Temperature is defined as the degree of hotness or coldness of a body. The natural flow of heat is from higher temperature to lower temperature.
Two bodies are said to be in thermal equilibrium with each other, when no heat flows from one body to the other. That is when both the bodies are at the same temperature.
(1) Temperature is one of the seven fundamental quantities with dimension \[[\theta ]\]. It is a scalar physical quantity with S.I. unit kelvin.
(2) When heat is given to a body and its state does not change, the temperature of the body rises and if heat is taken from a body its temperature falls i.e. temperature can be regarded as the effect of cause "heat".
(3) According to kinetic theory of gases, temperature (macroscopic physical quantity) is a measure of average translational kinetic energy of a molecule (microscopic physical quantity).
(4) Although the temperature of a body can to be raised without limit, it cannot be lowered without limit and theoretically limiting low temperature is taken to be zero of the kelvin scale.
(5) Highest possible temperature achieved in laboratory is about 108K while lowest possible temperature attained is \[{{10}^{-8}}K\].
(6) Temperature of the core of the sun is \[{{10}^{7}}K\] while that of its surface is 6000 K.
(7) Normal temperature of human body is 310. 15 K \[({{37}^{o}}C={{98.6}^{o}}F)\].
(8) NTP or STP implies 273.15K \[({{0}^{o}}C={{32}^{o}}F)\] 
(i) Directly proportional to the pressure difference (P).
(ii) Directly proportional to the fourth power of radius (r) of the capillary tube
(iii) Inversely proportional to the coefficient of viscosity \[(\eta )\] of the liquid.
(iv) Inversely proportional to the length \[(l)\] of the capillary tube.
i.e. \[V\propto \frac{P\,{{r}^{4}}}{\eta l}\] or \[V=\frac{KP\,{{r}^{4}}}{\eta l}\]
\[\therefore \] \[V=\frac{\pi P\,{{r}^{4}}}{8\eta l}\]
[Where \[K=\frac{\pi }{8}\] is the constant of proportionality]
This is known as Poiseuille's equation.
This equation also can be written as,
\[V=\frac{P}{R}\] where \[R=\frac{8\eta l}{\pi \,{{r}^{4}}}\]
R is called as liquid resistance.
(1) Series combination of tubes
(i) When two tubes of length \[{{l}_{1}},\,{{l}_{2}}\] and radii \[{{r}_{1}},\,\,{{r}_{2}}\] are connected in series across a pressure difference P,
Then \[P={{P}_{1}}+{{P}_{2}}\] ...(i)
Where \[{{P}_{1}}\] and \[{{P}_{2}}\] are the pressure difference across the first and second tube respectively
(ii) The volume of liquid flowing through both the tubes i.e. rate of flow of liquid is same.
Therefore \[V={{V}_{1}}={{V}_{2}}\]
i.e., \[V=\frac{\pi {{P}_{1}}r_{1}^{4}}{8\eta {{l}_{1}}}=\frac{\pi {{P}_{2}}r_{2}^{4}}{8\eta {{l}_{2}}}\] ...(ii)
Substituting the value of P1 and P2 from equation (ii) to equation (i) we get
\[P={{P}_{1}}+{{P}_{2}}\]\[=V\left[ \frac{8\eta {{l}_{1}}}{\pi r_{1}^{4}}+\frac{8\eta {{l}_{2}}}{\pi r_{2}^{4}} \right]\] \[\therefore \] \[V=\frac{P}{\left[ \frac{8\eta {{l}_{1}}}{\pi r_{1}^{4}}+\frac{8\eta {{l}_{2}}}{\pi r_{2}^{4}} \right]}\]\[=\frac{P}{{{R}_{1}}+{{R}_{2}}}=\frac{P}{{{R}_{eff}}}\]
Where \[{{R}_{1}}\] and \[{{R}_{2}}\] are the liquid resistance in tubes
(iii) Effective liquid resistance in series combination
\[{{R}_{eff}}={{R}_{1}}+{{R}_{2}}\]
(2) Parallel combination of tubes
(i) \[P={{P}_{1}}={{P}_{2}}\]
(ii) \[V={{V}_{1}}+{{V}_{2}}\]\[=\frac{P\pi r_{1}^{4}}{8\eta {{l}_{1}}}+\frac{P\pi r_{2}^{4}}{8\eta {{l}_{2}}}\]
\[=P\left[ \frac{\pi r_{1}^{4}}{8\eta {{l}_{1}}}+\frac{\pi r_{2}^{4}}{8\eta {{l}_{2}}} \right]\]
\[\therefore \]\[V=P\left[ \frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}} \right]\,=\frac{P}{{{R}_{eff}}}\]
(iii) Effective liquid resistance in parallel combination
\[\frac{1}{{{R}_{eff}}}=\frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}\] or \[{{R}_{eff}}=\frac{{{R}_{1}}{{R}_{2}}}{{{R}_{1}}+{{R}_{2}}}\] 
Force on the body
(i) Weight of the body (W) = mg = (volume \[\times \] density) \[\times \,g=\frac{4}{3}\pi {{r}^{3}}\rho g\]
(ii) Upward thrust (T) = weight of the fluid displaced
= (volume \[\times \] density) of the fluid \[\times g=\frac{4}{3}\pi {{r}^{3}}\sigma g\]
(iii) Viscous force \[(F)=6\pi \eta r\upsilon \]
When the body attains terminal velocity the net force acting on the body is zero.
