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Fluid Mechanics is a branch of Physics that studies the behavior of fluid (liquids, gases, blood, and plasmas) at rest and in motion. Fluid mechanics has a wide range of applications in chemical engineering, mechanical, biological systems, and astrophysics. Fluid mechanics help us figure out how fluids behave under different forces and in different atmospheric conditions. Breathing and blood flow, oceanography, hydrology, and energy generation are examples of fluid mechanics. Fans, turbines, pumps, missiles, and aircraft are more technical uses. Fluid mechanics deals with three aspects of the fluid namely fluid statics, fluid kinematics, and fluid dynamics.
Read More: NCERT Solutions For Class 11 Physics Mechanical Properties of Fluids
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Key Terms: Fluid Mechanics, Fluid Mechanics Formula, Flow, Turbulent Flow, Fluids, Fluid Kinematics, Pressure, Density, Viscosity
What is Fluid Mechanics?
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Fluid Mechanics is the branch of engineering science that looks at how fluids behave both when they are moving and when they are at rest. Civil Engineering, Mechanical Engineering, and Chemical Engineering all study the concept of fluid mechanics in detail. Fluid is the word used for something that moves. Fluids are all substances that are liquid or gas.
- Fluid mechanics is one of the four parts of mechanics.
- The other three are quantum mechanics, relative mechanics, and elastic body mechanics.
- There are three different classes in fluid mechanics. These are fluid statics, fluid kinematics, and fluid dynamics.
- From the statics and dynamics, a compressible flow and an incompressible flow can be made.
- These are further categorized into turbulent and laminar forms.
- Fluid mechanics looks at three aspects of fluids: how they stay still, how they move, and how they change over time.
Read More:
| Relevant Concepts | ||
|---|---|---|
| Mechanical Properties of Fluids | Froude Number Formula | Pascal's Law and Its Applications |
| Continuity Equation | Mechanical Properties of Fluids Formula | Hydraulic Lift |
Laws of Fluid Mechanics
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Fluid mechanics is based on five laws of physics, which are listed below:
- The Law of Conservation of Linear Momentum
- The Law of Conservation of Angular momentum
- The Law of Conservation of Energy
- The Law of Conservation of Mass
- The Law of Conservation of Thermodynamics
In fluid mechanics, if a liquid is moving at a speed that is not too fast, it usually moves in layers with speeds that change gradually. This is called streamlined flow. If the flow rate is too fast, though, there are a lot of bumps and the water doesn't move in layers. This is called turbulent flow. After the critical velocity, which is given by relation, the turbulence starts.
Where,
- Vc = Critical velocity
- k = Reynolds number
- η = coefficient of viscosity of the fluid
- σ = density of the fluid
- r = radius of the tube
When the fluid's speed is high enough that the Reynolds number is greater than 4000, the flow is said to be turbulent.
Fluid Mechanics
Read More: Conservation Laws in Physics
Applications of Fluid Mechanics
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Fluid Mechanics has a wide range of applications which are as follows:
- Water-driven power plants
- Hydraulic machines
- Automobiles
- Air conditioners and refrigerators
- Thermoelectricity
- Fluids as a Source of Clean Energy
- Heat Motors
Read More: Mechanical Properties of Fluids MCQ
Fluid Mechanics Formula
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Some important Fluid Mechanics Formula are as follows:
Density of a Sample at Constant Density
The density of a sample at constant density is given as:
Ρ = m/V
Where
- ρ: density of fluid
- m: Mass
- V: Volume
Pressure
The formula for pressure is
P = F/A
- P: Pressure
- F: Force applied
- A: Area affected
Pressure at a Depth in a Fluid of Constant Density
The pressure at a depth h in a fluid of constant density is calculated using the formula:
p = p0 +ρgh
- p: Pressure at height h
- p0: Pressure at zero height
- g: Acceleration due to gravity
- ρ: Fluid density
Read More: Mechanical Properties of Fluids Important Questions
Volume Flow Rate
The formula for volume flow rate is:
Q = dV/ dt
- Q: Flow Rate
- dV: Change in Volume
- dt: Time period.
Viscosity
The viscosity formula is given as
η = FL / vA
- η: Fluid viscosity
- F: Force
- L: Distance between the plates
- V: Constant velocity
- A: Area of the plate
Read More: Unit of Viscosity
Solved Examples on Fluid Mechanics Formula
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Example 1: Two pistons are spaced apart by 0.015 mm, and the viscous fluid passing through them exerts a force of 1.2 N per square meter to maintain the motion of the two plates at 35 cm/s. What will be the viscosity of the fluid between the plates? Use the formula for fluid mechanics.
