# Fluid Dynamics

Coefficient of viscosity;

Hence, the coefficient of viscosity is defined as the viscous force acting between two liquid layers each of unit area to maintain unit velocity gradient.

Its unit 1 deca.pose = 1NSm-2

The effect of pressure on the coefficient of viscosity of a fluid:

The coefficient of viscosity is defined as the viscous force acting between two liquid layers each of unit area to maintain unit velocity gradient. For most liquids, viscosity increases with increasing pressure because the amount of free volume in the internal structure decreases due to compression. Consequently, the molecules can move less freely and the internal friction forces increase. The result is an increased flow resistance.

The difference in velocity between adjacent layers of the fluid is known as a veloci

ty gradient. And the SI unit of Velocity Gradient is s−1

Coefficient of viscosity and Newton’s formula:

let's take a viscous liquid flowing through the surface Q in which P and Q are two different layers flowing with velocity 'V' and '${\rm{v}} + {\rm{dv}}$' respectively. Let 'A' be the area of both of the layer and 'x' and '${\rm{x}} + {\rm{dx}}$' be the distance of the layers from the ground.

Let 'F' be the viscous force acting between the liquid layers then, according to Newton, the viscous force is directly proportional to the area i.e.

${\rm{F}} \propto {\rm{A}}$ ----------- (i)

And, it is directly proportional to the velocity gradient i.e.

${\rm{F}} \propto \frac{{{\rm{dv}}}}{{{\rm{dx}}}}$ ---------- (ii)

Where,

$\frac{{{\rm{dv}}}}{{{\rm{dx}}}} = {\rm{velocity\: gradient}}$

$= \frac{{{\rm{change\: in\: velocity}}}}{{{\rm{change\: in\: distance}}}}$

Combining equation (i) and (ii), we get,

${\rm{F}} \propto {\rm{A}}\frac{{{\rm{dv}}}}{{{\rm{dx}}}}$

Or, ${\rm{F}} = - \eta {\rm{A}}\frac{{{\rm{dv}}}}{{{\rm{dx}}}}$ --------------- (iii)

Where, $\eta$ is a constant called coefficient of viscosity and the negative sign (-) indicates that the viscous force acts opposite to the flow of liquid.

Equation (iii) is the Newton’s formula for viscosity.

In magnitude,

${\rm{F}} = \eta {\rm{A}}\frac{{{\rm{dv}}}}{{{\rm{dx}}}}$

If ${\rm{A}} = 1{{\rm{m}}^2}{\rm{and\: }}\frac{{{\rm{dv}}}}{{{\rm{dx}}}} = 1{{\rm{s}}^{ - 1}}$, then

${\rm{F}} = \eta {\rm{*}}1{\rm{*}}1$

$\eta = {\rm{F}}$

Hence, the coefficient of viscosity is defined as the viscous force acting between two liquid layers each of unit area to maintain unit velocity gradient.

Unit and dimension of $\eta$

We have,

${\rm{F}} = \eta {\rm{A}}\frac{{{\rm{dv}}}}{{{\rm{dx}}}}\left( {{\rm{in\: magnitude}}} \right)$

or , $\eta = \frac{{\rm{F}}}{{{\rm{A*}}\frac{{{\rm{dv}}}}{{{\rm{dx}}}}}}$

$= \frac{{\rm{N}}}{{{{\rm{m}}^2}{\rm{*}}{{\rm{s}}^{ - 1}}}}$

$= {\rm{Ns}}{{\rm{m}}^{ - 1}}$

$= 1{\rm{deca}} - {\rm{poise}}\left( {{\rm{S}}.{\rm{I}}} \right)$

And,

$\eta = \left[ {{{\rm{M}}^1}{{\rm{L}}^1}{{\rm{T}}^{ - 2}}{\rm{*}}{{\rm{T}}^1}{{\rm{L}}^{ - 2}}} \right]$