\[\therefore \] \[W-T-F=0\]or \[F=W-T\]
\[\Rightarrow \] \[\,6\pi \eta rv=\frac{4}{3}\pi \,{{r}^{3}}\rho \,g-\frac{4}{3}\pi \,{{r}^{3}}\sigma \,g\]\[=\frac{4}{3}\pi {{r}^{3}}(\rho -\sigma )\,g\]
\[\therefore \] Terminal velocity \[\,v=\frac{2}{9}\frac{{{r}^{2}}(\rho -\sigma )\,g}{\eta }\]
(i) Terminal velocity depend on the radius of the sphere so if radius is made \[n-\] fold, terminal velocity will become \[{{n}^{2}}\] times.
(ii) Greater the density of solid greater will be the terminal velocity
(iii) Greater the density and viscosity of the fluid lesser will be the terminal velocity.
(iv) If \[\rho >\sigma \]then terminal velocity will be positive and hence the spherical body will attain constant velocity in downward direction.
(v) If \[\rho <\sigma \] then terminal velocity will be negative and hence the spherical body will attain constant velocity in upward direction. Example : Air bubble in a liquid and clouds in sky.
(vi) Terminal velocity graph :
Consider the two layers CD and MN of the liquid at distances \[x\] and \[x+dx\] from the fixed surface AB, having the velocities \[\upsilon \] and \[\upsilon +d\upsilon \] respectively. Then \[\frac{dv}{dx}\] denotes the rate of change of velocity with distance and is known as velocity gradient.
According to Newton's hypothesis, the tangential force F acting on a plane parallel layer is proportional to the area of the plane A and the velocity gradient \[\frac{dv}{dx}\] in a direction normal to the layer,
i.e., \[F\propto A\] and \[F\propto \frac{dv}{dx}\]
\[\therefore \] \[F\propto A\frac{dv}{dx}\]
or \[F=-\eta A\frac{dv}{dx}\]
Where \[\eta \] is a constant called the coefficient of viscosity. Negative sign is employed because viscous force acts in a direction opposite to the flow of liquid.
If \[A=1,\,\frac{dv}{dx}=1\] then \[\eta =F\].
Hence the coefficient of viscosity is defined as the viscous force acting per unit area between two layers moving with unit velocity gradient.
(1) Units : \[dyne-s-c{{m}^{2}}\] or Poise (C.G.S. system);
\[Newton-s-{{m}^{2}}\] or Poiseuille or decapoise (S.I. system)
1 Poiseuille = 1 decapoise = 10 Poise
(2) Dimension : \[[M{{L}^{1}}{{T}^{1}}]\]
(3) Viscosity of liquid is much greater (about 100 times more) than that of gases i.e. \[{{\eta }_{L}}>{{\eta }_{G}}\]
Example : Viscosity of water = 0.01 Poise.
While of air \[=200\mu \]Poise
(4) With increase in pressure, the viscosity of liquids (except water) increases while that of gases is practically independent of pressure. The viscosity of water decreases with increase in pressure.
(5) Difference between viscosity and solid friction : Viscosity differs from the solid friction in the respect that the viscous force acting between two layers of the liquid depends upon the area of the layers, the relative velocity of two layers and distance between two layers, but the friction between two solid surfaces is independent of the area of surfaces in contact and the relative velocity between them.