Solution: Given parameters:
Distance between plates,
L = 0.015mm = 0.015×10−3m
Viscous force = F / A =1.2 N per square meter area
Speed (v) = 35 cm per sec = 0.35 m per sec.
Now, applying the formula,
η= FL / vA
= F/A ×L/v
=0.0514×10 −3 N sec per Sq m
Thus fluid viscosity is 0.0514×10−3 N sec per Sq m.
Example 2: A submarine experiences a pressure of 5.05 x106 Pa at a given depth of d1 in a sea. When it goes further to a depth of d2, it then experiences a pressure of 8.08 x 106 Pa. Then what would be d1 – d2 approximately? (density of water = 103ms-2 and acceleration due to gravity = 10 ms-2)
(A) 300 m
(B) 400 m
(C) 600 m
(D) 500 m
Solution: (A) 300 m
Explanation:
P1 = P0 + ρgd1
P2 = P0 + ρgd2
ΔP = P2 – P1 = ρgΔd
(8.08 x 106 – 5.05 x106) = 103 x 10 x Δd
3.03 x 106 = 103 x 10 x Δd
Δd = 303 m ≈ 300 m
Read More:
| Related Topics | ||
|---|---|---|
| Hydrostatic Pressure | Relation Between Viscosity and Density | Difference between Kinematic and Dynamic Viscosity |
| Mass Flow Rate Formula | Fluid Flow | Poiseuille's Law Formula |
Things to Remember
- A fluid is defined as a state of matter that yields to sideways or shearing forces such as liquids and gases.
- Density is the mass per unit volume of a substance or object, defined as ρ= m/V.
- Pressure is defined as the force per unit perpendicular area over which the force is applied, P= F/A.
- Pressure due to the weight of a liquid of constant density is given by p =ρgh, where p is the pressure, h is the depth, ρ is the density, and g is the acceleration due to gravity.
- Gauge pressure refers to the extra pressure above the atmospheric pressure.
- A fluid at rest obeys hydrostatic equilibrium where its pressure increases with depth to balance its weight.
Previous Years Questions (PYQs)
- With the addition of impurities surface tension of a liquid… (COMEDK UGET 2009)
- A liquid is filled up to a height of 20cm in a cylindrical vessel. The speed… (KEAM)
- An air bubble of radius 0.1cm is in a liquid having surface tension… (JEE Main 2014)
- A small spherical droplet of density d is floating exactly half… (JEE Main 2020)
- Pressure applied to an enclosed fluid is transmitted…
- If the diameter of a capillary tube is doubled, the rate of flow… (COMEDK UGET 2007)
- A soap bubble, having a radius of 1mm, is blown from a detergent solution… (NEET 2019)
- Viscosity decreases with an increase in temperature is the… (COMEDK UGET 2009)
- When the temperature increases, the viscosity of… (JKCET 2006)
- The viscosity of liquid decreases with rising temperature. Viscosity…
- Fluid is in streamlined flow across a horizontal pipe of variable… (NEET 2013)
- Consider a water jar of radius R that has water filled up to height… (JEE Main 2016)
Sample Questions
Ques. When a solid sphere with a radius of R falls through a viscous fluid with a viscosity coefficient of η, it gets a final speed of v1. The sphere is split into 27 pieces that are all the same size. If each of these falls through the same fluid with a final speed of v2, the ratio (v1/v2) equals (3 Marks)
(a) 1/9
(b) 27
(c) 9
(d) 1/ 27
Ans. (c) 9
Explanation: 27 x (4/3)πr3 = (4/3)πR3
Or r = R/3
Terminal velocity, v ∝ r3
Therefore, (v1/v2) = (R2/r2)
v1/v2= [R/(R/3)]2= 9
(v1/v2) = 9
Ques. Water flows into a large tank with a flat bottom at the rate of 10-4m3s-1, however, it is also leaking out of a hole of area 1 cm2 at its button. What will be its height if the height of the water in the tank remains steady? (3 Marks)
Ans. Given that the height of the water column is constant, so,
Water inflow rate (Qin) = Water outflow rate (Qout)
Qin = 10-4m3s-1
Qout = 10-4 x √(2gh)
10-4 = 10-4 x √20 xh
h = (1/20) m = 5 cm
Thus, the height is 5 cm.
Ques. List the properties of Fluids. (3 Marks)
Ans. The important properties of fluids are as follows:
- Kinematic Properties: Kinematic Properties help in understanding fluid motion. Velocity and acceleration are included in the kinematic properties of the fluids.
