Mechanical Properties of Fluids: Pascal's Law and Bernoulli's Principle

The mechanical properties of fluids determine the behaviour of the fluid under applied forces.

• Fluids is a substance that flows under the action of applied force
• They do not have a shape of its own.
• Both liquids and gases are fluids.
• Fluids contain physical properties that control the way they react to external forces.
• They have three physical properties, density, viscosity and surface tension which are mainly examined based on molecular mass.
• The science of the mechanical properties of fluids is called Hydrostatics. 

Key Terms: Fluids, density, Viscosity, Surface tension, Molecular mass, Mass, Kinematic properties, Hydrostatics, Velocity, Bernoulli’s principle, Conservation of energy, Conservation of mass

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Fluids

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Anything that does not have a definite shape is referred to as a fluid.

• It has no resistance to changing its shape.
• Furthermore, the fluid's shape is determined by the shape of the container in which it is stored.
• A fluid is also incompressible and has its own free surface.
• Gas, on the other hand, is compressible and expands to fill all available space.
• The study of the mechanical properties of fluids is known as hydrostatics.
• The pressure acting on a liquid or gas determines its volume.
• Because liquids have a fixed volume, the volume change caused by changes in external pressure is minimal.
• Gases, on the other hand, are not the same because they do not have a defined volume.

Note: Fluids are liquids and gases with the property to flow in a certain direction on the application of external force.

Characteristics of Fluids

The different characteristics of fluids are

• They have a very small shearing stress.
• A little application of force brings them in motion.
• Gases are highly compressible.
• An external pressure in liquids causes a very small change in their volume.
• Liquids have free surfaces of their own while gases do not.

Also Read: Surface Energy

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Mechanical Properties of Fluids

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Even though each fluid is unique in terms of composition and specific attributes, there are several characteristics that all fluids share.

These qualities can be classified as follows:

• Kinematic properties: These properties help in the understanding of the motion of a fluid. The velocity and acceleration of fluids are kinematic properties.
• Thermodynamic properties: These characteristics aid in determining the fluid's thermodynamic state. The thermodynamic properties of fluids are temperature, density, pressure, and specific enthalpy.
• Physical properties: The physical properties help in determining the physical state of any fluid, such as colour and odour.

Also Read: Poisson’s Ratio

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Concept of Pressure

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Pressure is defined as the normal force acting per unit area of a surface.

The formula of pressure is given by

\(P=\frac{F}{A}\)

Where

• P is the pressure
• F is the force applied
• A is the area of the surface

Pressure is a scalar quantity since the force that appears in the numerator is the normal component of force.

• The SI unit of pressure is N m^-2, also called Pascal (Pa).
• Fluid pressure can be measured by Manometer, Pressure gauge, and Pressure transducer.
• The air around us also exerts pressure, known as Atmospheric pressure.
• The unit of atmospheric pressure is “atmosphere (atm)”.
• At sea level, the atmospheric pressure is 1 atm i.e. equal to 1.013 x 10⁵ Pa.

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Pascal’s Law

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Pascal’s law states that “Pressure in fluid at rest is same at all points which are at the same height”.

• The change in pressure applied to the enclosed fluid can be transmitted to every point of the fluid and the container vessel's walls without loss.
• The volume of an incompressible fluid moving through a conduit with a non-uniform cross-section will be the same as the steady flow.

Formula of Pascal Law

The formula of Pascal’s law is given by

F = PA

Where

• F is the force
• P is the pressure
• A is the area of the cross-section

Example of Pascal’s Law

Ques. Due to a force applied to a piston, a pressure of 2000 Pa is transmitted throughout a liquid column. What force is applied to a piston with a surface area of 0.1 m²?

Ans. The force applied to the piston can be calculated using Pascal's Law.

F = PA

Given,

• P = 2000 Pa = 2000 N/m²
• A = 0.1 m²

Substituting values, we get

F = 2000 x 0.1 = 200 N

Also Read: Temperature

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Flow of Fluids

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The study of fluids in motion is called fluid dynamics. When a fluid is in flow, its motion can be either smooth or irregular depending on its velocity of flow.

Various types of flow of fluids are

Streamline Flow

A streamline refers to a curve where it forms a tangent at any point giving the direction of the fluid velocity at that particular point.

