Flow of Liquid | Definition, Types – Hydrodynamics

Flow of Liquid Definition:
The movement of liquids and gases is generally referred to as “flow,” a concept that describes how fluids behave and how they interact with their surrounding environment. for example, water moving through a channel or pipe, or over a surface. Flow can be either steady or unsteady.

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Flow of Liquid | Definition, Types – Hydrodynamics

Types of Fluid Flow

  • Streamline Flow
  • Laminar Flow
  • Turbulent Flow

(i) Streamline Flow Definition:
The flow of liquid in which each of its particle follows the same path as followed by the preceding particles is called streamline flow.

Two streamlines cannot cross each other and the greater the crowding of streamlines at a place, the greater is the velocity of liquid at that place and vice-versa.

(ii) Laminar Flow Definition:
The steady flow of liquid over a horizontal surface in the form of layers of different velocities is called laminar flow.

The laminar flow is generally used synonymously with streamline flow of liquid.

(iii) Turbulent Flow Definition:
The flow of liquid with a velocity greater than its critical velocity is disordered and called turbulent flow.

In case of turbulent flow, maximum part of external energy is spent for producing eddies in the liquid and small part of external energy is available for forward flow.

Hydrodynamics:
In physics, hydrodynamics of fluid dynamics explains the mechanism of fluid such as flow of liquids and gases. It has a wide range of applications such as evaluating forces and momentum on aircraft, prediction of weather, etc.

Flow of liquid Reynold’s Number
Equation of Continuity Energy of a Liquid
Bernoulli’s Principle Venturimeter
Torricelli’s Theorem Viscosity
Poiseuille’s Law Rate of Flow of Liquid
Stoke’s Law and Terminal Velocity Critical Velocity

Torricelli’s Theorem Formula and Derivation – Hydrodynamics

Torricelli’s theorem, is a theorem in fluid dynamics relating the speed of fluid flowing from an orifice to the height of fluid above the opening.

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Torricelli’s Theorem Formula and Derivation – Hydrodynamics

Torricelli’s Theorem
Velocity of efflux (the velocity with which the liquid flows out of a orifice or narrow hole) is equal to the velocity acquired by a freely falling body through the same vertical distance equal to the depth of orifice below the free surface of liquid.
Torricelli’s Theorem

Torricelli’s Law Formula:

Velocity of efflux, v = \(\sqrt{2 g h}\)

where,
h = depth of orifice below the free surface of liquid.

t = \(\sqrt{\frac{2(H-h)}{g}}\)
Horizontal range, S = \(\sqrt{4 h(H-h)}\)

where, H = height of liquid column.

Horizontal range is maximum, equal to height of the liquid column H, when orifice is at half of the height of liquid column.
If the hole is at the bottom of the tank, then time required to make the tank empty is

t = \(\frac{A}{A_{0}} \sqrt{\frac{2 H}{g}}\)

where, A is area of the container and Ao is area of orifice.
Volume of liquid coming out from the orifice per second

= VAo = Ao\(\sqrt{2 g h}\)

Hydrodynamics:
In physics, hydrodynamics of fluid dynamics explains the mechanism of fluid such as flow of liquids and gases. It has a wide range of applications such as evaluating forces and momentum on aircraft, prediction of weather, etc.

Flow of liquid Reynold’s Number
Equation of Continuity Energy of a Liquid
Bernoulli’s Principle Venturimeter
Torricelli’s Theorem Viscosity
Poiseuille’s Law Rate of Flow of Liquid
Stoke’s Law and Terminal Velocity Critical Velocity

Total Energy of a Liquid | Pressure, Kinetic, Potential Energy – Hydrodynamics

Total Energy of a Liquid | Pressure, Kinetic, Potential Energy – Hydrodynamics

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Energy of a Liquid
A liquid in motion possess three types of energy
(i) Pressure Energy
Pressure energy per unit mass

= \(\frac{p}{\rho}\)

where, p = pressure of the liquid
and ρ = density of the liquid.
Pressure energy per unit volume = p

Pressure Energy SI Unit
SI unit of pressure energy is Joule(J).

(ii) Kinetic Energy
Kinetic energy per unit mass

= \(\frac{1}{2}\) v²

Kinetic energy per unit volume

= \(\frac{1}{2}\) ρv²

Kinetic Energy Units:
The SI unit of kinetic energy is Joule
The CGS unit of kinetic energy is erg.

(iii) Potential Energy
Potential energy per unit mass =gh
Potential energy per unit volume = ρgh

Potential Energy Unit:
The SI unit of kinetic energy is Joule.