(6) From kinetic theory point of view viscosity represents transport of momentum, while diffusion and conduction represents transport of mass and energy respectively.
(7) The viscosity of thick liquids like honey, glycerin, coaltar etc. is more than that of thin liquids like water.
(8) The cause of viscosity in liquids is cohesive forces among molecules where as in gases, it is due to diffusion.
(9) The viscosity of gases increases with increase of temperature, because on increasing temperature the rate of diffusion increases.
(10) The viscosity of liquid decreases with increase of temperature, because the more... 
Which is same as the speed that an object would acquire in falling from rest through a distance h and is called velocity of efflux or velocity of flow.
This result was first given by Torricelli, so this is known as Torricelli's theorem.
(i) The velocity of efflux is independent of the nature of liquid, quantity of liquid in the vessel and the area of orifice.
(ii) Greater is the distance of the hole from the free surface of liquid, greater will be the velocity of efflux [i.e., \[v\propto \sqrt{h}]\]
(iii) As the vertical velocity of liquid at the orifice is zero and it is at a height \[(H-h)\] from the base, the time taken by the liquid to reach the base-level \[t=\sqrt{\frac{2(H-h)}{g}}\]
(iv) Now during time t liquid is moving horizontally with constant velocity \[\upsilon ,\] so it will hit the base level at a horizontal distance \[x\] (called range) as shown in figure.
Such that \[x=vt=\sqrt{2gh}\times \sqrt{[2(H-h)/g]}=2\sqrt{h(H-h)}\]
For maximum range \[\frac{dx}{dh}=0\]
\[\therefore \] \[h=\frac{H}{2}\]
i.e., range \[x\] will be maximum when
\[h=\frac{H}{2}\].
\[\therefore \] 
Maximum range \[{{x}_{\text{max}}}=2\sqrt{\frac{H}{2}\left[ H-\frac{H}{2} \right]}=H\]
(v)
If the level of free surface in a container is at height H from the base and there are two holes at depth h and y below the free surface, then
\[x=2\sqrt{h(H-h)}\] and \[{x}'=2\sqrt{y(H-y)}\]
Now if \[x={x}'\], i.e., \[h(H-h)=y(H-y)\]
i.e., \[{{y}^{2}}-Hy+h(H-h)=0\]
or \[y=\frac{1}{2}[H\pm (H-2h)]\],
i.e., \[y=h\] or \[(H-h)\]
i.e., the range will be same if the orifice is at a depth h or \[(H-h)\] below the free surface. Now as the distance \[(H-h)\] from top means \[H-(H-h)=h\] from the bottom, so the range is same for liquid coming out of holes at same distance below the top and above the bottom.
(vi)
If \[{{A}_{0}}\] is the area of orifice at a depth y below the free surface and A is that of container, the volume of liquid coming out of the orifice per second will be \[(dV/dt)=v{{A}_{0}}={{A}_{0}}\sqrt{2gy}\]
[As \[v=\sqrt{2gy}\]]
Due to this, the level of liquid in the container will decrease and so if the level of liquid in the container above the hole changes from y to \[y-dy\] in time t to \[t+dt\] then \[-dV=A\,dy\]
So substituting this value of \[dV\] in the above equation
\[-A\frac{dy}{dt}={{A}_{0}}\sqrt{2gy}\]
i.e., \[\int_{{}}^{{}}{dt=-\frac{A}{{{A}_{0}}}}\frac{1}{\sqrt{2g}}\int_{{}}^{{}}{{{y}^{-1/2}}}dy\]
So the time taken for the level more... 
When two boats or buses move side by side in the same direction, the water (or air) in the region between them moves faster than that on the remote sides. Consequently in accordance with Bernoulli's principle the pressure between them is reduced and hence due to pressure difference they are pulled towards each other creating the so called attraction.
(ii) Working of an aeroplane
This is also based on Bernoulli's principle. The wings of the aeroplane are of the shape as shown in fig. Due to this specific shape of wings when the aeroplane runs, air passes at higher speed over it as compared to its lower surface. This difference of air speeds above and below the wings, in accordance with Bernoulli's principle, creates a pressure difference, due to which an upward force called 'dynamic lift' (= pressure difference \[\times \] area of wing) acts on the plane. If this force becomes greater than the weight of the plane, the plane will rise up.