$= \left[ {{{\rm{M}}^1}{{\rm{L}}^{ - 1}}{{\rm{T}}^{ - 1}}} \right]$

Also,

$\eta = \frac{{\rm{F}}}{{{\rm{A*}}\frac{{{\rm{dv}}}}{{{\rm{dx}}}}}}$

= $\frac{{{\rm{dyne}}}}{{{\rm{c}}{{\rm{m}}^2}{\rm{*}}{{\rm{s}}^{ - 1}}}}$

$= {\rm{poise}}\left( {{\rm{C}}.{\rm{G}}.{\rm{S}}} \right)$

Hence,

$1{\rm{deca}}.{\rm{poise}} = 1{\rm{NS}}{{\rm{m}}^{ - 2}}$

$= \frac{{1{\rm{*}}{{10}^5}{\rm{dyne*sec}}}}{{{{\rm{m}}^2}}}$

$= \frac{{1{\rm{*}}{0^5}{\rm{dyne*sec}}}}{{{{10}^4}{\rm{c}}{{\rm{m}}^2}}}$

$= 10{\rm{\: dyne*sec*c}}{{\rm{m}}^{ - 2}}$

$= 10{\rm{\: poise\: }}$

Hence, if a force of 1 dyne is acting between two liquid layers of area 1cm2 to maintain unit velocity gradient then the value of that force is called 1poise.

Stroke’s law dimensional analysis:

Let's take a bob of certain weight 'w' which falls down through a viscous medium like air. Initially the weight of the object becomes more than up thrust due to which it moves with certain acceleration given by

${\rm{a}} = \frac{{{\rm{w}} - {\rm{U}}}}{{\rm{m}}}$

Where, m is the mass of the object. When, the object travels some distance with acceleration, then liquid layers or air layers start to form. As soon as the layers are formed the acceleration becomes zero and then the object start to fall with constant velocity called terminal velocity. The velocity with which an object through a viscous medium after vanishing the acceleration in which the weight and up thrust become equal is called terminal velocity.

The object start to move with terminal velocity then viscous force appears between the object and liquid layers. The magnitude of this viscous force is given by stroke's law which states that the viscous force depends on radius of the object, the terminal velocity and the coefficient of viscosity of the medium i.e.

${\rm{F}} \propto {{\rm{r}}^{\rm{x}}}$ -------- (i)

${\rm{F}} \propto {{\rm{v}}^{\rm{y}}}$ -------- (ii)

${\rm{F}} \propto {\eta ^{\rm{z}}}$ -------- (iv)

Where, r, v and $\eta$ be the radius, terminal velocity and coefficient of viscosity of the medium. Also, x, y and z are the integers to be found dimensionally.

Combining equation (i) and (ii) and (iii) we get,

${\rm{F}} \propto {{\rm{r}}^{\rm{x}}}{{\rm{v}}^{\rm{y}}}{\eta ^{\rm{z}}}$

Or, ${\rm{F}} = {\rm{K}}{{\rm{r}}^{\rm{x}}}{{\rm{v}}^{\rm{y}}}{\eta ^{\rm{z}}}$ ----------------- (iv)

Where, ${\rm{K}} = \sigma {\rm{\pi \: in\: S}}.{\rm{I}}$

In terms of dimensions, equation (iv) becomes

$\left[ {{{\rm{M}}^1}{{\rm{L}}^1}{{\rm{T}}^{ - 2}}} \right] = {\left[ {{{\rm{L}}^1}} \right]^{\rm{x}}}{\left[ {{{\rm{L}}^1}{{\rm{T}}^{ - 1}}} \right]^{\rm{y}}}{\left[ {{{\rm{M}}^1}{{\rm{L}}^{ - 1}}{{\rm{T}}^{ - 1}}} \right]^{\rm{z}}}$

