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FLUID IN MOTION
HYDRODYNAMICS
Hydrodynamics is the branch of physics which deals with the study of properties of fluids in motion.
Viscosity of the fluid
Is the property of a moving fluid (liquid or gas) to oppose the relative motion between its layers.
Thus viscosity is that property of a fluid that indicates its internal friction.
The greater the viscosity of a fluid, the grater is the force required to cause one layer of fluid to slide past another.
For example:
The viscosity of honey is very large as compared to that of water. This means that for the same applied external force, the rate of flow of honey will be very small as compared to that of water.
The viscosity of a fluid not only retards its own motion but it also retards the motion of a solid through it.
The greater the viscosity of a fluid, the harder it is for a solid to move through it, imagine the difference between swimming in water and honey.
CAUSE OF VISCOSITY
Viscosity is the internal friction of a fluid which opposes the motion of one layer of fluid past another.
The forces of attraction between the molecules of a moving fluid determine the viscosity of the fluid.
Viscous force
Is the tangential force that tends to destroy the relative motion between different fluid layers.
Viscous Fluid
Is the fluid which offers a resistance to the motion through it of any solid body.
Non Viscous fluid
Is the fluid which does not offer a resistance to the motion through it of any solid.
Velocity Gradient
Is the change of velocity divided by the distance in a direction perpendicular to the velocity.
NEWTON’S LAW OF VISCOSITY
Newton’s law of viscosity states that “the frictional force F between the layers is directly proportional to area A of the layers in contact and to the velocity gradient”
is a constant of proportionality and is called coefficient of viscosity. Note that the negative sign shows that the direction of viscous drag F is opposite to the direction of motion of the liquid.
From,
The coefficient of viscosity is the tangential force required to maintain a unit velocity gradient between two parallel layers each of unit area.
or
Is the tangential force per unit area of a layer, required to maintain unit velocity gradient normal to the direction of flow.
Coefficient of viscosity 
of the liquid is a measure of the degree to which the fluid exhibits viscous effects.
of the liquid is a measure of the degree to which the fluid exhibits viscous effects.
Units of
The SI units of F is 1N, The SI unit or A is 1
and that of velocity gradient is 
SI unit of is NSM-2. It also called
.
and that of velocity gradient is SI unit of is NSM-2. It also called
.The coefficient of viscosity of a liquid is if a tangential force of 1N is required to maintain a velocity gradient of between two parallel layers each of area 
. 
. is also called Dynamic viscosity or Absolute Viscosity The viscosity of an ideal liquid is zero. Dimensional formula of 
= The coefficient of viscosity of a liquid decrease with the increases in temperature and vice versa. However, the coefficient of viscosity of gases increases with the increase in temperature.
Fluidity
Is a measure of re ability of a fluid of flow and is equal to the reciprocals of
Dimensional formula of fluidity is 
NEWTONIAN AND NON NEWTONIAN FLUID
Newtonian fluid
Is the fluid with which the velocity gradient is proportional to the tangential stress.
These fluids obeys Newton’s law of viscosity
Non Newtonian fluid
Is the fluid with which the velocity gradient is not proportional to the tangential stress.
These fluids does not obey Newton‘s law of viscosity.
They don‘t have constant values of , Oil-paint is an example of a non-Newtonian liquid
, Oil-paint is an example of a non-Newtonian liquidVARIATION OF COEFFICIENT OF VISCOSITY WITH TEMPERATURE
- For liquids
In the case of liquids the viscosity is due to the attraction among molecules within the liquid and also between the molecules of the liquid and those of solids in contact.
With rise in temperature, the molecular attractions get weakened and hence viscosity decreases.
- For Gases
The molecules are much furt
her apart and the viscosity is due to the collisions between the fast moving (flowing) molecules and those flowing at lower velocities.
her apart and the viscosity is due to the collisions between the fast moving (flowing) molecules and those flowing at lower velocities.
During collisions the fast molecules give up momentum to the slow molecules and are retarded in their flow.
As temperature increases molecular activity increases and this led to the increase of viscosity with rise in temperature.
Differences between friction and viscosity.
Friction | Viscosity |
| – The viscosity of a liquid decreases with the increase in temperature. |
surfaces between the solids. | – Heat is generated within the fluid and not at the interface of the solid and the fluid |
– Friction between two surfaces of solids is independent of the area of contact and of the relative | – Viscosity depends upon the area of contact and the velocity gradient between the layers. |
velocity.
Similarities between friction and viscosity
- Both come into play wherever there is a relative motion
- Both oppose the relative motion (iii) Both arise from intermolecular forces (iv) Both depend on nature of surfaces.
STOKE’S LAW
Stoke’s law state that “for steady motion of a small spherical body, smooth and rigid moves slowly in a fluid of infinite extent, the viscous drag force experienced on the body is given by,
F – Viscous drag force
– Coefficient of viscosity of the fluid
r – Radius of spherical body
v – Terminal velocity
Derivation of the Formula
Consider a sphere of radius 
moving with velocity v through a fluid whose coefficient of viscosity.
moving with velocity v through a fluid whose coefficient of viscosity.It is desired to find the expression for the viscous force F of the sphere.