Hydrodynamics:
In physics, hydrodynamics of fluid dynamics explains the mechanism of fluid such as flow of liquids and gases. It has a wide range of applications such as evaluating forces and momentum on aircraft, prediction of weather, etc.

Flow of liquid Reynold’s Number
Equation of Continuity Energy of a Liquid
Bernoulli’s Principle Venturimeter
Torricelli’s Theorem Viscosity
Poiseuille’s Law Rate of Flow of Liquid
Stoke’s Law and Terminal Velocity Critical Velocity

Equation of Continuity | Definition, Derivation – Hydrodynamics

Equation of Continuity Physics:
If a liquid is flowing in streamline flow in a pipe of non-uniform cross-sectional area, then rate of flow of liquid across any cross-section remains constant.

A continuity equation in physics is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity.

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Equation of Continuity | Definition, Derivation – Hydrodynamics

Equation of Continuity Derivation:
i.e. a1v1 = a2v2 ⇒ av = constant
or
a ∝\(\frac{1}{v}\)
Equation of Continuity
The velocity of liquid is slower where area of cross-section is larger and faster where area of cross-section is smaller.
Equation of Continuity
The falling stream of water becomes narrower, as the velocity of falling stream of water increases and therefore its area of cross-section decreases. Deep water appears still because it has large cross-sectional area.

Hydrodynamics:
In physics, hydrodynamics of fluid dynamics explains the mechanism of fluid such as flow of liquids and gases. It has a wide range of applications such as evaluating forces and momentum on aircraft, prediction of weather, etc.

Flow of liquid Reynold’s Number
Equation of Continuity Energy of a Liquid
Bernoulli’s Principle Venturimeter
Torricelli’s Theorem Viscosity
Poiseuille’s Law Rate of Flow of Liquid
Stoke’s Law and Terminal Velocity Critical Velocity

Stoke’s Law and Terminal Velocity | Definition, Formula – Hydrodynamics

Stoke’s Law Definition in Physics:
Stokes Law, named after George Gabriel Stokes, describes the relationship between the frictional force of a sphere moving in a liquid and other quantities. If a sphere or a body moves through a fluid, a friction force must be overcome.

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Stoke’s Law and Terminal Velocity | Definition, Formula – Hydrodynamics

Stoke’s Law Formula:
When a small spherical body falls in a liquid column with terminal velocity, then viscous force acting on it is

F = 6πηrv

where,
r = radius of the body,
v = terminal velocity and
η = coefficient of viscosity. This is called Stoke’s law.

Terminal Velocity Definition:
When a small spherical body falls in a long liquid column, then after sometime it falls with a constant velocity, called terminal velocity.

Terminal Velocity Formula in terms of Viscosity:

Terminal velocity, v = \(\frac{2}{9} \frac{r^{2}(\rho-\sigma) g}{\eta}\)

where,
ρ = density of body,
a = density of liquid,
η = coefficient of viscosity of liquid and
g = acceleration due to gravity.

(i) If ρ > σ, the body falls downwards.
(ii) If ρ < σ, the body moves upwards with the constant velocity.
(iii) If ρ << ρ, v = \(\frac{2 r^{2} \rho g}{9 \eta}\)

  • Terminal velocity depends on the radius of the sphere in such a way that, if radius becomes n times, then terminal velocity will become n² times.
  • Terminal velocity-Time/distance graph

Stoke’s Law and Terminal Velocity

Importance of Stoke’s Law

(i) This law is used in the determination of electronic charge by Millikan in his oil drop experiment.
(ii) This law helps a man coming down with the help of parachute.
(iii) This law accounts for the formation of clouds.

Hydrodynamics:
In physics, hydrodynamics of fluid dynamics explains the mechanism of fluid such as flow of liquids and gases. It has a wide range of applications such as evaluating forces and momentum on aircraft, prediction of weather, etc.

Flow of liquid Reynold’s Number
Equation of Continuity Energy of a Liquid
Bernoulli’s Principle Venturimeter
Torricelli’s Theorem Viscosity
Poiseuille’s Law Rate of Flow of Liquid
Stoke’s Law and Terminal Velocity Critical Velocity

What is Viscosity? | Definition, Formula, Units – Hydrodynamics

Viscosity Definition Physics:
The property of a fluid by virtue of which an internal frictional force acts between its different layers which opposes their relative motion is called viscosity.

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What is Viscosity? | Definition, Formula, Units – Hydrodynamics

This internal frictional force is called viscous force.

Viscous forces are intermolecular forces acting between the molecules of different layers of liquid moving with different velocities.

