Fluid Dynamics

 Viscosity

The property of fluid in motion by virtue of which it opposes relative motion between different layers and it opposes motion of another body through that is known as viscosity.

The velocity of different layers of flowing liquid is different because of viscosity.

Difference in velocity between laminar plates, that are unit distance from each other, is called velocity gradient (dv/dx).

Ideal fluids have their viscosity and compressibility both zero.

Newton’s Formula

The viscous force between two layers of liquids in motion is:

$F= -\eta A (\dfrac{dv}{dx})$

where, $\dfrac{dv}{dx} = \dfrac{(v_1-v_2)}{(x_1-x_2)} = \dfrac{Δv}{Δx}$ is velocity gradient and A is the area of laminar plate.

$\eta = \dfrac{-F}{(A(\frac{dv}{dx})}$

Hence, the coefficient of viscosity ($\eta$) is defined as the viscous force per unit area to produce a velocity gradient of unity.

S.I unit of ƞ is Pascal second i.e poiseuille.

1 poiseuille = 1 pascal sec = 1Ns/m$^2$ = 10 Poise.

1 decalpoise = 10 poise.

The dimension of ƞ is $[ML^{-1} T^{-1}]$

Reynold’s Equation

The maximum velocity of liquid flowing through a tube upto which the flow is streamline is called critical velocity.

Reynold’s number indicates the nature of flow (i.e., streamline or turbulent).

Mathematically, $k = \dfrac{\rho v r}{\eta}$

where, $\rho$ = density of liquid, r = radius of tube, $\eta$ = coefficient of viscosity, v = critical velocity, K = Reynold’s number

K is dimensionless pure member

Beyond critical velocity (v), the flow will be turbulent.

(i) If K < 2000, the flow is laminar.

(ii) K > 3000, the flow will be turbulent

(iii) 2000 < K < 3000, the flow will be unstable.

Equation of Continuity

For an ideal liquid flowing through a non-uniform tube, under streamlined condition, the mass flowing per sec is same at each cross-section.

i.e., $\dot{m}$ = Av = Constant=

or, $A_1v_1= A_2v_2$

This is the equation of continuity.

Clearly, the greater the cross-section, the velocity of flowing liquid will be smaller and vice-versa. Due to this reason,

i. Deepwater appears to be still.

ii. Falling stream water becomes narrower.

iii. Fine jet is ejected by making cross-section narrower.

Equation of continuity is based on conservation of mass.

Equation of continuity also refers that volume of liquid flowing per second through any cross- section is constant.

Equation of continuity is applied only in case of ideal liquid (non-viscous and incompressible).

For laminar flow through non-uniform cylindrical tube,

$v \propto \dfrac{1}{A} \propto \dfrac{1}{r^2}$

Bernoulli’s Theorem and Energy of Fluid in Motion

A flowing liquid has three types of energies

a. Pressure energy = pressure × volume = PV

b. Kinetic energy = $\dfrac{1}{2} mv^2$

c. Potential energy = $mgh$

According to Bernoulli’s theorem, the total energy of an liquid under streamline flow remains constant i.e., $PV + 1/2 mv^2+ mgh$ = constant ……….. (i)

Conventionally, these energies are expressed per unit volume. So, this equation may be written as, $P + 1/2 ρv^2+ ρgh$ = constant ………… (ii) ($ρ$=density)

or, $\dfrac{P}{ρg} + \dfrac{v^2}{2g}+ h$ = constant …………. (iii)

where,

$\dfrac{P}{ρg}$=Pressure head

$\dfrac{v^2}{2g}$=Velocity head

h = Gravitational head

In case of horizontal flow ( i.e., ρgh=constant ), $P + 1/2 ρv^2$= Constant

Bernoulli’s theorem is based on Principle of Conservation of mechanical energy.

Pressure decreases liquid flows from broader to narrower portion of pipe.

Applications of Bernoulli’s Principle

1. Blowing of roofs by storms

2. Attraction between two closely parallel moving boats in opposite direction.

3. Repulsion between two closely parallel moving boats in opposite direction.

4. Magnus effect in spinning ball

5. Action of aspirator, carburetor, paint gun, scent spray, insect sprayer.

Torricelli’s Theorem and Velocity of Efflux

It states that the velocity of efflux through an orifice at depth from liquid’s surface is the same as that of a freely falling body from height i.e.,

$v = \sqrt{2gh}=v \propto \sqrt{h}$

Torricelli’s theorem can be derived from Bernoulli’s principle.

The velocity of efflux is independent of nature of liquid, quantity of liquid in the container and area of cross section of hole.

Time taken for the liquid stream to reach the base of the tank (ground) is,

$T= \sqrt{\dfrac{2 (H-h)}{g}} \propto \sqrt{(H-h )} \propto \sqrt{x}$

Where (H-h) = x is distance of hole from base of tank.

Range of liquid stream at base level is $R = \sqrt{(2 h (H-h))}$

The range is maximum if h= H/2, The maximum range Rmax = H

If a hole of cross-sectional area $A_0$ is made at the bottom of the tank of cross-sectional area A, then

(i) Time taken to lower the height from $H_1$ to $H_2$ is,

(ii) Time taken to empty is, 

$t = \dfrac{A}{A_0} \sqrt{\dfrac{2h}{g}}$

When a vertical tank is filled with a liquid up to height $H$ and a hole at the bottom is opened then the time taken to lower the level by each $H/n$ from the top are in the ratio,

$(\sqrt{n}-\sqrt{(n-1)}:………(\sqrt{3}-\sqrt{2}):(\sqrt{2}-1):1$

When a vertical tank is filled with liquid and several holes are made in vertical wall simultaneously then

i. Velocity of efflux increases downward but ranges first increase becomes maximum for hole at centre and then decrease.

ii. Range for stream escaping from two holes at equal distance from bottom and top are equal.