Or, $\left[ {{{\rm{M}}^1}{{\rm{L}}^1}{{\rm{T}}^{ - 2}}} \right] = \left[ {{{\rm{M}}^{\rm{z}}}{{\rm{L}}^{{\rm{x}} + {\rm{y}} - {\rm{z}}}}{{\rm{T}}^{ - {\rm{y}} - {\rm{z}}}}} \right]$

Equating for M, we get

$1 = {\rm{z}}$

Equating for T, we get

$- {\rm{y}} - 1 = - 2$

Or, $- {\rm{y}} = - 2 + 1$

Or, ${\rm{y}} = 1$

Equating for L, we get

$1 = {\rm{x}} + 1 - 1$

Or, $1 = {\rm{x}}$

Putting the value of x, y and z in equation (iv), we get

${\rm{F}} = 6{\rm{\pi }}{{\rm{r}}^1}{{\rm{v}}^1}{\eta ^1}$

Or, ${\rm{F}} = 6{\rm{\pi }}\eta {\rm{rv}}$ ---------- (v)

Equation (v) gives the mathematical expression for stroke's law

An expression for the terminal velocity of a rain drop falling through air:

Let's take a bob of certain weight 'w' which falls down through a viscous medium like air. Initially the weight of the object becomes more than up thrust due to which it moves with certain acceleration given by

${\rm{a}} = \frac{{{\rm{w}} - {\rm{U}}}}{{\rm{m}}}$

Where, m is the mass of the object. The object start to move with terminal velocity then viscous force appears between the object and liquid layers. The magnitude of this viscous force is given by stroke's law which states that the viscous force depends on radius of the object, the terminal velocity and the coefficient of viscosity of the medium i.e.

${\rm{F}} \propto {{\rm{r}}^{\rm{x}}}$ -------- (i)

${\rm{F}} \propto {{\rm{v}}^{\rm{y}}}$ -------- (ii)

${\rm{F}} \propto {\eta ^{\rm{z}}}$ -------- (iv)

Where, r, v and $\eta$ be the radius, terminal velocity and coefficient of viscosity of the medium. Also, x, y and z are the integers to be found dimensionally.

Combining equation (i) and (ii) and (iii) we get,

${\rm{F}} \propto {{\rm{r}}^{\rm{x}}}{{\rm{v}}^{\rm{y}}}{\eta ^{\rm{z}}}$

Or, ${\rm{F}} = {\rm{K}}{{\rm{r}}^{\rm{x}}}{{\rm{v}}^{\rm{y}}}{\eta ^{\rm{z}}}$ ----------------- (iv)

Where, ${\rm{K}} = \sigma {\rm{\pi \: in\: S}}.{\rm{I}}$

In terms of dimensions, equation (iv) becomes

$\left[ {{{\rm{M}}^1}{{\rm{L}}^1}{{\rm{T}}^{ - 2}}} \right] = {\left[ {{{\rm{L}}^1}} \right]^{\rm{x}}}{\left[ {{{\rm{L}}^1}{{\rm{T}}^{ - 1}}} \right]^{\rm{y}}}{\left[ {{{\rm{M}}^1}{{\rm{L}}^{ - 1}}{{\rm{T}}^{ - 1}}} \right]^{\rm{z}}}$

Or, $\left[ {{{\rm{M}}^1}{{\rm{L}}^1}{{\rm{T}}^{ - 2}}} \right] = \left[ {{{\rm{M}}^{\rm{z}}}{{\rm{L}}^{{\rm{x}} + {\rm{y}} - {\rm{z}}}}{{\rm{T}}^{ - {\rm{y}} - {\rm{z}}}}} \right]$

Equating for M, we get

$1 = {\rm{z}}$

Equating for T, we get

$- {\rm{y}} - 1 = - 2$

Or, $- {\rm{y}} = - 2 + 1$

Or, ${\rm{y}} = 1$

Equating for L, we get

$1 = {\rm{x}} + 1 - 1$

Or, $1 = {\rm{x}}$

Putting the value of x, y and z in equation (iv),

We get

${\rm{F}} = 6{\rm{\pi }}{{\rm{r}}^1}{{\rm{v}}^1}{\eta ^1}$

Or, ${\rm{F}} = 6{\rm{\pi }}\eta {\rm{rv}} \ldots \ldots ..$  this is required expression

Poiseuille’s formula for the flow of a liquid through a capillary tube;

Poiseuille's formula gives the expression for volume of liquid flowing through pipe per second in which the flow of time is uniform or streamlined.

