from a tube, I tighten a clamp on the tube, narrowing the space through which the liquid must pass. The result has become something laboratory workers expect.

Werner J. Gatzek, a physician in Waynesboro, Va., recently surprised me with a related demonstration that ruffled my certainty. He was trying to model the flow of liquid through a human kidney because some properties of the flow had confounded medical researchers. In his experiment water drained through a garden hose from an elevated reservoir that was kept full. The lower end of the hose was connected to a rubber tube that included a loop along its course to simulate the arrangement of the input and output vessels of the kidney. The loop was held in place by a short corset consisting of three sections cut from the end of the tube; Gatzek forced the tub through the corset by stretching the sections with two needle-nose pliers. A screw clamp was mounted at the free end of the tube to simulate the variable resistance experienced by the output vessel of the kidney. The tube lay flat on a workbench, so that there were no variations in height along its length to affect the flow.

Gatzek measured the rate of flow through the system by collecting water in a small container for a measured time and then weighing the water on a dietary scale. Initially the flow was a trickle because of the constriction from the corset. While continuing to measure the flow rate Gatzek gradually tightened the clamp by repeatedly turning the screw through a measured angle. Initially the flow rate remained constant. Then, when the tube had been narrowed appreciably, the rate suddenly and dramatically increased. Eventually it reached a level more than five times higher than it was when the clamp was fully open. Further tightening of the clamp reduced the rate, which reached zero when the clamp was fully closed.

How could closing the clamp increase the flow rate? Surely the tightening of the clamp must introduce turbulence, thereby reducing the energy of the flow and the flow rate. Nevertheless, constraining the flow somehow increased it.

Once the clamp was fully closed Gatzek started opening it by specific increments, again monitoring the flow rate. This time a wider opening of the clamp was necessary to achieve a peak rate, which was higher than the peak rate attained while he was closing the clamp. When the peak was reached, further opening of the clamp suddenly decreased the flow rate until it had again become a trickle. When Gatzek plotted his measurements of flow rate versus clamp aperture, the graph revealed a hysteresis (a lag) in the pattern. Why did the system behave in a different way when the clamp was being closed?

The details of Gatzek's experiment offered no easy solution to the puzzle. The water reservoir was 5.2 meters above the workbench. The tube, made of amber latex rubber, had a thin wall and an outside diameter of five millimeters. (The characteristics of the tube Z are not critical, but it should be thin-walled and expandable.) The loop was 22 centimeters long, and the sections of the tube that served as input to and output from the loop were respectively nine and 10 centimeters long. The three sections of the corset were each one millimeter wide.

To measure the flow rate Gatzek collected samples of water for 30 seconds. After measuring the weight of a sample on the scale he converted the reading into milliliters and calculated the flow rate. (He could have measured the time the flow took to fill a container marked in milliliters.) The screw of the clamp was rotated five degrees between measurements. The tube showed no signs of fatigue under the water pressure.

Another experiment done by Gatzek compounded the puzzle. In that experiment he removed the clamp and gradually raised the height of the reservoir. At first, as intuition would suggest, the flow rate increased. After the reservoir passed a certain height, however, raising it farther either left the flow rate unchanged or decreased it as much as 30 percent. When Gatzek continued raising the reservoir beyond that region, the rate of flow began to increase again.

Gatzek originally did his experiments with a rotary pump that pushed pulses of water through the looped and clamped tube. He thought the pulsation might be responsible for the flow's perplexing properties. In 1986, however, Steven C. Wells, then a graduate student at Old Dominion University in Virginia, discovered that the paradox can be demonstrated with a steady flow. Since then Gatzek has substituted the elevated reservoir for the pump.

In the experiment in which the clamp is tightened Gatzek believes the sudden increase in the flow rate results from competition for space in the corset. The section of the input tube within the corset is under pressure from the water. When the clamp is open, the pressure causes that section to bulge against the adjacent section of the output tube within the corset, reducing the flow through that part of the output tube to a trickle.

When the clamp is gradually closed, the narrowing introduces a resistance to the flow, increasing the pressure in the region between the corset and the clamp and in the output tube within the corset. Eventually the pressure in the output tube is sufficient to counteract the pressure from the input tube. The output tube expands within the corset. Although it does so at the expense of the input tube, the result is a much greater flow rate. When the peak rate is reached, further tightening of the clamp introduces so much resistance that the flow gradually decreases and then stops.