Viscous Force Formula:

Viscous force (F) = -η A\(\frac{d v}{d x}\)
or
η = –\(\frac{F}{A\left(\frac{d v}{d x}\right)}\)

where,
\(\frac{d v}{d x}\) = rate of change of velocity with distance called velocity gradient,
A = area of cross-section and
η = coefficient of viscosity.

Viscosity Units:
SI unit of η is Nsm-2 or pascal-second or decapoise.

Viscosity Dimensional Formula:
Dimensional formula is [ML-1T-1].

1. The knowledge of the coefficient of viscosity of different oils and its variation with temperature helps us to select a suitable lubricant for a given machine.

2. The cause of viscosity in liquid is due to cohesive force between liquid molecules, while in gases, it is due to diffusion.

3. Viscosity is due to transport of momentum. The value of viscosity (and compressibility) for ideal liquid is zero.

4. The viscosity of air and of some liquids is utilised for damping the moving parts of some instruments.

5. The knowledge of viscosity of some organic liquids is used in determining the molecular weight and shape of large organic moleculars like proteins and cellulose.

6. In any layer of liquid, the pulling of lower layers backwards while upper layers forward direction is known as laminar flow.

Variation of Viscosity

1. The viscosity of liquids decreases with increase in temperature

ηt = \(\frac{\eta_{0}}{\left(1+\alpha t+\beta t^{2}\right)}\)

where,
η0 and ηt are coefficient of viscosities at 0°C and t°C, α and β are constants.

2. The viscosity of gases increases with increase in temperatures as

η ∝ \(\sqrt{T}\)

3. The viscosity of liquids increases with increase in pressure but the viscosity of water decreases with increase in pressure.

4. The viscosity of gases increases with increase of temperature because when temperature of gas increases, then rate of diffusion increases.

Hydrodynamics:
In physics, hydrodynamics of fluid dynamics explains the mechanism of fluid such as flow of liquids and gases. It has a wide range of applications such as evaluating forces and momentum on aircraft, prediction of weather, etc.

Flow of liquid Reynold’s Number
Equation of Continuity Energy of a Liquid
Bernoulli’s Principle Venturimeter
Torricelli’s Theorem Viscosity
Poiseuille’s Law Rate of Flow of Liquid
Stoke’s Law and Terminal Velocity Critical Velocity

Reynold’s Number | Definition, Formula – Hydrodynamics

Reynold’s Number Definition:
Reynold’s number is a pure number. It is equal to the ratio of the inertial force per unit area to the viscous force per unit area for a flowing fluid.

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Reynold’s Number | Definition, Formula – Hydrodynamics

Reynold’s Number Formula:
Reynold number,

K = \(\frac{\text { Inertial force }}{\text { Force of viscosity }}=\frac{v_{c} \rho r}{\eta}\)

where, vc = critical velocity.

For pure water flowing in a cylindrical pipe, K is about 1000.

  • When 0 < K < 2000, the flow of liquid is streamlined.
  • When 2000 < K < 3000, the flow of liquid is variable between streamlined and turbulent.
  • When K > 3000, the flow of liquid is turbulent.

It has no unit and dimension.

Hydrodynamics:
In physics, hydrodynamics of fluid dynamics explains the mechanism of fluid such as flow of liquids and gases. It has a wide range of applications such as evaluating forces and momentum on aircraft, prediction of weather, etc.

Flow of liquid Reynold’s Number
Equation of Continuity Energy of a Liquid
Bernoulli’s Principle Venturimeter
Torricelli’s Theorem Viscosity
Poiseuille’s Law Rate of Flow of Liquid
Stoke’s Law and Terminal Velocity Critical Velocity

Bernoulli’s Principle & Equation | Applications, Definition – Hydrodynamics

Bernoulli’s Principle Definition:
If an ideal liquid is flowing in streamlined flow, then total energy, i.e. sum of pressure energy, kinetic energy and potential energy per unit volume of the liquid remains constant at every cross-section of the tube.

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Bernoulli’s Principle & Equation | Applications, Definition – Hydrodynamics

Bernoulli’s Principle Formula:
Mathematically,

p + \(\frac{1}{2}\) ρv² + ρgh = constant

It can be expressed as, \(\frac{p}{\rho g}+\frac{v^{2}}{2 g}\) + h = constant
where,
\(\frac{p}{\rho g}\) = pressure head,
\(\frac{v^{2}}{2 g}\) = velocity head,
and h = gravitational head or potential head.

For horizontal flow of liquid, p + \(\frac{1}{2}\) ρv² = constant
where,
p is called static pressure and \(\frac{1}{2}\) ρv² is called dynamic pressure.