• It is parallel to the line of force present in an electric or magnetic field.
• In a constant flow of the streamline, its pattern becomes static with time giving the real path for the fluid particle.
• A constant flow can also be referred to as a streamlined or laminar flow.
• Two streamlines can never intersect each other, if they do then there will be diversions created in the flow of the fluid making it unsteady.

Tube of Flow

A tube of flow is a tubular region of fluid surrounded by a streamline-based barrier. Because no fluid may pass through the borders of a tube of flow, it behaves like a pipe of the same shape.

Laminar Flow

If the liquid flows over a horizontal surface in the form of layers of different velocities, then the flow of liquid is called Laminar flow.

• The motion of the fluid particles in laminar flow is very ordered, with particles near a solid surface travelling in straight lines parallel to that surface.
• Laminar flow can be recognized by strong momentum diffusion and low momentum convection.

Turbulent Flow

The flow of fluid in which the velocity of all particles crossing a given point is not the same and the motion of the liquid becomes disordered or irregular is called Turbulent flow.

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Equation of Continuity

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The equation of continuity can be stated as

“For the streamlined flow of an incompressible fluid through a pipe of carrying cross-section, the product of the velocity of the fluid (V) at a point and cross-sectional area (A) at that point remains constant”. 

AV = Constant

• It is a statement of conservation of mass.
• This equation shows that the fluid velocity at a point is inversely proportional to the cross-sectional area of the pipe.
• It means liquid flows faster at the section where streamlines are spaced closer and vice-versa.

Read More: Dynamic Lift

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Applications of Pressure

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Fluids include all liquids and gases. Fluid pressure is the force exerted typically at a unit area of a fluid's surface.

• Consider a liquid that is contained in a sealed container.
• Assume the liquid's temperature is 20°C and the pressure is atmospheric.
• At 100°C, this liquid will evaporate. When a liquid evaporates, the molecules escape from the liquid's free surface.
• The space between the free liquid surface and the vessel's top collects these vapour molecules.
• The pressure of the accumulating vapours exerts itself on the liquid surface.
• The liquid's vapour pressure is referred to as pressure. Alternatively, this is the pressure at which the liquid turns into vapours.

Read More: Venturimeter

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Surface Tension

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It is the phenomenon that occurs when the surface of a liquid comes into contact with another phase, according to the definition (it can be a liquid as well). Liquids prefer to have as little surface area as feasible. The liquid's surface behaves like an elastic sheet.

• The forces of attraction between particles within a liquid, as well as the forces of attraction of solids, liquids, and gases in contact with it, determine surface tension.
• The work or energy required to remove a unit area's surface layer of molecules is roughly equivalent to the energy responsible for surface tension.
• Surface tension is usually expressed in dynes/cm, which is the force necessary to break a 1 cm film.

The surface tension of several liquids is listed in the table below:

Surface tension can be stated mathematically as follows:

T = F/L

Where,

• F is the force per unit length
• L is the length of the force's action.
• T is the liquid's surface tension.

Dimension of Surface Tension

The formula for surface tension is, as we all know.

Surface tension = F/L

We know that F = ma, so we may substitute the value in the equation.

=ma/L

We get the equation by equating in the fundamental quantities.

=MLT^-2L^-1

Solving further, we get

=MT^-2

Hence, the dimensional formula of surface tension is MT^-2.

Also Read: Specific Heat Capacity

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Viscosity

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Viscosity refers to a fluid's resistance to progressive deformation caused by shear or tensile stress.

• To put it another way, viscosity is the reluctance of a fluid to flow.
• Simply explained, honey is more viscous than water because it is thicker than water.
• Example: The viscosity of syrup is higher than that of water.

Viscosity Formula

The viscosity of a fluid is determined by the ratio of shearing stress to velocity gradient.

A sphere can be dropped into a fluid to determine its viscosity, which can then be calculated using the formula:

η = 2ga² (Δρ)/9v

Where,

• Δρ is the density difference between the fluid and the sphere tested
• a is the radius of the sphere
• g is the acceleration due to gravity
• v is the velocity of the sphere

The viscosity of a substance is measured in Pascal seconds (Pa s).

• The viscosity increases as the sphere's speed slows, as this equation shows.
• The higher a fluid's viscosity, the more resistance it causes to any object moving through it.
• Although all liquids have some viscosity, the four liquids are typically classed as high or low viscosity, with water as a baseline.

