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Relation Between Pressure and Velocity

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Last updated date: 17th Apr 2024
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What is Pressure?

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 Relation Between Pressure and Velocity

1. Explain the Term Velocity?

The term velocity gives us a basic knowledge of the speed and direction in which an object moves. In situations where we have to compare two objects moving in the opposite direction and have variable speeds, the term velocity becomes helpful. The velocity of an object can be defined as the rate of change of speed with respect to the object’s position. In a simpler sense, velocity means the speed of an object in a specific direction. Velocity is a vector quantity as it requires both magnitude and direction to define it. The S.I. unit of velocity is m/s-1.

2. Define the Relation of Flow, Pressure and Velocity.

Flow, pressure, and velocity are the three most important terms of fluid dynamics. Before knowing the relation between each of them, we need to know their meaning. 


Flow can be defined as the measure of air output in terms of volume per unit of time. On the other hand, pressure can be defined as the force exerted on a unit area. And velocity can be defined as the rate of change of an object’s speed with respect to its position. Now the relationship between flow, pressure and velocity can be explained by applying Bernoulli’s principle. According to Bernoulli’s equation, when a fluid flows through a tube and if we decrease the diameter of the tube, then the velocity of the fluid increases and the pressure also decreases.

3. What is the principle of conservation of energy for fluids?

The principle of conversion of law for a system with fluids, states that while a steady flow, the sum of all types of energy in a fluid will remain the same at all points on the streamlined flow. It includes the sum of all the forms of energies - kinetic energy, potential energy, and internal energy will remain the same in a steady flow or will be acting as a constant term. When the speed of flow of the fluid increases then the kinetic energy of the fluid will increase too and to counter this the potential energy and the internal energy have to decrease equally, in order to maintain the constant status of the sum of all forms of the energy.


The principle of conservation of energy can be stated as the energy per unit volume in a water body like a reservoir, will always remain the same in every place in it.

4. What was the Correction for Newtonian Formula given by Laplace?

Pierre-Simon Laplace challenged Newton's formula and suggested that the change in pressure and volume of gas because of the passage of sound waves through the gas is not isothermal as stated by Newton but it is adiabatic. Thus, it is more appropriate to use the adiabatic bulk modulus rather than the other modulus of elasticity. i.e. E = Kadi. Thus Newton's explanation of Boyle's law was extended and updated. Hence the velocity of sound in a gas is given by - 

\[V=\sqrt{\frac{\gamma P}{\rho }}\]

5. Why is pressure inversely proportional to velocity?

Pressure is basically the measurement of forces exerted per unit area of an object. And the velocity refers to the rate with which the object is changing its location over the course of time. 


Both of these terms share an inversely proportional relationship between, which means when the velocity decreases the pressure will start to increase in accordance.


This relationship between the pressure and velocity can be explained by the principle of conservation of energy at a molecular level. It can be shown with the equation given as

P V  =  constant


Where P is the pressure and the V is used for velocity. All at a constant temperature.


And by this equation, one can say that if velocity increases then the pressure decreases.

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