(iii) Action of atomizer
The action of carburetor, paint-gun, scent-spray or insect-sprayer is based on Bernoulli's principle. In all these, by means of motion of a piston P in a cylinder C, high speed air is passed over a tube T dipped in liquid L to be sprayed. High speed air creates low pressure over the tube due to which liquid (paint, scent, insecticide or petrol) rises in it and is then blown off in very small droplets with expelled air.
(iv) Blowing off roofs by wind storms
During a tornado or hurricane, when a high speed wind blows over a straw or tin roof, it creates a low pressure (P) in accordance with Bernoulli's principle.
However, the pressure below the roof (i.e., inside the room) is still atmospheric \[(={{P}_{0}})\]. So due to this difference of pressure, the roof is lifted up and is then blown off by the wind.
(v) Magnus effect :
When a spinning ball is thrown, it deviates from its usual path in flight. This effect is called Magnus effect and plays as important role in tennis, cricket and soccer, etc. as by applying appropriate spin the moving ball can be made to curve in any desired direction.
If a ball is moving from left to right and also spinning about a horizontal axis perpendicular to the direction of motion as shown in fig. then relative to the ball, air will be moving from right to left.
The resultant velocity of air above the ball will be \[(v+r\omega )\] while below it \[(v-r\omega )\]. So in accordance with Bernoulli's principle pressure above the ball will be less than below it. Due to this difference of pressure an upward force will act on the ball and hence the ball will deviate more... 
Mathematically for unit volume of liquid flowing through a pipe.
\[P+\rho gh+\frac{1}{2}\rho {{v}^{2}}=\]constant
To prove it, consider a liquid flowing steadily through a tube of non-uniform area of cross-section as shown in fig. If \[{{P}_{1}}\] and \[{{P}_{2}}\] are the pressures at the two ends of the tube respectively, work done in pushing the volume V of incompressible fluid from point B to C through the tube will be
\[W={{P}_{1}}V-{{P}_{2}}V=({{P}_{1}}-{{P}_{2}})V\] ...(i)
This work is used by the fluid in two ways. (a) In changing the potential energy of mass m (in the volume V ) from \[mg{{h}_{1}}\] to \[mg{{h}_{2}},\]
i.e., \[\Delta U=mg({{h}_{2}}-{{h}_{1}})\] ...(ii)
(b) In changing the kinetic energy from \[\frac{1}{2}mv_{1}^{2}\] to \[\frac{1}{2}mv_{2}^{2}\],
i.e., \[\Delta K=\frac{1}{2}m(v_{2}^{2}-v_{1}^{2})\] ...(iii)
Now as the fluid is non-viscous, by conservation of mechanical energy
\[W=\Delta U+\Delta K\]
i.e., \[({{P}_{1}}-{{P}_{2}})\,V=mg({{h}_{2}}-{{h}_{1}})+\frac{1}{2}m(v_{2}^{2}-v_{1}^{2})\]
or \[{{P}_{1}}-{{P}_{2}}=\rho g({{h}_{2}}-{{h}_{1}})+\frac{1}{2}\rho (v_{2}^{2}-v_{1}^{2})\] [As \[\rho =m/V\]]
or \[{{P}_{1}}+\rho g{{h}_{1}}+\frac{1}{2}\rho v_{1}^{2}={{P}_{2}}+\rho g{{h}_{2}}+\frac{1}{2}\rho v_{2}^{2}\]
or \[P+\rho gh+\frac{1}{2}\rho {{v}^{2}}=\]constant
This equation is the so called Bernoulli's equation and represents conservation of mechanical energy in case of moving fluids.
(i) Bernoulli's theorem for unit mass of liquid flowing through a pipe can also be written as:
\[\frac{P}{\rho }+gh+\frac{1}{2}{{v}^{2}}=\]constant
(ii) Dividing above equation by g we get \[\frac{P}{\rho g}+h+\frac{{{v}^{2}}}{2g}\]= constant
Here \[\frac{P}{\rho g}\] is called pressure head, h is called gravitational head and \[\frac{{{v}^{2}}}{2g}\] is called velocity head. From this equation Bernoulli's theorem can be stated as.