For this purpose, we shall use dimensional analysis.
Stokes observed that in case of a slowly moving small sphere, the viscous force F depends on
- The radius r of the spherical body
- The coefficient of viscosityof the fluid.
- The velocity v of the spherical body.
- Shape and size of the solid body.
Equating the indices of M, L, T
1 = y , 1 = x – y + z, -2 = y – z Solving gives y = 1 z = 1
x = 1
The value of k was found to be 
Limitation of the Stoke’s law
- Strictly, the law applies to a fluid of infinite extent.
- The law does not hold good if the spherical body is moving so fast that conditions are not streamline.
- The spherical body must be smooth, small and rigid
Importance of Stoke’s law
- It accounts for the formation of goods
- It is used in Millikan‘s experiment for the measurement of charge on an electron
- It is used to find the size of small particles
- It explains why large rain drops hurt much more than small ones when falling on you. It is not just that they are heavier but they are actually falling faster.
VERTICAL MOTION OF SPHERICAL BODY ON VISCOUS FLUID
Consider a small sphere falling freely from rest through a large column of a viscous fluid as shown below.
The forces acting on the sphere are
- Weight W of the sphere acting vertically downwards.
- Upthrust U equal to the weight of the liquid displaced.
- Viscous drag F (In direction opposite to motion).
When the sphere body falls with terminal velocity the body is at the equilibrium.
W – (U + F) = 0
W = U + F
Consider a small sphere of radius r falling, freely through a viscous fluid Let,
= Density of the sphere body
= Density of the fluid = Coefficient of viscosity of the fluidWeight of sphere W
Upthrust on sphere U
Viscous drag
From, W = U + F
Stokes made the following assumptions
- The medium through which the sphere falls is infinite in extent.
- The spherical body is perfectly rigid and smooth.
- There is no slip between the spherical body and medium.
- When the body moves through the medium no eddy current or waves should be set up in the medium.
- The body must move through the medium slowly.
- The diameter of spherical body must be large compared with the spaces between the molecules of the medium.
- Terminal velocity of the body must be less the critical velocity of the medium.
Terminal velocity is the maximum constant velocity acquired by a body while falling freely through a viscous medium.
- The terminal velocity of a spherical body falling freely through a viscous fluid is directly proportional to the square of its radius.
From,
This means that for a given medium, the terminal velocity of a large sphere is greater than that of a small sphere of the same material.
For this reason, bigger raindrops fall with greater velocity as compared with smaller ones.
- The terminal velocity of a spherical body is directly proportional to the difference in the densities of the body and the fluid
POISEUILLE’S FORMULA
When liquid flows through a horizontal pipe with wall of the pipe remains at rest while the velocity of layers goes on increasing towards the centre of the pipe, as the result the rate of flow of the liquid is slowest near the pipe walls and fastest in the centre of the pipe.
Poiseuille’s studied the liquid flow through horizontal pipes and concluded that the rate of flow Q of a liquid through a pipe varies as,
Where
Q – Volume of liquid flowing per second P = Pressure difference across ends of the pipe r – Radius of the pipe
– Length of the pipe – Viscosity coefficient of liquid Combining these factors
Here K is a constant of proportionality and is found to be 
Consider the steady flow of liquid through a capillary tube AB or radius R and length 
.
.
Pressure difference at its ends
P = pA – PB
Poiseulle’s formula
V – Volume of liquid collected in a time t
Poiseulle’s made the following assumptions.
- The flow of liquid is steady and parallel to the axis of the tube.
- The pressure over any cross section at right angles to the tube is constant.
- The velocity of liquid layer in contact with the sides of the tube is zero and increases in a regular manner as the axis of the tube is approached.
DERIVATION OF POISEULLE’S FORMULA BY DIMENSIONAL ANALYSIS
Consider a viscous liquid undergoing steady flow through a horizontal pipe of circular cross section as shown in the Fig. below.
Because of viscous drag, the velocity varies from a maximum at the centre of the pipe to zero at the walls.
We shall use dimensional analysis to derive an expression for the rate of flow of liquid Q through the pipe.
It is reasonable to suppose that the rate of flow of liquid through the pipe depends on
- The coefficient of viscosity
- The radius r of the pipe
- The pressure gradient
• Pressure gradient
Is the pressure difference between the ends of the pipe per unit length of the pipe.
Pressure gradient = 
We can express the rate of flow of the liquid as
Equating the induces of M, L, T
On solving
The value of k cannot be found by using dimensional analysis.
This formula for Q is called Poiseulle’s formula
Limitation of Poiseuille’s Law
The law is true only for the steady flow of a liquid through horizontal pipe.
The formula applies only to Newtonian fluids which are undergoing steady flow.
Speed of Bulk flow.
Speed of Bulk flow is the rate of volume flow divided by the cross-sectional area of the pipe.