Stroke’s Law and Terminal Velocity

When a body falls in a viscous medium, its velocity increases firstly and finally it attains a constant value called the terminal velocity.

The viscous force on a spherical body of radius r and moving with velocity v is given by,

F= 6πƞrv=F $\propto$ v

Where ƞ = coefficient of viscosity

When a body attains terminal velocity (v) then it effective weight downward equals to viscous force upward. i.e.,

$Vρg-Vσg=6πƞrv_1$

Where, ρ=density of body and σ=density of viscous medium

or, $V(ρ-σ)g=6πƞrv_1$

or, $\dfrac{4}{3} πr^2 (ρ-σ)g=6πƞrv_1$

Therefore, $v_1 = \dfrac{2 (\rho- \sigma) gr^2}{9 \eta} \propto r^2 \propto \dfrac{1}{ƞ} (ρ-σ)$

i. If ρ>σ, then body falls with terminal velocity.

ii. If ρ=σ, then body remains stationary wherever is released.

iii. If ρ<σ, then the body rises upwards through medium.

iv. If σ is negligible, then, $v = \dfrac{2 \rho gr^2}{9 \eta}

$v_1 \propto r^2$ i.e., doubling the radius of the spherical body, quadrupled will be the terminal velocity.

Poiseuille’s Equation

The rate of flow of liquid through a capillary tube of radius r, lenth l, under streamlined motion for constant pressure difference ∆P across the capillary is given by,

$\dfrac{V}{t} = \dfrac{\pi \Delta P r^4}{8 \eta l}$

[V= Volume] Where,

$R = \dfrac{(8ƞ l)}{(πr^4)}$ is called the fluid resistance.

The velocity of liquid flowing at distance x from axis of tube is,

$v= \dfrac{P}{4ƞ l} (r^2-x^2)$

i. Series combination of tubes

• Rate of flow V/t will be same but pressure difference will be different.

$R_{eq}= R_1+R_2$

ii. Parallel combination of tubes

Rate of flow will be different but pressure difference will be equal.

$\dfrac{1}{R_{eq}} = \dfrac{1}{R_1} +\dfrac{1}{R_2}$ and

Better You Know

Viscous force is due to electromagnetic interaction.

In Torricelli’s theorem, velocity of efflux, is independent of the nature of liquid, quantity of liquid in container and area of cross-section of hole.

Viscosity of liquids decreases with increase of temperature, but viscosity of gases increases with increase in temperature. [IOM]

Viscosity of liquid increases with increase in density while viscosity of gases decreases with increase in density.

Viscosity of liquids is due to cohesive force.

Viscosity of gases is due to transfer of momentum between gas molecules.

Viscosity of liquid increase with increase in pressure for a certain range of pressure. But at very high and low pressure, viscosity is directly proportional to pressure.

The design of aeroplane wings is such to obey the Bernoulli’s principle, that upper surface is convex and lower one is concave downwards.

Viscosity arises due to intermolecular force which are basically electromagnetic forces.

Machine parts are jammed in winter due to increase in viscosity of lubricant.

The C.G.S. unit of coefficient of viscosity is poise.

1 centipoises = 1/100 poise=(1/100)×1/10

1 poiseuille = $0.001kgm^{-1} s^{-1}$

A spinning ball follows a curved path in air due to Magnus effect.

Between layer of liquid in relative motion, viscosity introduces tangential force.

The critical velocity $v_c$, for a non-viscous liquid is zero.

Coefficient of viscosity is also defined as shearing stress per unit velocity gradient.

The terminal velocity of droplet is $v_t$. If n such drops are coalesced to form a single drop, the terminal velocity will be, $n^{2/3} v_t$.

Viscosity of all the gasses increase with increase in temperature as per the relation ƞ α √T

Viscosity of water decreases with increase in pressure while that of other liquid increases with increases in pressure.

Viscosity of gases remains constant at high pressure.

Viscosity of gases is directly proportional to pressure at low pressure region.

Rain drops fall with constant velocity due to viscosity.

Clouds float in the sky due to their low density.

If viscosity of air is taken into account orbital velocity of satellite moving close to earth increases till the satellite falls back on earth.

Angle between viscous force and motion of liquid flow is π.

When the velocity of liquid flowing steadily through a tube increases its pressure decreases. [Hint: P + 1/2 ρv^2+ ρgh=constant]

Ideal fluid has zero viscosity and zero compressibility.

A hole is near the bottom of a tank. The volume of liquid emerging from the hole does not depend upon density of liquid. [Hint: V= av = a√2gh ]

A liquid moves through a horizontal pipe. The pressure is higher where diameter is large. [Hint: P + 1/2 ρv^2=constant;P α 1/V^2 , As Av= constant therefore, P α A]

Reynold’s number is low for high viscosity, low density and low velocity.

Poiseuille’s law for liquid can be compared with Ohm’s law for current flow where viscosity of liquid is equivalent to resistivity.

If viscosity of air is taken into account, orbital velocity of satellite moving close to earth increases till the satellite falls back on earth.

Cold syrup flows sluggishly because of viscosity.

A sphere is dropped from a height in viscous liquid. Then the velocity first increases and becomes constant.

Reynold number is related to critical velocity.