Pressure in a fluid

It is due the motion of atoms (or) molecules collide continuously with the walls of the container

It is defined as the magnitude of the normal force (applied by fluid) acting per unit surface area i.e.… P = \(\frac{F}{A}\)

Units : Pascal

1 Pa = 1 \(\frac{N}{m^2}\) 1 bar = \(10^5\) Pa

Atmospheric Pressure \(P_{atm}\): 1.013 x \(10^5\) Pa = 1.013 bar = 760 torr

Pressure is a scalar quantity because it acts perpendicular to any surface in the fluid no matter how the surface is oriented

Gauge Pressure= Absolute Pressure – Atmospheric Pressure

Gauge Pressure is the excess pressure above the atmospheric pressure and the total pressure is the absolute pressure

The force exerted by the fluid on the bottom surface of the container is enough to break but due to upward force by the underside of the bottom surface is same in magnitude to the downward force it won’t break (or) collapse

Density of a liquid

Defined as the mass per unit volume i.e.. D = \(\frac{Mass}{Volume}\)

Units: \(\frac{kg}{m^3}\)

Relative Density

The ratio of Density of liquid or substance to that of density of water at 4℃ i.e… RD = \(\frac{Density of liquid or substance}{density of water at 4℃}\)

Density of water at 4℃ is 1000 \(\frac{kg}{m^3}\)

Density for the mixture of two or more liquids

Case 1: Two liquids of densities \(D_1\)and \(D_2\) and masses \(M_1\) and \(M_2\). Then the density of the mixture is

D = \(\frac{m_1 + m_2}{(\frac{M_1}{D_1}) +(\frac{M_2}{D_2})}\)

If \(M_1\) = \(M_2\)= M then D = \(\frac{2D_1D_2}{D_1+D_2}\)

Case 1: Two liquids of densities \(D_1\)and \(D_2\)and masses \(V_1\)and \(V_2\) Then the density of the mixture is

D = \(\frac{D_1V_1+ D_2V_2}{V_1+ V_2}\)

If \(V_1\)and \(V_2\) = V then V = \(\frac{V_1 + V_2}{2}\)

Effect of Temperature and Pressure on Density;

1) Temperature

If Temperature increases the mass remains same the volume is increased and the density of the liquid decreases

Equation = \(D^{‘}\) = \(\frac{D}{1+γΔθ}\)

2) Pressure

If Pressure increases the mass remains same the volume is decreased and the density of the liquid increases

Equation = \(D^{‘}\) = \(\frac{D}{1+\frac{ΔP}{B}}\)

Variation of Pressure with depth

A fluid of density D is kept at rest in a cylindrical vessel of height h and we get

1) Pressure at the bottom by fluid is P = ρgh

2) Total Pressure at the bottom by fluid and atmosphere is P = P_{o} + ρgh

In a liquid Column two points A and B are situated at h1 and h2 respectively below free surface of liquid having height difference ‘h’ then

1) Pressure Difference = P_{B} – P_{A} = ρgh

Pascal Law

It states that Pressure applied to an enclosed fluid is transmitted to every portion of the fluid and the walls of the container

Hydraulic Lift

It is an application of Pascal law

For an area \(A_1\) the force applied is \(F_1\) and For Area \(A_2\) the force is \(F_2\) and the pressure is same on both of the areas so we can write it as

\(\frac{F_1}{A_1}\) = \(\frac{F_2}{A_2}\)

If \(A_2\) >>> \(A_1\) then \(F_2\) >>> \(F_1\)

Measurement of Pressure

1) Barometer

It is used to measure the atmospheric Pressure

As the two surfaces are of same height they have same pressure so we can say \(P_1\) = \(P_2\) and \(P_o\) = ρgh

We always take mercury as barometer fluid because of great density and we can the atmospheric pressure directly from the height of the mercury column

2) Manometer

It is used to find the pressure of a gas inside the container

It consists of a container and U shaped Tube (we take mercury)