Steady flow occurs o
nly when the speed of bulk flow is less than a certain critical velocity. Vc.
nly when the speed of bulk flow is less than a certain critical velocity. Vc.
Poiseuille’s Formula does not hold good when the speed of bulk flow exceed critical velocity.
SERIES CONNECTION OF THE TUBES
When the two tubes of different diameters and lengths are connected in series the rate of fluid flow is the same in all the tubes.
Expression of the rate of volume of the fluid flow.
For tube 1
For Tube 2
Adding the 2 equation
Let 
be the pressure difference at the end of the tubes
be the pressure difference at the end of the tubes
Expression of the pressure at the junction
Then
Applications of Viscosity
- The quality of ink is decided by the coefficient of viscosity of ink.
- The study of variation of viscosity with temperature helps us to pick up the best lubricant of a certain machine.
- Applied in the study of circulation of blood. The variation in the coefficient of viscosity of blood affects the pressure and that in turn affects the efficiency of our bloody.
- For damping the motion of certain instruments such as, shock absorber in car’s suspension system.
- Used in production and transportation of oils.
- Liquids having higher values of coefficient of viscosity are used as buffers at railway station.
FLUID DYNAMICS
Fluid dynamics is the study of fluids in motion.
While discussing fluid flow, we generally make the following assumptions.
- The fluid is non-viscous
- The fluid is incompressible
- The fluid motion is steady
Non viscous fluid.
Non viscous fluid is the fluid which does not offer a resistance to the motion through it of any solid body.
There is no internal friction between the adjacent layers of the fluid.
Incompressible fluid.
An incompressible fluid is the fluid in which changes in pressure produce no change in the density of the fluid
This means that density of the fluid is constant.
Steady flow of a fluid
This means that the velocity, density and pressure at each point in the fluid do not change with time.
TYPES OF LIQUID FLOW
The liquid flow is of two main types
- Streamline flow or steady flow
- Turbulent flow
Stream line flow.
Streamline flow is the flow of a fluid when all the fluid particles that pass any given point follow the same path at the same speed.
The fluid particles have the same velocity. This flow is also called orderly flow or uniform flow.
Characteristics of streamline.
- The velocity of a particle at any point is a constant and is independent of time.
- The liquid layer in contact with the solid surface will be at rest
- The motion of the fluid (liquid) follows Newton‘s law of viscous force For example, consider a liquid flowing through a pipe as shown in figure below
The flowing liquid will have a certain velocity v1 at a, a velocity v2 at b and so on.
As time goes, the velocity of whatever liquid particle happens pas to pass be at a is still v1, that at b is still v2, then the flow is said to be steady or streamline flow.
Every particle starting at a will follow the same path abc. The line abc is called streamline.
Streamline.
A streamline is a curve whose tangent at any point is along the direction of the velocity of the liquid particle at that point.
Streamlines never cross each other otherwise, particles reaching the intersection would not have a unique velocity at that point in space.
Tube of flow.
A tube of flow is a tabular region of a flowing fluid whose boundaries are defined by a set of streamlines.
Since the streamlines represent the path of particles, we see that no liquid conflation or out of the sides of a tube of flow.
In a steady flow, the velocity, density and pressure at each point in the fluid do not change with time.
Laminar flow.
Laminar flow is a special case of steady flow in which the velocities of all particles on any given streamline are the same, though the particles of different streamlines may move at different speeds.
Turbulent flow.
Turbulent flow is the flow of fluid when the speed and direction of fluid particles passing any point vary with time.
It is also known as disorderly flow.
Line of flow.
A line of flow is the path followed by a particle of the fluid.
Rotational flow
This is when the element of fluid at each point the angular velocity is equal to zero.
Irrotation flow.
Irrotation flow is the type of fluid flow by which the element of fluid at each irrotation no net angular velocity about that axis.
RATE OF FLOW
A rate of flow is the volume of a liquid that passes the cross-section of a vessel (pipe) in one second It is denoted by the symbol Q
The SI unit of rate of flow of liquid is m3/s
Consider a pipe of uniform cross-sectional area A as shown below.
If the liquid is flowing at an average velocity of v, then distance 
through which the liquid moves in time
through which the liquid moves in timet is
This may be regarded as the length of an imaginary cylinder of the liquid that has passed the section S in time t.
Then the volume of liquid that has passed section S in time t is V
This is called discharge equation.
The quantity Av is called the flow rate or volume flux.
It is clear that for constant rate of flow, the velocity v is inversely proportional to the area of cross sectional of the pipe.
If the cross sectional area of the pipe decrease the velocity of the liquid increases and vice verse.
REYNOLD’S NUMBER
Reynold’s number is a dimensionless ratio which determines whether the fluid flow is streamline or turbulent.
It is denoted by Re or NR
= Density of the liquid v = Average velocity of flow
= Diameter of the tube or pipe
= Coefficient of viscosity of the liquid If NR< 2000, the flow is laminar or steady
If NR> 3000, the flow is turbulent
It NR is between 2000 and 3000, the flow is unstable, it may change from laminar to turbulent and vice versa.
