Mechanical Properties of Fluids: Definition, Pascal's Law, Pressure and Law of Floatation

What is Fluid?

A substance that starts to flow when an external force is applied is referred to as a "fluid." Fluids have distinctive physical characteristics that control how they respond to applied forces.

What are the mechanical properties of fluids?

Hydrostatics is the study of a fluid's mechanical characteristics. A substance that gives in to even the smallest pressure is said to be fluid.

Fluids can be categorized into two groups: liquids and elastic fluids or gases, which later include the atmosphere's air and all the other types of air that chemistry has introduced us to. We will focus only on the mechanical characteristics of liquids or non-elastic fluids at this time.

When compared to solid objects, fluids exhibit the gravitational effects more accurately due to the strong cohesive attraction of the latter's particle population, which partially offsets the gravitational pull of gravity.

Fluid Mechanics

1. In general, both liquids and gases are referred to as fluids. In other words, it can be said that the substances which have the potential to flow are termed as fluids.
2. It is assumed that fluids are incompressible, meaning that their density is independent of changes in pressure and remains constant. 
3. The assumption that fluids are non-viscous means that no tangential force is exerted by the two liquid surfaces in contact.

Also read more about Magnetic Effect of Electric Current.

Fluid Pressure

The normal force acting per unit area is defined as fluid pressure, or p, at each point. Mathematically,

p = dFl

      —-

      dA

The S.I. unit of pressure is the pascal (Pa) and 1 pascal=1N/m²

Regardless of how the surface is oriented, fluid force acts perpendicularly to that surface. As a result, fluid pressure can be thought of as a scalar quantity because it lacks an inherent direction of its own.

Pressure

1. When the fluid is at rest or moving with constant speed, the pressures at two locations in a horizontal plane or at an equal level are the same.
2. When a fluid is at rest or moving with constant speed, pressures at two positions are separated at a depth of h, and these pressures are related by the expression: p₂−p₁=ρgh, where  ρ is the density of liquid.
3. When a fluid container experiences a constant horizontal acceleration, the pressures at two locations in a horizontal plane are related by the expression: p₁−p₂=lρa. Also, tanθ=a/g, where  θ is the angle between the liquid's free surface and the horizontal.
4. When the liquid container is accelerating up, pressures at two locations within the liquid at a vertical distance of h are related by the expression: p₂−p₁=ρ(g+a)h. When the container accelerates down, then p₂−p₁=ρ(g−a)h

Read more about the Resistance and Sugar Formula.

Atmospheric Pressure   

1. Atmospheric pressure refers to the force exerted by the earth's atmosphere. An average measurement of normal atmospheric pressure at sea level is 1 atmosphere (atm), which is equal to 1.013 × 10⁵Pa
2. Gauge pressure is the excess pressure above atmospheric pressure, and absolute pressure is the total pressure.
3. The barometer is a tool used to measure atmospheric pressure, while a manometer, also known as a U-tube manometer, is a tool used to measure gauge pressure.

Pascal's Law   

1. When applied to an enclosed fluid, a pressure change causes the fluid's entire volume to expand and contract, as well as the vessel's walls.
2. The hydraulic lift is just one of the numerous real-world uses for Pascal's law.

Archimedes Principle

1. A body loses some weight when immersed entirely or partially in a liquid. The weight of the liquid displaced by the part of the body that is submerged determines how much weight the body loses in the liquid.
2. When a body is submerged, the liquid that is displaced exerts an upward force known as buoyancy. Due to this, there is apparent loss in the weight experienced by the body.

Also read about- Father of Physics.

Law of Floatation

If the weight of the liquid displaced by the portion of the body that is submerged is equal to the weight of the body, the body floats. The various forces acting on a body when it is partially or completely submerged in a liquid are 

(i)upward thrust (T) acting at the centre of buoyancy and whose magnitude is equal to the weight of the liquid displaced and

(ii)the weight of the body (W), which acts vertically downward through its center of gravity.

(a) The body sinks in the liquid when W > T;

(b) The body will continue to be in equilibrium inside the liquid when W = T;

(c) If W < T, the body will rise to the liquid's surface in such a way that the weight of the body plus the weight of the liquid is displaced by its immersed portion. As a result, only a portion of the body will be submerged in the liquid, causing it to float.

1. If such a fluid particle is passing through the flow of a liquid, it is said to be steady or stream line flow.
2. passes through a given point at the same speed and along the same path as the particle that came before it.
3. Laminar flow is the term for a liquid's flow over a horizontal surface when it occurs as layers of fluid moving at various speeds. 
4. Turbulent flow is defined as a fluid flow where the speed of all particles passing through a given point is not the same and the fluid motion becomes erratic or disordered.

Equation of Continuity

1. The volume of liquid entering the flow tube at time t in a constant flow is given by: A₁V₁Δ t.
2. Given that the liquid is incompressible by nature, the same volume should discharge from the tube.The volume flowing out in t is given by A₂V₂Δ t.
3. Moreover, mass flow rate is provided by ρ AV, where  ρis the density of the liquid.

Read more about the Refractive Index Formula.