As the two surfaces are of same height they have same pressure so we can say \(P_1\) = \(P_2\) and P= \(P_o\) + ρgh because one of the vertical surface of U shaped Tube directly in contact to atmosphere)

Archimedes Principle

It states that “A body wholly or partially submerged in a fluid experiences an upward force which is equal to the weight of the displaced fluid (F = \(V_s\)\(ρ_l\)g)

The up thrust act vertically upwards through the Centre of gravity of displaced fluid

Apparent weight of body inside a liquid

\(W_app\) = \(V_s\)\(ρ_s\)g – \(V_s\)\(ρ_l\)g

\(W_{app}\) = \(W_{actual}\) (1 –\(\frac{D_l}{D_s})\)

Cases

1). \(D_s\) < \(D_l\) Body floats partially

2) \(D_s\) = \(D_l\) Body floats but completely submerged

3) \(D_s\) > \(D_l\) Body Sinks

Buoyant force in accelerating fluids in a lit

Here in the formula F =\(V_sρ_lg\)
we replace g as \(g_{eff}\)

If lift goes up \(g_{eff}\) = g + a

If lift goes down \(g_{eff}\) = g – a

If lift is freely falling \(g_{eff}\) = zero and this is the reason why the freely falling vessel filled with some liquid, the air bubbles don’t rise up

Flow of fluids

The motion can be either smooth or irregular depending on the velocity of flow

1) Steady flow

The type of flow in which the velocity at any point of the fluid at any point does not vary with time but at different points may be different

Also called streamlined or laminar flow

The path followed by a fluid particle in a steady flow is called streamline and velocity at any point of streamline is along the tangent

Properties

1) In streamline flow the two streamline can’t cross each other

2) The greater is crowding of streamlines the greater is the velocity

3) Streamline is possible if liquid velocity equal to critical velocity

4) The bundle of streamlines forming a tubular region is called a tube of flow

2) Turbulent flow

The type of flow in which the velocity crossing a given point of the fluid at any point and vary with time

The motion of fluid becomes irregular or disordered

Reynolds Number (\(R_e\))

Formula = \(\frac{Inertial force}{Viscous force}\)=\(\frac{ρvD}{η}\)

here ρ is density of fluids v is speed of fluid D is diameter of fluid and η is coefficient of viscosity

(\(R_e\)) < 1000 flow is streamline or laminar

(\(R_e\)) is 1000 to 2000 flow is unsteady

(\(R_e\)) > 2000 flow is turbulent

Equation of continuity

For incompressible and non-viscous fluid flows in a streamline motion through a tube of non-uniform then mass flow rate is same at every cross section

\(A_1V_1\) = \(A_2V_2\)

It is derived from law of conservation of mass

For non-incompressible fluids equation of continuity is \(ρ_1A_1V_1\) = \(ρ_2A_2V_2\)

Bernoulli’s Theorem

Statement is Sum of pressure energy per unit volume, kinetic energy per unit volume and potential energy per unit volume

Expression is P + \(\frac{1}{2}\)ρ\(v^2\) +ρgh

If we divide the expression by ρg we get \(\frac{P}{ρg} + \frac{\frac{1}{2}ρv^2}{ρg} + \frac{ρgh}{ρg}\) we get

\(\frac{P}{ρg}\) is Pressure head

\(\frac{1}{2}\)\(\frac{v^2}{g}\) is velocity head

h is gravitational head

Applications

1) Venturimeter

We apply Bernoulli’s Theorem to the wide and narrow points and parts of horizontal pipe with height h

Equation \(P_1\) + \(\frac{1}{2}\)ρ\(v_1^2\) = \(P_2\) + \(\frac{1}{2}\)ρ\(v_2^2\) and using \(A_1V_1\) = \(A_2V_2\) we get

\(V_1\) = \(\sqrt\frac{2gh}{\frac{A_1}{A_2}^2 – 1}\) or \(\sqrt\frac{2(P_1 –P_2)}{ρ(\frac{A_1}{A_2}^2 – 1)}\)

