Why Pressure Decreases as Velocity Increases: Bernoulli’s Principle Simplified
Before knowing the relation between velocity and pressure, we must know the meaning of both pressure and velocity. Now, we can define pressure as the physical and external force exerted or applied to an object. In scientific terms, the force exerted on a unit area can be termed pressure. As pressure can be explained in terms of force acting on a unit area, the formula for pressure is F/A.
What is Velocity?
Velocity can be defined as the rate of change of the position of an object in comparison to a given time frame. The definition of velocity can be confusing for some students, but velocity, in simple terms, means the speed of an object moving in a specific direction. Velocity is a vector quantity as both speed and direction are required to identify it. Meter per second is the S.I. unit of velocity.
Pressure and Velocity Relation
In the comparison of pressure velocity relations, one thing is common, and that is, both are macroscopic parameters that govern plenty of natural occurrences. On the one hand, pressure is the measurement of force per unit area. And on the other hand, velocity is the measure of the rate of change of displacement.
Two independent formulas explain the pressure velocity relationship in a more convincing manner.
Bernoulli’s Formula for Relation Between Pressure and Velocity
The first formula that defines the relationship between pressure and velocity is Bernoulli's principle. Daniel Bernoulli first gave this formula in his book Hydrodynamica, which was published in the year 1738.
In this formula, Bernoulli explains that in thermodynamics or fluid dynamics, the increase in the speed of any in-compressible or non-viscous fluid is a result of the decrease in the static pressure exerted on the fluid.
The Formula given by Bernoulli under this principle to explain the relation of pressure and velocity is:
\[P + \frac{1}{2}\rho v^{2}+\rho gh = \text{Constant}\]
In the above formula,
P denotes the pressure of the in-compressible, non-viscous fluid that is measured using N/m2.
ρ denotes the density of the non-viscous liquid, which is measured using Kg/m2.
v denotes the velocity of the in-compressible, non-viscous liquid, measured using m/s.
g stands for acceleration due to gravity, which is measured through m/s2.
Lastly, h denotes the height from a reference level where the fluid is contained. It is measured in meters (m).
In simple words, Bernoulli’s formula explains the relation of pressure and velocity is inversely proportional. It means that when pressure increases, the velocity decreases, keeping the algebraic sum of potential energy, kinetic energy, and pressure constant. In a similar way, when velocity increases, the pressure decreases.
Bernoulli’s principle for pressure and velocity relation can be applied to different types of fluid flow. But, it has to be in different forms. The simple form of Bernoulli’s equation is only applicable for in-compressible and non-viscous fluids flow.
Laplace Correction for Newtonian Formula
The Laplace equation was given as a transformation to Newton’s formula for the velocity of sound. The correction was given by Pierre-Simon Laplace, in which he transformed the equation taking into consideration the following:
There is no heat exchange as the compression and rarefaction of the sound wave takes place rapidly.
The temperature does not remain constant, and the movement of a sound wave in the air is an adiabatic process.
Laplace explains the relation of velocity and pressure by the following formula:
\[V=\sqrt{\frac{\gamma P}{\rho }}\]
In the above equation,
v denotes the sound waves’ velocity, which is measured using m/s.
P denotes the pressure of the medium measured using N/m2.
𝛾 denotes the adiabatic constant.
ρ stands for the density of the medium, measured using kg/m2.
Laplace’s correction for Newton’s formula explains the relation between velocity and pressure, as pressure is directly proportional to the square of velocity. Hence, when pressure increases, velocity also increases and vice versa.
Connection between Velocity, Pressure, and Area (In fluid dynamics)
Cross-section of the fluid’s body through which a fluid is flowing is inversely proportional to the velocity and pressure.
In order to prove this equation, Let us first assume that there are two bodies in which the fluid is flowing. Let us further assume that we got the different cross - sections as A1 and A2 respectively, and also the volume V1 and V2.
And under the further assumption that the given fluid is incompressible, which means the volume of that object at a given point is going to be the same. This phenomenon is commonly called as the “Continuity equation”
A 1 v 1 = A 2 v 2
To put this equation in simpler terms, the equation represents that if the cross section Area of the object carrying the fluid, increases, then the volume of that object will have to reduce.