Let's take a pipe of uniform radius through which a viscous liquid of coefficient of viscosity $\eta$ flows through it. Let 'l' be the length of the pipe and 'r' be its radius. Let P1 and P2 be the pressures at the two ends of the pipe in which the difference of pressure is

${\rm{P}} = {{\rm{P}}_1} - {{\rm{P}}_2}$

If V be the volume of liquid flowing through the pipe, then, volume of liquid flowing through the pipe per second depend on

a) Radius of the pipe

i.e. $\frac{{\rm{V}}}{{\rm{t}}} \propto {{\rm{r}}^{\rm{x}}}$

b) Coefficient of viscosity

i.e. $\frac{{\rm{V}}}{{\rm{t}}} \propto \eta {\rm{y}}$

c) Depends on pressure gradient

i.e. $\frac{{\rm{V}}}{{\rm{t}}} = {\left( {\frac{{\rm{P}}}{{\rm{l}}}} \right)^{\rm{z}}}$

Combining these all we get,

$\frac{{\rm{V}}}{{\rm{t}}} \propto {{\rm{r}}^{\rm{x}}}{\eta ^{\rm{y}}}{\left( {\frac{{\rm{P}}}{{\rm{l}}}} \right)^{\rm{z}}}$

or, $\frac{{\rm{V}}}{{\rm{t}}} = {\rm{k}}{{\rm{r}}^{\rm{x}}}{{\rm{n}}^{\rm{y}}}{\left( {\frac{{\rm{p}}}{{\rm{l}}}} \right)^{\rm{z}}}$----------(i)

Where x, y and z are the integers to be found dimensionally and K is a proportionality constant whose value in S.I. is $\frac{{\rm{\pi }}}{8}$

In terms of dimensions, equation (i) becomes

$\left[ {\frac{{{{\rm{L}}^3}}}{{{{\rm{T}}^1}}}} \right] = \left[ {{{\rm{L}}^{\rm{x}}}} \right]{\left[ {{{\rm{M}}^1}{{\rm{L}}^{ - 1}}{{\rm{T}}^{ - 1}}} \right]^{\rm{y}}}{\left[ {\frac{{{{\rm{M}}^1}{{\rm{L}}^{ - 1}}{{\rm{T}}^{ - 2}}}}{{{{\rm{L}}^1}}}} \right]^{\rm{z}}}$

Or, $\left[ {{{\rm{M}}^0}{{\rm{L}}^3}{{\rm{T}}^{ - 1}}} \right] = \left[ {{{\rm{M}}^{{\rm{y}} + {\rm{z}}}}{{\rm{L}}^{ - {\rm{y}} + {\rm{x}} - 2{\rm{z}}}}{{\rm{T}}^{ - {\rm{y}} - 2{\rm{z}}}}} \right]$

Equating for M we get,

$0 = {\rm{ytz}}$ ----------- (ii)

Equating for T, we get,

$- 1 = - {\rm{y}} - 2{\rm{z}}$

Or, $1 = {\rm{y}} + 2 + {\rm{z}}$

Or,$1 = {\rm{o}} + {\rm{z}}$

Or, ${\rm{z}} = 0$

And,

${\rm{y}} + 2 = 0$

Or, ${\rm{y}} + 1 = 0$

Or, ${\rm{y}} = - 1$

Equating for L, we get

$3 = {\rm{x}} - {\rm{y}} - 2{\rm{z}}$

Or,$3 = {\rm{x}} + 1 - 2{\rm{*}}1$

Or,$3 = {\rm{x}} - 1$

Or,${\rm{x}} = 4$

Putting the value of x, y and z in equation (i)