Bernoulli's Theorem

1. The total of the pressure energy, kinetic energy, and potential energy per unit mass is constant for an incompressible, non-viscous, irrotational liquid.
2. Bernoulli's equation is simplified as follows for non-viscous fluid flow along a horizontal pipe in steady state: 
3. p + ρgh + ρV²

                            —-   = const

                             2

Viscosity

Viscosity is a property of a fluid that prevents relative motion between its various layers, and the force that results from this property is known as the viscous force. Mathematically, viscous force is given by:

            F= −ηA dv

                         —-

                         dx

Where, η is a constant dependent on the nature of the liquid and is termed the coefficient of viscosity and

dv  is the velocity gradient.

—-                                                                                                                                                                     

dx                                                                                                                                                                    

Pa.s or Nsm-² is the S.I. unit of coefficient of viscosity.

CGS unit of viscosity is poise(1Pa.s=10 poise).

Stoke's Law

1. When a solid moves through a viscous medium, a viscous force that depends on the body's speed, shape, and size opposes its motion.
2. When a sphere of radius r is moving at a speed of v through a viscous medium with the viscosity η, the viscous drag on the sphere is given by the following equation Fviscous = 6πηrv. This formula is termed Stoke's law.

Also read more about the Electric Power Formula and Fluid Friction.

Importance of Stoke's law

1. This law is applied using Millikan's experiment to determine the electronic charge.
2. The creation of clouds is covered in this law.
3. This law explains why an object falling freely at constant speed from the height of clouds would travel more quickly than raindrops.
4. A person is permitted to use a parachute to fly to the ground under this law.

Terminal Velocity

In a free fall through a viscous medium, it describes the highest constant velocity a body can achieve. Mathematically,  

vr= 2r²(ρ− ρ₀)g

       —-----------

             9η

Poiseuille's Formula

In capillary tubes, Poiseuille discovered the liquid's stream-line flow. The volume of liquid leaving the tube each second is equal to

πPr⁴

-—--.

 8ηl

Reynold Number

1. Viscosity forces help to maintain the stability of laminar flow. However, it is evident that when the flow rate is high, laminar or steady flow is disrupted. High flow rates result in turbulence, which is an irregular, unsteady motion.

2. Reynold's number is a dimensionless number whose value gives a rough idea of the flow rate and whether or not it will be turbulent. Mathematically, it is given by

Re= ρvD

        —---.

          η  

where ρ is the density of the fluid flowing at speed v, D is the diameter of the tube, and η=  the coefficient of viscosity of the fluid.                                                                                                                                                                                                                                            

3. When Re is less than 1000, the flow is observed to be streamline or laminar, whereas when Re > 2000, the flow is turbulent. Additionally, between 1000 and 2000, the flow for Re becomes erratic.

Also read about- SI Unit of Electric Flux.

Surface Tension

1. The force per unit length in the plane of a liquid surface perpendicular to either side of a fictitious line drawn on that surface is referred to as the surface tension of a liquid. Mathematically,

S = F

      —, where S is the surface tension of a liquid.

       l                                                                                                                                                                                                                                                                             

2. Unit of surface tension in MKS system: N/m or J/m².Unit of surface tension in CGS system: dyne/cm or erg/cm².

Surface Energy

Work must be done over the liquid's surface in order to increase surface area. The potential energy of this work is stored in the liquid surface. Therefore, the excess potential energy per unit area of a liquid's surface is referred to as the liquid's surface energy. Mathematically, W= SΔA; where ΔA= increase in surface area.

Angle of Contact

1. Angle of contact is the angle formed by the solid surface inside the liquid and the tangent to the liquid surface at the point of contact.
2. The force of cohesion is the intermolecular force between molecules of the same substance. However, the force that exists between molecules of various materials is known as the adhesion force.

Torricelli's Theorem

This theorem states that the velocity of efflux, or the speed at which liquid exits an orifice (a small hole), is equivalent to the speed that a body falling freely would reach after falling through a vertical distance equal to the orifice's depth below the liquid's free surface.

The velocity is determined by

V = √2gh

Magnus Effect

A ball will follow a curved path that is convex toward the side of greater pressure when it is spun while it is in a streamline of air molecules. The ball getting a lift and areodynamics from spin bowlers are based on this theory.

Also read about- Specific Heat of Water.

Capillary Tube and Capillary Action      

A very narrow glass tube with a fine borehole and open at both ends is called a capillary tube. When a capillary tube is dipped in a liquid, the liquid will rise or fall in the tube, and this reaction is known as capillarity.

Mathematically, capillary rise or fall (h) is given by

            2Scos⁡θ        2S

 h =       —-----    =  —----- 

               Rρg          Rρg

Where, 

S= surface tension

θ= angle of contact

r= radius of capillary tube

R= radius of meniscus

ρ= density of liquid

Capillary rise in a tube of insufficient length

A capillary tube can be referred to as having "insufficient length" if its length is less than the height to which a liquid would actually rise in it (h).

In this instance, liquid ascends to the top of the l(l

Read more about the Law of Variable Proportion.