Volume flow rate = \(A_1\)\(\sqrt\frac{2gh}{\frac{A_1}{A_2}^2 – 1}\) or \(A_1 \sqrt\frac{2(P_1 –P_2)}{ρ(\frac{A_1}{A_2}^2 – 1)}\)

2) Speed of efflux

It is defined as the outflow of a fluid is called efflux and the speed of the liquid coming out is called Speed of Efflux

A vessel of Height H having a small hole is made in the wall of h depth from the surface of liquid then the speed of efflux is \(\sqrt{2gh}\) and from this we can see that it equals to freely falling through vertical height between the liquid surface and orifice and this statement is known as Torricelli’s theorem

Range is the horizontal distance covered by liquid coming out of the hole is given by R = 2\(\sqrt{h(H-h)}\) and it is maximum at h = \(\frac{H}{2}\) and \(R_{max}\) = H

Time taken to empty a tank is given by T = \(\frac{A}{a}\sqrt\frac{2H}{g}\) and Time taken by liquid to fall from H1 to H2 is T = \(\frac{A}{a}\)\(\sqrt\frac{2}{g}\sqrt{H_1}-\sqrt{H_2}\)

Viscosity

The character of fluid by virtue of which relative motion between different layers is opposed is called viscosity

It is an internal friction in a liquid as it occurs due to diffusion of molecules of one layer into another and opposes the motion of one portion of a liquid relative to the other

The flow of speeds of intermediate layers of fluid increase uniformly from bottom to top

Formula F = – ηA\(\frac{dv}{dx}\) Here η is coefficient of viscosity and depends on the nature of the fluid and negative sign indicates the direction of viscous force is opposite to relative velocity of layer

Unit Ns\(m^{-2}\) or decapoise or Pascal second

1 decapoise = 1 Pascal second = 1 Ns\(m^{-2}\) = 10 poise

For water coefficient of viscosity is 1.3 x \(10^{-13}\) Ns\(m^{-2}\)

Flow of liquid through a cylindrical pipe

Formula is \(\frac{(P_1-P_2)(R^2– r^2)}{4ηL}\)

\(P_1\) and \(P_2\) are the pressures at the two ends

L is Pipe Length

The speed is greatest along the axis r = 0 so \(\frac{(P_1-P_2)(R^2)}{4ηL}\)

At the walls r = R v = 0

Volume Flow Rate (Q )or \(\frac{dV}{dt}\)

Q = \(\frac{dV}{dt}\) =\(\frac{(P_1-P_2)(πR^4)}{8ηL}\) and is known as Poiseuille Equation

Stokes Law

It is defined as the viscous forced on the sphere of radius r moving with velocity v

Force is F = 6 πηrv

Terminal Velocity

When a sphere starts falling from rest through the column of viscous fluid and forces acting are weight (w) downwards Buoyant force (\(F_t\)) upwards and viscous force (\(F_v\)) upwards

At the beginning it accelerates downwards and after some time all three forces will become equal and move with constant velocity called Terminal velocity

Terminal Velocity is \(\frac{2}{9}\frac{ (ρ– σ) r^2 }{g}\)

We can see from above equation Terminal Velocity is proportional to (ρ– σ) and If (ρ < σ) Terminal Velocity is negative and due to this air bubble starts rising upwards in water

Surface Tension

It is the property of liquid at rest by virtue of which a liquid surface tends to occupy a minimum surface area and behaves like stretched membrane

It is defined as the force acting per unit length of an imaginary line drawn on the liquid surface ,the direction of force being perpendicular to this line and tangential to the liquid surface and denoted by S and it is scalar

Formula Surface Tension =\(\frac{ Force }{Length}\)

As the Temperature increases Surface Tension Decreases

Surface Energy

It always has a tendency to contract and possess minimum surface area

It is defined as the amount of work done in increasing the area of the surface film through unity