$\frac{{\rm{V}}}{{\rm{t}}} = {\rm{k*}}{{\rm{r}}^4}{\eta ^{ - 1}}{\left( {\frac{{\rm{p}}}{{\rm{l}}}} \right)^1}$

Or,$\frac{{\rm{V}}}{{\rm{t}}} = \frac{{\rm{\pi }}}{8}{\rm{*}}\frac{{\mathop {\Pr }\limits^4 }}{{{\eta ^1}}}{\rm{\: }}$

Or, $\frac{{\rm{V}}}{{\rm{t}}} = \frac{{{\rm{\pi }}\mathop {\Pr }\limits^4 }}{{8{\rm{ml}}}}$ ----------- (iii)

Equating (iii)is the expression for Poiseuille's formula.

Bernoulli’s theorem:

It states that, "For the stream line flow of incompressible and non-viscous fluid or liquid, the sum of pressure energy per unit mass, kinetic energy per unit mass and the potential energy per unit mass is always constant. i.e.

$\frac{{\rm{P}}}{\rho } + \frac{{{{\rm{V}}^2}}}{2} + {\rm{gh}} = {\rm{constant}}$

Where,

$\frac{{\rm{P}}}{\rho } = {\rm{pressure\: energy\: per\: unit\: mass}}$

$\frac{{{{\rm{V}}^2}}}{2} = {\rm{K}}.{\rm{E\: per\: unit\: mass}}$

${\rm{gh}} = {\rm{P}}.{\rm{E\: per\: unit\: mass}}$

Let's take a pipe of non-uniform diameter through which a non viscous liquid flows through it with streamline motion. The liquid is incompressible and its density is $\sigma$. Let A1, V1, P2, and h1 be the area of cross section, velocity of liquid, pressure of liquid, and height from ground at end A. Similarly, A2, V2, P2, and h2, be the area of cross section, velocity of liquid, pressure of liquid and be the height from the end B  from the ground.

Let 'm' be the mass of liquid of flowing per second through the pipe.

Then, work done per second on entering the liquid at end A is

${{\rm{W}}_1} = {\rm{force*distance\: per\: second}}$

$= {{\rm{P}}_1}{\rm{*}}{{\rm{A}}_1}{\rm{*}}{{\rm{V}}_1}$ -------------- (i)

Similarly, work done per second on leaving the liquid at end B

${{\rm{W}}_2} = {{\rm{P}}_2}{\rm{*}}{{\rm{A}}_2}{\rm{*}}{{\rm{V}}_2}$ -------------- (ii)

Now, the difference in energy per second on the flow of liquid is,

$\Delta {{\rm{w}}_1} = {{\rm{w}}_1} - {{\rm{w}}_2}$

$= {{\rm{P}}_1}{\rm{*}}{{\rm{A}}_1}{\rm{*}}{{\rm{V}}_1} - {{\rm{P}}_2}{\rm{*}}{{\rm{A}}_2}{\rm{*}}{{\rm{V}}_2}$

But, ${{\rm{A}}_1}{\rm{*}}{{\rm{V}}_1} = {{\rm{A}}_2}{\rm{*}}{{\rm{V}}_2}\left( {{\rm{from\: equation\: of\: continuity}}} \right)$

so,

$\Delta {{\rm{w}}_1} = {{\rm{P}}_1}{\rm{*}}{{\rm{A}}_1}{\rm{*}}{{\rm{V}}_1} - {{\rm{P}}_2}{\rm{*}}{{\rm{A}}_1}{\rm{*}}{{\rm{V}}_1}$