I took another approach to the puzzle by focusing on the energy. The point that intrigued me was that although tightening the clamp is certain to remove energy from the flow, the flow actually then becomes more energetic. To follow my steps in solving this paradox you must first understand several general features of water flow. The flow rate through a tube is equal to the product of the speed of the flow and the cross-sectional area of the tube. Along the tube the flow rate must be constant, and so there can be no magical appearance or disappearance of water. If the tube narrows, the water must flow faster; if it widens, the water must flow slower. If the tube does not change in width, the speed of flow will not change.

When the flow lacks turbulence, it is said to be laminar. In such a state it can be traced by streamlines that show where individual parcels of water travel. Every time a new parcel travels through any particular point in the system, it thereafter follows the path taken by all the parcels that preceded it through that point.

Flowing water has two types of pressure: static and dynamic. Static pressure exists in a conduit even if the water is stationary; only where the water is in direct contact with the atmosphere, as at the end of a hose, is there no static pressure. Dynamic pressure is proportional to the square of the flow speed. The pressures are related to energy. The static pressure is a form of potential energy per unit of volume; the dynamic pressure is the kinetic energy per unit of volume. If the flow changes in height, the gravitational potential energy per unit of volume must also be considered.

The Bernoulli equation, named after the 1 8th-century Swiss mathematician Daniel Bernoulli, states that in laminar flow the sum of those energies must be constant along a streamline. Suppose the water flows through a horizontal tube that narrows. As water enters the narrow section its speed and kinetic energy (dynamic pressure) increase because energy is transferred from the static pressure. If the tube returns to its former width, the energy is transferred back to static pressure. At all times the sum of the static and dynamic pressures is unchanged.

Suppose the tube descends without any change in width. As water flows through the tube, gravitational potential energy is converted into static pressure. Again the sum of the energies is unchanged.

The Bernoulli equation and the requirement that the flow rate be constant through a system are not sufficient to explain Gatzek's experiment. If the flow were laminar, the flow rate through the corset would be the same as it is when the corset is removed. The fact that the flow is only a trickle indicates the corset introduces enough turbulence to remove nearly all the energy in the flow. I believe that when the clamp is tightened the flow rate increases because the loss of energy in the output tube within the corset is greatly reduced.

To develop my explanation I began with a simple system of laminar flow in which water drains through a horizontal tube of constant width. It is attached near the bottom of a container, which is kept full. Since the water that issues from the tube is directly exposed to the atmosphere, the static pressure at the open end must be zero. Since the tube's radius is constant, the speed of the flow must be the same throughout its length. Therefore, according to the Bernoulli equation, the static pressure must be zero at any point in the tube.

Suppose you are looking at a parcel of water in the container near the drain. The pressure on the drain side is lower than the pressure on the side where the tube leaves the drain. The pressure difference accelerates the water parcel toward the drain, giving the parcel the speed of the rest of the water flowing through the tube.

I next worked with a drain tube that has a vertical section connecting two horizontal sections. Again the static pressure in the final horizontal section must be zero. Moreover, the flow through the tube must have a constant speed, even in the vertical section, because the radius is constant. What differs in this setup from the preceding arrangement is that when the water flows through the vertical section, gravitational potential energy is converted into static pressure. At the top of the vertical section the static pressure is negative.

Focus again on a parcel of water about to enter the drain. Since this time the parcel has a negative static pressure on one side, it accelerates 19 more and enters the drain at a higher speed than the parcel in the preceding example. With the faster flow the flow rate is higher.

Note that the kinetic energy and the speed of the water are derived from the static pressure and the gravitational potential energy of the water in the container. Suppose a constriction in the final horizontal section of the tube introduced turbulence that removed energy from the flow. Then the kinetic energy of the water leaving the end of the tube would be lower and the water speed throughout the system would be reduced. In Gatzek's experiment the corset introduces two constrictions and the tightened clamp gives rise to a third.

To review Gatzek's experiment I worked with a schematic representation of the tube that included two short, narrow regions and a clamp in the final horizontal section [see Figure 4]. The narrow regions represent the input and output tubes at the point where they pass through the corset. The static pressure at the tube's open end and in the regions numbered 3 and 4 is zero. In region 1 the static pressure is high, collapsing the output section of the tube in the corset and allowing only a trickle to flow through the system. Region 2 represents the loop.