- Thermodynamic Properties: They help us in understanding the thermodynamic state of the fluid. Temperature, density, pressure, and specific enthalpy are included in the thermodynamic properties of fluids.
- Physical Properties: The physical properties help in understanding the physical state of the fluid such as color and odor.
Ques. What is Streamline Flow? (3 Marks)
Ans. A streamline refers to a curve the tangent to which at any point gives the direction of the fluid velocity at that point. Streamline is analogous to a line of force in an electric or magnetic field. In steady flow, the pattern of the streamline is stationary with time and thus, a streamline gives the actual path of a fluid particle.
A steady flow is therefore also known as a streamlined flow or laminar flow. It must be noted that no two streamlines can ever cross one another, for if they did, a fluid particle arriving at that point could go one way or the other and the flow won’t be steady.
Ques. A group of spheres with a radius R is falling through a viscous fluid with a velocity v. The slowing force of viscosity on the round ball is (3 Marks)
(a) directly related to R but oppositely related to v
(b) directly related to both its radius R and its speed v
(c) the opposite of both the radius R and the speed v
(d) inversely proportional to R but directly proportional to v
Ans. (b) directly related to both its radius R and its speed v
Explanation: Force that slows down flow = 6πηRv
Thus, (b) is the best choice. It is directly related to both the radius R and the speed v
Ques. A capillary tube that is 20 cm long is put in water. Up to 8 cm of water comes up. If the whole thing is put in an elevator and allowed to fall freely, the height of the water column in the capillary tube will be (3 Marks)
(a) 4 cm
(b) 10 cm
(c) 8 cm
(d) 20 cm
Ans. (d) 20 cm
Explanation: In a free-falling elevator, g = 0. This means that water will rise to the full length of the tube, which is 20 cm.
Ques. Explain how viscosity affects surface tension. (3 Marks)
Ans. Surface tension is defined as a contractive tendency of a liquid’s surface, that allows it to withstand external force. Viscosity increases with an increase in temperature however the surface tension decreases. The finer the liquid would be, the greater would be the viscosity. Thus, both the viscosity and surface tension in a liquid diminishes.
Ques. Half of a long, cylindrical container is filled with liquid. When the vessel is turned around its own vertical axis, the liquid rises up near the wall. If the vessel's diameter is 5 cm and it turns 2 times per second, then the height difference between the middle and the sides, in cm, will be (3 Marks)
(a) 1.2
(b) 0.4
(c) 0.1
(d) 2.0
Ans. (d) 2.0
Explanation: The Liquid’s linear speed at the sides is rω. So, the difference in height is given as follows
2gh = ω2r2
h = ω2r2/2g
here ω = 2πf
Therefore, h = [(2 x 2π)2 X (52 x 10-2 X2/(2×10)) = 2 cm
Ques. Distinguish between laminar flow and turbulent flow. (3 Marks)
Ans. The difference between laminar flow and turbulent flow is as follows:
| Laminar Flow | Turbulent Flow |
|---|---|
| Laminar flow also called streamline flow, is when a fluid moves through a pipe or tube in layers that don't touch each other. | Turbulent flow is a flow regime in which the properties change in a random way. |
| When the speed of the fluid is low, the layers tend to move past each other like playing cards, without mixing. | This includes pressure and flow speeds that change quickly in space and time. |
| There are no eddies or swirls of fluid that go in the opposite direction of the flow. In laminar flow, all of the fluid's particles move in straight lines parallel to the pipe walls. | In contrast to laminar flow, the fluid no longer moves in layers, and it mixes very well across the tube. Flows with Reynolds numbers above 4000 are usually, but not always, turbulent, while flows with Reynolds numbers below 2300 are usually smooth. |
Ques. The top of a water tank is open and its water level is maintained. The tank is giving out 0.74 m3 water per minute through a circular opening of a 2 cm radius in its wall. The depth of the center of the opening from the level of water in the tank is close to which one of the following? (3 Marks)
(a) 6.0 m
(b) 4.8 m
(c) 9.6 m
(d) 2.9 m
Ans. (b) 4.8 m
Explanation: It is given that, volumetric flow rate = (0.74/60) = πr2v = (π x 4 x 10-4) x √2gh
√2gh =[ (74 x 100)/240π)]
√2gh= 740/24π
2gh = (740/24π)2
h = [(740 x 740)/24 x 24 x 10)] (as π2 =10)
h = 4.8 m
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