$= \left( {{{\rm{P}}_1} - {{\rm{P}}_2}} \right){\rm{*}}{{\rm{A}}_1}{{\rm{V}}_1}$ ---------- (iii)

But,${{\rm{A}}_1}{\rm{*}}{{\rm{V}}_1} = {\rm{volume\: of\: liquid\: per\: second}} = \frac{{{\rm{mass\: per\: second}}}}{{{\rm{density}}}}$

$= \frac{{\rm{m}}}{\rho }$

Then, equation (iii) gives,

$\Delta {{\rm{w}}_1} = \left( {{{\rm{P}}_1} - {{\rm{P}}_2}} \right){\rm{*}}\frac{{\rm{m}}}{\rho }$ -------------- (iv)

Since, the height h2 is less than h1, the potential energy at the end be is less than that of A. So, the last in P.E. per second is

$\Delta {\rm{P}}.{\rm{E}} = {\rm{p}}.{\rm{e\: per\: second\: at\: A}} - {\rm{p}}.{\rm{e\: per\: second\: at\: B}}$

$= {\rm{mg}}{{\rm{h}}_1} - {\rm{mg}}{{\rm{h}}_2}$ ----------- (v)

Again, using equation of continuity,

${{\rm{A}}_1}{{\rm{V}}_1} = {{\rm{A}}_2}{\rm{*}}{{\rm{V}}_2}$

Since, A2 is less than A1, the velocity V2 becomes more than V1. So, there is gain in kinetic energy of the following liquid. Then, the gain in K.E per second is,

$\Delta {\rm{K}}.{\rm{E}} = {\rm{K}}.{\rm{E\: per\: second\: at\: B}} - {\rm{K}}.{\rm{E\: per\: second\: at\: A}}$

$= \frac{1}{2}{\rm{mv}}_2^2 - \frac{1}{2}{\rm{mv}}_1^2$ ------------ (vi)

For the flow of liquid, the net gain in energy per second is

$\Delta {{\rm{w}}_2} = \left( {\frac{1}{2}{\rm{mv}}_2^2 - \frac{1}{2}{\rm{mv}}_1^2} \right) - \left( {{\rm{mg}}{{\rm{n}}_1} - {\rm{mg}}{{\rm{n}}_2}} \right)$ --------- (vii)

From equation (iv) and (vii), the value of $\Delta {{\rm{w}}_1}{\rm{and\: }}\Delta {{\rm{w}}_2}$ are equal for the conversation of energy. So we have

$\Delta {{\rm{w}}_1} = \Delta {{\rm{w}}_2}$

Or, $\left( {{{\rm{P}}_1} - {{\rm{P}}_2}} \right){\rm{*}}\frac{{\rm{m}}}{\rho } = \left( {\frac{1}{2}{\rm{mv}}_2^2 - \frac{1}{2}{\rm{mv}}_1^2} \right) - \left( {{\rm{mg}}{{\rm{h}}_1} - {\rm{mg}}{{\rm{h}}_2}} \right)$

Or, $\frac{{{{\rm{P}}_1}}}{\rho } - \frac{{{{\rm{P}}_2}}}{\rho } = \frac{{{\rm{V}}_2^2}}{2} - \frac{{{\rm{V}}_1^2}}{2} - {\rm{g}}{{\rm{h}}_1} + {\rm{g}}{{\rm{h}}_2}$

Or, $\frac{{{{\rm{P}}_1}}}{\rho } + \frac{{{\rm{V}}_1^2}}{2} + {\rm{g}}{{\rm{h}}_1} = \frac{{{{\rm{P}}_2}}}{\rho } + \frac{{{\rm{V}}_2^2}}{2} + {\rm{g}}{{\rm{h}}_2}$

In general,

$\frac{{\rm{P}}}{\rho } + \frac{{{\rm{V}}_1^2}}{2} + {\rm{gh}} = {\rm{constant}}$ ------------ (viii)