"In stream line flow of an ideal liquid, the sum of pressure head, gravitational head and velocity head of every cross section of the liquid is constant." | Pressure Energy | Potential energy | Kinetic energy |
| It is the energy possessed by a liquid by virtue of its pressure. It is the measure of work done in pushing the liquid against pressure without imparting any velocity to it. | It is the energy possessed by liquid by virtue of its height or position above the surface of earth or any reference level taken as zero level. | It is the energy possessed by a liquid by virtue of its motion or velocity. |
| Pressure energy of the liquid PV | Potential energy of the liquid mgh | Kinetic energy of the liquid \[\frac{1}{2}m{{v}^{2}}\] |
| Pressure energy per unit mass of the liquid \[\frac{P}{\rho }\] | Potential energy per unit mass of the liquid gh | Kinetic energy per unit mass of the liquid \[\frac{1}{2}{{v}^{2}}\] |
| Pressure energy per unit volume of the liquid P | Potential energy per unit volume of the liquid \[\rho gh\] | more...
The equation of continuity is derived from the principle of conservation of mass.
A non-viscous liquid in streamline flow passes through a tube AB of varying cross section. Let the cross sectional area of the pipe at points A and B be \[{{a}_{1}}\] and \[{{a}_{2}}\] respectively. Let the liquid enter with normal velocity \[{{v}_{1}}\] at A and leave with velocity \[{{v}_{2}}\] at B. Let \[{{\rho }_{1}}\] and \[{{\rho }_{2}}\] be the densities of the liquid at point A and B respectively.
Mass of the liquid entering per second at A = Mass of the liquid leaving per second at B
\[{{a}_{1}}{{v}_{1}}{{\rho }_{1}}={{a}_{2}}{{v}_{2}}{{\rho }_{2}}\] and \[{{a}_{1}}{{v}_{1}}={{a}_{2}}{{v}_{2}}\] [If the liquid is incompressible \[{{\rho }_{2}}={{\rho }_{1}}\]] or \[av=\]constant
or \[a\propto \frac{1}{v}\]
This expression is called the equation of continuity for the steady flow of an incompressible and non-viscous liquid.
(1) The velocity of flow is independent of the liquid (assuming the liquid to be non-viscous)
(2) The velocity of flow will increase if cross-section decreases and vice-versa. That is why :
(a) In hilly region, where the river is narrow and shallow (i.e., small cross-section) the water current will be faster, while in plains where the river is wide and deep (i.e., large cross-section) the current will be slower, and so deep water will appear to be still.
(b) When water falls from a tap, the velocity of falling water under the action of gravity will increase with distance from the tap\[(i.e.,\,{{v}_{2}}>{{v}_{1}})\]. So in accordance with continuity equation the cross section of the water stream will decrease \[(i.e.,\,{{A}_{2}}<{{A}_{1}}),\]i.e., the falling stream of water becomes narrower.
The critical velocity is that velocity of liquid flow upto which its flow is streamlined and above which its flow becomes turbulent.
Reynold's number is a pure number which determines the nature of flow of liquid through a pipe.
It is defined as the ratio of the inertial force per unit area to the viscous force per unit area for a flowing fluid.
\[{{N}_{R}}=\frac{\text{Inertial force per unit area}}{\text{Viscous }\,\text{force per unit area}}\]
If a liquid of density \[{{P}_{2}}\] is flowing through a tube of radius r and cross section A then mass of liquid flowing through the tube per second \[\frac{dm}{dt}=\] volume flowing per second \[\times \] density = \[{{P}_{1}}\]
\[\therefore \] Inertial force per unit area = \[P+\rho gh+\frac{1}{2}\rho {{v}^{2}}=\] = \[\frac{1}{2}\rho {{v}^{2}}\] = \[{{v}^{2}}\rho \]
Viscous force per unit area \[F/A=\frac{\eta v}{r}\]
So by the definition of Reynolds number
\[{{N}_{R}}=\frac{\text{Inertial force per unit area}}{\text{Viscous force per unit area}}\]\[=\frac{{{v}^{2}}\rho }{\eta v/r}=\frac{v\rho r}{\eta }\]
If the value of Reynold's number
(i) Lies between 0 to 2000, the flow of liquid is streamline or laminar.
(ii) Lies between 2000 to 3000, the flow of liquid is unstable and changing from streamline to turbulent flow. (iii) Above 3000, the flow of liquid is definitely turbulent.
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