Examine the graph below the diagram of the schematic tube. The top line represents the sum of the pressures at points along the length of the tube. The line below it represents the static pressure. The distance between the two lines represents the dynamic pressure, or kinetic energy, of the flow. Since the water speed is constant and small, the distance between the lines is also constant and small.

In the vertical section of the tube the static pressure increases at the expense of the gravitational potential energy. Both lines in the graph rise, but the kinetic energy of the flow does not increase. Both the input and the output tube within the corset introduce turbulence that removes energy, reducing the static pressure. Since the input tube is relatively wide, the energy loss there is small. The energy loss in the partially collapsed output tube is considerably larger, reducing the static pressure to zero.

Next I drew a schematic diagram and a graph representing the maximum flow rate. Again the static pressure in region 4 is zero. Since the size of the tube in region 4 is unchanged, the fact that the flow rate is now greater means that the water speed is higher. Hence the two pressure lines in the graph are more widely spaced than they are in the preceding example.

Note the energy losses at the simulated input and output of the corset. Although the loss at the input is somewhat greater than it was before, the loss at the output is considerably reduced owing to the increased size of the output tube within the corset. The combined loss of energy at the corset and at the clamp is less than it was when the clamp was fully open. The reduced loss shows up as the added kinetic energy in the flow.

When does the corset allow maximum flow? Gatzek suggested that the flow rate is maximum when the input and output tubes within the corset are equal in size. To check his suggestion I assumed that the energy loss from a constriction in the tube is proportional to the ratio of the tube's normal crosssectional area and its cross-sectional area within the constriction. Next I added the losses at the input and output points of the corset-under the requirement that the sum of the areas of the constricted tubes had to be the cross-sectional area of the corset. I then checked for the case in which the combined loss is least. Indeed it is least when the input and output tubes within the corset are of equal size. Thus part of the puzzle is solved. When the clamp is tightened, the increased static pressure in the ouput tube within the corset widens the output tube, greatly reducing the energy loss there and allowing more of the energy of the flow to end up as kinetic energy.

Why is the transition to the peak flow rate sudden? I believe the suddenness arises from the shape of the walls where the input and output tubes touch in the corset. When the clamp is being tightened, the wall of the input tube is convex and the wall of the output tube is concave, leaving the output tube at a disadvantage. As pressure builds in that tube it suddenly becomes high enough to overcome the concave shape of the wall, whereupon the wall flexes outward. Then the walls of the input and output tubes are flat against each other and the flow rate suddenly increases to its peak value.

Why does the system exhibit hysteresis? Suppose the clamp has been tightened to the point where the flow rate is maximum. Since the walls of the input and output tubes are then flat against each other, the energy loss at the corset is either at the minimum or nearly so. If the clamp is opened somewhat, the walls of the input and output tubes remain flat and the energy loss in the corset is unchanged. Since the energy loss at the clamp is reduced, however, the combined energy loss in the system is also reduced. For this reason more of the energy in the flow is kinetic energy and the flow rate is at a higher peak value.

One more piece of the puzzle remains. What accounts for the behavior of the flow rate when the clamp is left open and the reservoir is gradually raised? Gatzek's explanation serves well. Initially the flow rate increases because the added height of the reservoir pushes water through the system faster. Soon, however, the increased pressure in the input tube of the corset begins to collapse the output tube. Once the reservoir is raised beyond a certain height, the collapse either holds the flow rate constant or decreases it. If the reservoir is raised even more, the rate begins to increase again because the high pressure in the input tube expands the corset.

Gatzek's experiments may help to explain some characteristics of the flow through the glomerulus in the kidney. The afferent vessel (the input arteriole) and the efferent vessel (the output arteriole) pass through the same opening in Bowman's capsule and may compete for space there. The efferent vessel does not have a clamp, but it is subject to a vasoconstricting hormone called angiotensin, a derivative of the renin produced by the kidney. In some physiological conditions the kidney may produce angiotensin in order to constrict the efferent vessel, increasing its resistance to flow. The increased resistance may then widen the vessel in the opening in Bowman's capsule, leading to a noteworthy increase in the flow rate.

Bibliography

A NEW CONCEPT OF RENAL HEMODYNAMICS. W. J. Gatzek in Medical Hypotheses, Vol. 4, No. 3, pages 221-230; May-June, 1978.

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