Equation (viii) is the mathematical expression for Bernoulli's principle

For horizontal pipe, ${{\rm{n}}_1} = {{\rm{n}}_2}$

So, $\frac{{{{\rm{P}}_1}}}{\rho } + \frac{{{\rm{V}}_1^2}}{2} + {\rm{g}}{{\rm{h}}_1} = \frac{{{{\rm{P}}_2}}}{\rho } + \frac{{{\rm{V}}_2^2}}{2} + {\rm{g}}{{\rm{h}}_1}$

Or, $\frac{{{{\rm{P}}_1}}}{\rho } + \frac{{{\rm{V}}_1^2}}{2} = \frac{{{{\rm{P}}_2}}}{\rho } + \frac{{{\rm{V}}_2^2}}{2}$

Or, ${{\rm{P}}_1} + \frac{1}{2}\rho {\rm{v}}_1^2 = {{\rm{P}}_2} + \frac{1}{2}\rho {\rm{v}}_2^2$

In general,

${\rm{P}} + \frac{1}{2}\rho {{\rm{v}}^2} = {\rm{constant\: }}$ ------------ (ix)

Equation (ix) is the expression for Bernoulli's principle for horizontal pipe.

Stream line;

When the flow of liquid is such that the velocity, v of every particle at any point of the fluid is constant, then the flow is said to be steady or streamline flow. In fluid dynamics, laminar flow (or streamline flow) occurs when a fluid flows in parallel layers, with no disruption between the layers. At low velocities, the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards.

Figure14_1: stream line (laminar) flow and turbulent flow

Where, when a liquid moves with the velocity greater than its critical velocity, the motion of the particles of liquid becomes disorderly or irregular. Such a flow is called turbulent flow. In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and flow velocity in space and time.

The factors on which the terminal velocity of an object, moving in a fluid, depends:

We know that weight affects the terminal velocity because it affects the force of gravity on the object. Another factor, the one we will be testing, is the amount of air that the object will be exposed to. The position in which the object falls changes the surface area and in turn changes the terminal velocity The terminal velocity of a body through a fluid is given by an object through liquid is proportional to square if its radius. Then

${V_t} = \frac{{2{r^2}\left( {P - \sigma } \right)g}}{{g\eta }}$

This tells that if radius increases, terminal velocity will also increase. If the radius is twice the original value, ${r_n} = 2r$, then new terminal velocity is given by ${V_{tn}} = 2r_n^2$

The velocity of water in a river less on the bank but great at the middle:

The velocity of a river is the speed at which water flows along it. The velocity of water in a river less on the bank but great at the middle because it follow the equation of continuity which states that if the area of cross-section of the tube becomes larger the liquid’s speed becomes smaller and vice versa.

Equation of continuity is it valid for compressible liquid:

The equation of continuity states that if the area of cross-section of the tube becomes larger the liquid’s speed becomes smaller and vice versa.

Where,

If the liquid is incompressible then, the density remains the same and

${p_1} = {p_2}$

Therefore;

${a_1}{v_1} = {a_2}{v_2}$

$a{\rm{\: }}v = constant$

When water flowing into a broader pipe enters into a narrower pipe the pressure decreases:

${A_1}{V_1} = {A_2}{V_2}$

Or, $A{\rm{*}}V = constant$

For narrow pipe, area of cross section is less due to which velocity of liquid becomes more. Therefore water flowing into a broader pipe enters into a narrower pipe the pressure decreases.

An airplane requires a long run on the ground before taking off:

An airplane requires a long run on the ground before taking off because to take off from the ground, an airplane must reach a sufficiently high speed. The velocity required for the takeoff, the takeoff velocity, depends on several factors, including the weight of the aircraft and the wind velocity. But the velocity of air below the wing is less due to which the pressure becomes more. The difference of pressure below and above the wing produces up thrust which balances the weight of the plane and hence, the plane flies with constant or uniform velocity.

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