Bernoullian Thoughts

Bernoullian Thoughts Daniel Bernoulli was born into a family of mathemeticians on February 8, 1700. He was the only person in his family to make an impressive mark on physics. Bernoulli became a Swiss physicist and mathmatician who made enourmous contributions to the world of physics. He uncovered many significant phenomena in hydrodynamics, and in 1738, published his most famous work, Hydrodynamica, which was a study of equilibrium, pressure, and velocity of fluids. He proved that as the velocity of fluid flow increases, its pressure decreases. Bernoullis principle was an early formulation of the subsequent idea of the conservation of energy. Bernoullis Hydrodynamica was also the first attemt to explainof the behavior of gasses with changing pressure and temperature.

This was the beginning of the kinetic theory of gasses. His gas model has been revived and transformed into a powerful theory regarding the thermal and mechanical properties of gases using the atomic hypothesis. Bernoulli thought of the corpuscles of the gas as so minute that there were practically an infinite number under ordinary conditions, even in a small container. In their rapid motion, these corpuscles collide with each other and also with the rigid walls of the closed vessel. The collisions, however, can be assumed to be perfectly elastic; therefore, the kinetic energy of the particles is conserved and the motion can continue undiminished. Therefore, the pressure which the gas is expected to exert against all sides of the container is caused by the incessant impact of millions of high speed particles; hence the name impact theory of gas pressure.

Imagine a gas filled cylindrical container with the top end that is able to slide up and down, in and out like a piston. If the volume is slowly decreased, the corpuscles are more crowded in the progressively smaller space and the number of collisions per second with the walls would be larger (i.e., the pressure should become greater, as observed). Bernoulli even calculated the magnitude of this expected increase and found that it corresponded to Boyles experimental law. At the time of Bernoullis discovery, his work was generally ignored. The lack of general attention was due to the unclear knowledge of gases. Yet more than a century later, his work simultaneously clarified the main problems of the nature of gases, heat and chemistry.

For Bernoulli, in effect, had made two enormous leaps in his thinking for which most scientists were not ready to take. First, he illucidates, the direct equivalence of heat and internal molecular motion, ignoring any interactions between the two. Second, he confirmed the idea that a well-defined numerical relationship, such as Boyles simple law, could be deduced from a chaotic picture of randomly moving particles. Bernoullis principle was centerd around the notion that we suppose a small portion of liquid flow from one point to another point, and that change of position is affected without incurring any waste of energy. From the principle of conservation of energy, it may be asserted that the total energy is not changed during the displacement.

This statement is known as Bernoullis theorem and is often expressed as: P + 1/2 pv2 + pgy = constant Bernoullis equation states that the sum of the pressure (P), the kinetic energy per volume (1/2 pv2), and the potential energy per unit volume (pgy) have the same value at all points along a streamline. Using Bernoullis law, because there is no waste of energy during the passage of the liquid, the total energies at each three places are equal. If the fluid is incompressible then the internal energy is the same, which proves, in turn, that Bernoullis equation holds true along any streamline. Bernoullis foregoing principle explains a number of phenomena about the behavior of liquids which, at first, seem strange. Suppose two ships are steaming side by side in still water: The relative motion of the ships with respect to the water will remain unchanged if the ships are imagined to be stationary and the water imagined to flow with the same velocity in the opposite direction.

the water entrapped between the ships will speed up because of the narrow space. As a consequence, the pressure in the water between the ships will be reduced and will become less than the water pressure on the far sides of the ships. The excess pressure will cause the ships to become closer in proximity. Bernoullis theorem, when applied to gasses instead of liquids, explains such effects as the curved flight of a tennis ball that is spinning when served, the action of an atomizer in dividing a jet of liquid into a fine spray, the reduction of gas pressure in a container by using an aspirator connected to a water faucet and the propulsion of a ship by wind power using cylindrical rotors instead of sails. Bernoullis theorem provides a means for measuring the flow of a liquid through a pipe.

A section of pipe containing a constriction or throat is inserted in the pipe line and the pressures are measured both at the throat and in the pipe by pressure gauges or their equivalent. The rise of liquid in small tubes, called manometers, indicate the pressure. The pipe beyond the throat flares out slowly so that the velocity of the liquid can be reduced without disturbing the streamline flow. Since the velocity of the liquid is greater at the throat than in the pipe, the pressure at the throat will be less than that in the pipe, as prescribed by Bernoullis equation, and consequently, the liquid in the throat manometer elevations, together with a knowledge of the cross-sections of pipe and throat, permit the liquid flow to be measured. This device is known as a Venturi meter. Bernoullis theorem is not only applicable for liquids, but also for gasses.

In this case, the mathematical treatment is complicated by the fact that gases are highly compressible, but the general effect is the same as previously described; namely, that when a flowing stream of gas speeds up, its pressure decreases, and vice versa. The lift on an aircraft wing can be explained by this effect. Airplane wings are designed so that the air speed above the wing is greater than that below the wing. As a result, the air pressure above the wing is less than the pressure below, and there is a net upward force on the wing called the lift. In conclusion, Bernoulli contributed much to the world and to the realm of physics.

Daniel Bernoulli derived a fundamental expression that relates pressure to fluid speed and elevation. Bernoullis equation is not a freestanding law of physics, but instead a consequence of energy conservation as applied to the ideal fluid.