For example: Water has a viscosity of 0.001 Pa s, air has a viscosity of 0.000019 Pa s, and motor oil has a viscosity of 1. So, you can very much figure it out on your own. In addition, the viscosity of liquids reduces as the temperature rises, but the viscosity of gases rises as the temperature rises.

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Bernoulli’s Principle and Equation

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For fluids in an ideal state, Bernoulli's principle, commonly known as Bernoulli's equation, will apply.

• As a result, the relationship between pressure and density is inverse.
• This suggests that a fluid travelling slowly exerts more pressure than a fluid moving quickly.

Bernoulli's principle states that

“As we move along a streamline, the sum of pressure (P), the kinetic energy per unit volume (ρv²/2), and potential energy per unit volume (ρgh) remain a constant”.

P + ρv²/2 + ρgh = Constant

According to Bernoulli's principle, the total energy of water remains constant throughout time, hence when the flow of water in a system increases, the pressure must drop.

• When water begins to flow in a hydraulic system, the pressure drops, and when the water flow ceases, the pressure rises.
• As a result, the overall energy head in a hydraulic system is equal to the sum of three individual energy heads.

This can be expressed as follows-

Total Head = ElevationHead + Pressure Head + Velocity Head

Where,

• Elevation head is the pressure caused by the water's elevation
• The pressure head is the maximum height of a water column that a system's hydrostatic pressure can support.
• Velocity head is the energy present as a result of the water's velocity.

General Expression of Bernoulli’s Equation

Let's look at two separate regions in the diagram above.

• The first will be known as BC, and the second will be known as DE.
• Consider the fluid that existed earlier between B and D.
• This fluid, on the other hand, will move in a minute (infinitesimal) interval of time (Δt).
• If the fluid speed at point B is v₁ and v₂ at point D.
• As a result, if the fluid starts at B and flows to C, the distance is v1Δt.
• However, v₁Δt is quite small, and we can assume it is constant in the region BC over the cross-section.
• Similarly, within the same time frame, the fluid that was previously present at point D has moved to point E.
• As a result, the distance travelled is v₂Δt. P₁ and P₂ pressures will act in the two regions, A₁ and A₂, binding the two portions together.

The entire diagram should resemble the figure below.

Finding the Work Done

First, we'll figure out how much work (W1) has been done on the fluid in the BC region.

Work done is

W₁ = P₁A₁ (v₁Δt) = P₁ΔV

Furthermore, the same volume of fluid will pass through BC and DE if the equation of continuity is used. As a result, the fluid's work on the right side of the pipe, or in the DE region, is significant.

W₂ = P₂A₂ (v₂Δt) = P₂ΔV

As a result, we can denote the fluid work as – P₂V. As a result, the total amount of work done on the fluid is

W₁ – W₂ = (P₁ − P₂) ΔV

The total work done aids in the conversion of the fluid's gravitational potential energy and kinetic energy. Now, in the Δt interval of time, consider the fluid density and the mass travelling through the pipe.

Hence, Δm = ρA₁ v₁Δt = ρΔV

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Things to Remember

• Mechanical properties of fluids are a part of CBSE class 11 physics syllabus 2022.
• It is a part of unit 7 properties of bulk matter carrying a total of 7 to 9 periods and 4 to 6 marks. 
• Mechanical properties of fluids help to understand whether a liquid is appropriate for a particular purpose.
• Fluids don't have a definite shape and can take the shape of a container.
• According to the mechanical properties of fluids they show very minimum resistance to shear pressure. 
• Hydrostatics deal with the properties of liquids at rest.
• Hydrostatics help in the understanding of the force exerted by any liquid or on any body which is immersed in it. 

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Previous Years Questions

• A metal wire of density ′ρ′′ρ′ floats on water surface horizontally. If it is NOT to sink in water then maximum radius of wire is proportional to
• If the densities of the two constituent metals are ρ1ρ1 and ρ2ρ2 respectively, then the weight of the first metal in the sample is 
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• If the coefficient of viscosity of the liquid is η then the sphere encounters an opposing force of​
• What will be the approximate terminal velocity of a rain drop of diameter​
• At another point, where the pressure is p/2p/2, the velocity of flow is [density of water = ??]​
• For liquid to rise in a capillary tube, the angle of contact should be​
• Then the ratio of the final surface energy to the initial surface energy of all the drops together is.​
• The height above the surface of water up to the ball will jump is​
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• The water rises up to 8cm. If the entire arrangement is put in a freely falling elevator, the length of water column in the capillary tube will be:​
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