Pressure and velocity are also bound in an inversely proportional relation, according to Bernoulli's principle (which is considered as just a formulation of the conservation of energy). The pressure reduced on the velocity increases in a scenario where a fluid flows into a narrower cross-section.
Important Things to Remember
Listed down here are some of the most important things that the students have remembered from this topic.
The definition of velocity: In physics, the velocity refers to the rate with which the object is changing its location over the course of time
The physical force exerted per unit of the volume is referred to as Pressure,
There is inversely proportionality relation between the two terms of Velocity and pressure, which means when one of those two increases then the other one will start to fall or decrease.
All energy in a present in a fluid is respecting the principle of conservation of the fluids, which entail that the sum of all the energy will remain the same.
FAQs on Understanding the Relation Between Pressure and Velocity
1. What is the fundamental relationship between pressure and velocity for a moving fluid?
The fundamental relationship, described by Bernoulli's principle, states that for an ideal fluid in a steady flow, pressure and velocity are inversely proportional. This means that in a region where the fluid's velocity is high, its pressure is low, and where the velocity is low, the pressure is high. This principle is a direct consequence of the conservation of energy for a flowing fluid.
2. What is the mathematical formula that connects pressure, velocity, and height for a fluid in motion?
The mathematical relationship is given by Bernoulli's equation. For any two points along a streamline in an ideal fluid, the equation is:
P + ½ρv² + ρgh = constant
Where:
- P is the static pressure of the fluid.
- ρ (rho) is the density of the fluid.
- v is the velocity of the fluid flow.
- g is the acceleration due to gravity.
- h is the height of the fluid above a reference point.
3. Why does the pressure of a fluid decrease when its velocity increases?
This phenomenon is explained by the conservation of energy. When a fluid is forced to speed up, for example, by flowing into a narrower section of a pipe, its kinetic energy increases. Since the total energy of the fluid system must remain constant (according to Bernoulli's principle), this increase in kinetic energy must be balanced by a decrease in another form of energy. In this case, the fluid's internal or pressure energy decreases, resulting in lower pressure.
4. How do pressure and velocity change when a fluid flows through a pipe with a narrowing section (a nozzle)?
When a fluid flows from a wider section of a pipe into a narrower section, its behaviour is governed by two principles:
- Equation of Continuity (A₁v₁ = A₂v₂): As the cross-sectional area (A) decreases, the fluid's velocity (v) must increase to maintain a constant flow rate.
- Bernoulli's Principle: Because the velocity has increased in the narrow section, the pressure in that section must decrease to conserve the total energy of the fluid.
5. What is a common real-world example of the pressure-velocity relationship?
A common example is a paint sprayer or a perfume atomiser. In these devices, air is blown at high speed across the top of a small tube (a dip tube) that leads down into the liquid. This high-velocity airstream creates a region of low pressure above the tube. The higher atmospheric pressure on the surface of the liquid in the container then pushes the liquid up the tube and into the airstream, where it is dispersed as a fine mist.
6. How does an airplane wing generate lift using this principle?
An airplane wing, or airfoil, is designed with a curved upper surface and a flatter lower surface. As the plane moves, air splits to flow over and under the wing. The air flowing over the curved top surface has to travel a longer distance in the same amount of time, so it moves at a higher velocity. According to Bernoulli's principle, this higher velocity results in lower pressure on the top of the wing. The air below the wing moves slower, creating higher pressure. This pressure difference between the bottom (high pressure) and top (low pressure) of the wing generates an upward force called lift.
7. Is it a misconception that higher velocity always means lower pressure in all fluid scenarios?
Yes, it can be a misconception if not properly contextualised. The inverse relationship described by Bernoulli's principle applies specifically along a single streamline in a steady, non-viscous, incompressible flow. It does not necessarily apply when comparing different, unrelated flows or in situations with significant energy addition (like from a pump) or energy loss (due to friction or turbulence). The principle is an energy conservation statement, not a universal law for any two points in any fluid.
