The branch of physics that studies the features of fluid motion is called hydrodynamics. One of the main mathematical expressions of hydrodynamics is the Bernoulli equation for an ideal fluid. This article is devoted to this topic.
What is a perfect fluid?
Many people know that a liquid substance is such an aggregate state of matter that retains volume under constant external conditions, but changes its shape with the slightest effect on it. An ideal fluid is understood to mean a fluid substance that has no viscosity and is incompressible. These are the two main properties that distinguish it from real fluids.
Note that almost all real liquids can be considered incompressible, since a huge external pressure is needed for a small change in their volume. For example, if you create a pressure of 5 atmospheres (500 kPa), then water will increase its density by only 0.024%. Regarding the issue of viscosity, for a number of practical problems, when water is considered as a working fluid, it can be neglected. For completeness of information, we note that the dynamic viscosity of water at 20 o C is 0.001 Pa * s 2 , which in comparison with this value for honey (> 2000) is a meager value.
It is important not to confuse the concepts of an ideal fluid and an ideal gas, since the latter is easily compressible.
Continuity equation
In hydrodynamics, they begin to consider the motion of an ideal fluid by studying the equation of continuity of its flow. To understand the essence of the issue, it is necessary to consider the movement of fluid through the pipe. Imagine that at the inlet the pipe has a cross-sectional area A 1 , and at the outlet A 2 .
Now suppose that the fluid flows at the beginning of the pipe at a speed of v 1 , which means that over time t, a stream of volume V 1 = A 1 * v 1 * t will pass through section A 1 . Since the fluid is ideal, i.e. incompressible, the exact same volume of water should exit the end of the pipe in time t, we obtain: V 2 = A 2 * v 2 * t. From the equality of the volumes V 1 and V 2 follows the continuity equation for the flow of an ideal fluid:
A 1 * v 1 = A 2 * v 2 .
From the resulting equation it follows that if A 1 > A 2 , then v 1 should be less than v 2 . In other words, by decreasing the cross section of the pipe, we thereby increase the speed of the fluid flow emerging from it. Obviously, this effect was observed by every person in life who at least once watered flower beds or a garden from a hose, so, covering a hole in the hose with oneβs finger, one can observe how the stream of water that spouts from it becomes stronger.
The equation of continuity for a branched pipe
It is interesting to consider the case of the movement of an ideal fluid through a pipe that has not one but two or more outlets, that is, it is branched. For example, the cross-sectional area of ββthe pipe at the inlet is A 1 , and at the exit it branches into two pipes with sections A 2 and A 3 . We define the flow rates v 2 and v 3 if it is known that water enters the inlet at a speed v 1 .
Using the continuity equation, we obtain the expression: A 1 * v 1 = A 2 * v 2 + A 3 * v 3 . To solve this equation for unknown speeds, you need to understand that at the output, no matter what pipe the stream is in, it moves at the same speed, that is, v 2 = v 3 . This fact can be understood intuitively. If the outlet pipe is divided into two parts by a certain partition, the flow velocity will not change. Given this fact, we obtain the solution: v 2 = v 3 = A 1 * v 1 / (A 2 + A 3 ).
Bernoulli equation for perfect fluid
Daniil Bernoulli, a Swiss physicist and mathematician of Dutch origin , in his work "Hydrodynamics" (1734) presented the ideal fluid equation describing its motion. It is written in the following form:
P + Ο * v 2/2 + Ο * g * h = const.
This expression reflects the law of conservation of energy in the case of fluid flow. So, the first term (P) is the pressure directed along the fluid displacement vector that describes the flow, the second term (Ο * v 2/2) is the kinetic energy of the fluid substance, and the third term (Ο * g * h) is its potential energy.
Recall that this equation holds for an ideal fluid. In reality, there is always friction of a fluid substance on the walls of the pipe and inside its volume; therefore, an additional term is introduced into the Bernoulli equation that describes these energy losses.
Using the Bernoulli Equation
It is interesting to cite some inventions that use conclusions from the Bernoulli equation:
- Chimney and hoods. From the equation it follows that the greater the speed of movement of a fluid substance, the lower its pressure. The air velocity at the top of the chimney is greater than at its base, so the flow of smoke always tends upward due to pressure differences.
- Water pipes. The equation helps to understand how the pressure of the water in the pipe changes if the diameter of the latter changes.
- Aircraft and Formula 1. The angle of the wings of the aircraft and the wing "Formula 1" provides the difference in air pressure above and below the wing, which creates a lifting and pressing force, respectively.
Fluid flow regimes
The Bernoulli equation does not take into account the regime of fluid motion, which can be of two types: laminar and turbulent. The laminar flow is characterized by a calm flow, in which the fluid layers move along relatively smooth paths and do not mix with each other. The turbulent regime of fluid motion is characterized by the random movement of each molecule that makes up the flow. A feature of the turbulent regime is the presence of turbulences.
How the liquid will flow depends on a number of factors (system features, for example, the presence or absence of roughness on the inner surface of the pipe, the viscosity of the substance and its speed of movement). The transition between the considered modes of motion is described by Reynolds numbers.
A striking example of a laminar flow is the slow movement of blood through smooth blood vessels. An example of a turbulent flow is the strong pressure of water from a tap.