Formula = \(\frac{Work done in increasing surface area}{Increase in surface area}\)

Relation Between Surface energy and surface tension : Surface Energy = Work Done = Surface Tension X Area

Work Done for Coalesce of Bubbles is ΔU = E (\(n^{\frac{2}{3}}\) -1 ) ΔU is Negative because Energy is released ( E is the energy of single drop)

Work Done for Splitting of Bubbles is ΔU = E (\(n^{\frac{1}{3}}\) -1 ) ΔU is Positive because Energy is absorbed ( E is the energy of single drop)

Here E = S X 4π\(r^2\)

Excess Pressure

Due to surface tension a drop or bubble tends to contract and so compresses the matter and equilibrium is attained. At equilibrium the pressure inside the bubble or drop is greater than the outside and this difference in pressure between two sides of the liquid surface is excess pressure

Excess Pressure inside soap bubble is \(\frac{4T}{R}\)

Excess Pressure inside liquid drop is \(\frac{2T}{R}\)

Excess Pressure due to coalesce of two soap bubbles of radius \(r_1\) and \(r_2\) is

4T(\(\frac{r_1r_2}{r_1-r_2}\))

Points regarding excess pressure

1) For Air bubble the pressure in the concave side is greater than the convex side by an factor of \(\frac{2T}{R}\)

2) For Spherical liquid surface the pressure in the concave side is greater than the convex side by an factor of \(\frac{2T}{R}\)

3) For Soap bubble the pressure in the concave side is greater than the convex side by an factor of \(\frac{4T}{R}\)

4) If two bubbles (small and large) are connected through a thin tube so that air flows from smaller bubble to larger bubble due to high pressure in smaller bubble

Angle of contact

It is defined as the angle that the tangent to the liquid surface at the point of contact makes with the liquid surface at the contact makes with the solid surface. Denoted by θ

It denotes whether the liquid will spread on the surface of solid or it will form droplets on it

Case – 1

1) Surface tension of Solid Liquid interface is greater than the Liquid Air interface

2) θ > 90 and angle of contact is obtuse angle

3) Molecules of liquid are attracted strongly to themselves and weakly to solid

4) Liquid does not wet the solid

5) Examples are Water- Leaf or glass-mercury interface

Case – 2

1) Surface tension of Solid Liquid interface is lesser than the Liquid Air interface

2) θ < 90 and angle of contact is acute angle

3) Molecules of liquid are attracted strongly to solid and weakly to themselves

4) Liquid does wet the solid

5) Examples are when soap or detergent is added to water

Shape of liquid surface

The curved surface of the liquid is called meniscus and shapes is determined by relative strength of cohesive and adhesive forces

Cohesive force is the force between the molecules of same material

Adhesive force is the force between the molecules of different kinds of material

Case 1

1) Adhesive > Cohesive forces

2) Angle of contact is less than 90

3) Shape is Concave

Case 2

1) Adhesive < Cohesive forces

2) Angle of contact is greater than 90

3) Shape is Convex

Capillarity

The phenomenon of rise or fall in the capillary is called capillarity

Examples: Some capillaries in fibers of towels rocks water our skin and in trees sap rises in stem due to capillary action

Formula of Capillary rise

The phenomenon of rise or fall in the capillary is called capillary action

Formula (h) = \(\frac{2T cosθ}{rρg }\) where h is the height of liquid column rises or falls r is the radius of capillary tube θ is angle of contact ρ is density of liquid g is the acceleration due to gravity T is surface tension

If θ > 90 the term cosθ is negative hence h is negative and this gives depression of the liquid

If capillary tube is held vertically in a liquid which gives concave meniscus then capillary rise is hR = \(\frac{2T}{ρg}\)

If the tube makes an angle &rhi; units vertical then rise will be\( h^{‘} =\frac{ h}{cosφ}\)

# Fluid Mechanics

CategoriesNEET Physics