  • Therefore in horizontal flow of liquid, if p increases, v decreases and vice-versa.
  • This theorem is applicable to ideal liquid, i.e. a liquid which is non-viscous incompressible and irrotational.

Applications of Bernoulli’s Theorem:

  1. The action of carburetor, paintgun, scent sprayer, atomiser and insect sprayer is based on Bernoulli’s theorem.
  2. The action of Bunsen’s burner, gas burner, oil stove and exhaust pump is also based on Bernoulli’s theorem.
  3. Motion of a spinning ball (Magnus effect) is based on Bernoulli’s theorem.
  4. Blowing of roofs by wind storms, attraction between two closely parallel moving boats, fluttering of a flag, etc are also based on Bernoulli’s theorem.
  5. Bernoulli’s theorem helps in explaining blood flow in artery.
  6. The working of an aeroplane is based on Bernoulli’s theorem.

Hydrodynamics:
In physics, hydrodynamics of fluid dynamics explains the mechanism of fluid such as flow of liquids and gases. It has a wide range of applications such as evaluating forces and momentum on aircraft, prediction of weather, etc.

Flow of liquid Reynold’s Number
Equation of Continuity Energy of a Liquid
Bernoulli’s Principle Venturimeter
Torricelli’s Theorem Viscosity
Poiseuille’s Law Rate of Flow of Liquid
Stoke’s Law and Terminal Velocity Critical Velocity

What is Pascal’s Law? | Definition, Formula, Examples – Hydrostatics

Pascal’s Law Definition:
The increase in pressure at a point in the enclosed liquid in equilibrium is transmitted equally in all directions in liquid and to the walls of the container.

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What is Pascal’s Law? | Definition, Formula, Examples – Hydrostatics

The working of hydraulic lift and hydraulic brakes are based on Pascal’s law.

Pascal’s Law Formula:

F = PA

Where,
F is the force applied
P is the pressure transmitted
A is the cross-sectional area

Pascal’s Law Examples:
One of the most common examples of Pascal’s Law is the hydraulic braking system present in the automobiles. Every time you see a car come to a halt.

Pascal’s law finds numerous examples in our daily life such as

  • Automobiles.
  • Hydraulic brake system.
  • Hydraulic jack.
  • Hydraulic press.
  • Hydraulic machines.

Hydrostatics:
Hydrostatics is a property of liquid or fluid in mechanics. A fluid is a material which flows at room temperature, because its upper molecule overlaps the inner molecule, which tends to flow the liquid in forward direction. In hydrostatics, we will learn about the condition of fluids when it is in rest or exerted by an external force. Here we will study the fluids in motion.

Properties of Fluids Thrust
Pressure Pressure Exerted by Liquid
Buoyant Force Pascal’s Law
Archimedes’ Principle Law of Floatation
Density Relative Density
Density of a Mixture

What is Buoyancy in Physics? | Definition, Formula, Examples, Types – Hydrostatics

Buoyancy Definition Physics:
When a body is partially or fully immersed in a fluid, an upward force acts on it, which is called buoyant force, the phenomena is called buoyancy.

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What is Buoyancy in Physics? | Definition, Formula, Examples, Types – Hydrostatics

The buoyant force acts at the centre of gravity of the liquid displaced by the immersed part of the body and this point is called the centre of buoyancy.

The Magnitude of Buoyant Force Formula:
The magnitude of buoyant force, F = uρg.

Buoyancy is caused by the differences in pressure acting on opposite sides of an object immersed in a static fluid. It is also known as the buoyant force.

3 Types Buoyancy:
The three types of buoyancy are,

  • Positive buoyancy
  • Negative buoyancy
  • Neutral buoyancy.

Buoyancy Examples in Real Life

  • A boat or a ship floating in water is an example of buoyancy.
  • Immersing a cork in water will result in floating of it on the surface and this is because of the buoyancy.
  • An example of buoyancy is when salt water has the ability to help things float.
  • An example of buoyancy is a cheerful person who is grateful for her job and who is generally happy and optimistic.

Hydrostatics:
Hydrostatics is a property of liquid or fluid in mechanics. A fluid is a material which flows at room temperature, because its upper molecule overlaps the inner molecule, which tends to flow the liquid in forward direction. In hydrostatics, we will learn about the condition of fluids when it is in rest or exerted by an external force. Here we will study the fluids in motion.

Properties of Fluids Thrust
Pressure Pressure Exerted by Liquid
Buoyant Force Pascal’s Law
Archimedes’ Principle Law of Floatation
Density Relative Density
Density of a Mixture