Fluid Mechanics- A Detailed Notes

Density:

• NCERT Reference: Class 11, Chapter 11: Properties of Matter

Definition: Density is a measure of how closely packed the particles of a substance are together. It is defined as the mass per unit volume. Units: (kg/m^3) Variations: Density varies with temperature and pressure. Generally, density increases with decreasing temperature and increasing pressure. Applications:

• Archimedes’ Principle: Buoyancy force upthrust on an object immersed in a fluid is equal to the weight of the fluid displaced by the object. Ships float on water due to this principle.
• Equilibrium of fluids: When the net force acting on an object immersed in a fluid is zero, the object is said to be in equilibrium.

Viscosity:

• NCERT Reference: Class 11, Chapter 11: Properties of Matter

Definition: Viscosity refers to the resistance of a fluid to flow. It is a measure of the fluid’s internal friction. Units: (Pa \cdot s) (pascal-seconds) or (N \cdot s/m^2) Newton’s Law of Viscosity: The viscous force acting between two parallel layers of fluids is proportional to the area of contact between the layers, the velocity gradient between them, and the coefficient of viscosity of the fluid. Applications:

• Poiseuille’s law describes the flow rate of a liquid through a capillary tube under laminar conditions.
• Stokes’ law explains the motion of a spherical object falling through a viscous fluid.

Surface Tension:

• NCERT Reference: Class 11, Chapter 11: Properties of Matter

Definition: Surface tension is the property of a liquid surface that causes it to behave like a stretched elastic membrane. It is the force required to increase the surface area of a liquid. Units: (N/m) (newton per meter) or (J/m^2) (joule per square meter) Factors Affecting Surface Tension: Temperature and impurities influence the surface tension of liquids. Applications: Capillary action: Liquids rise or fall in narrow tubes due to surface tension, which is vital for plants’ water transport.

• Formation of droplets and bubbles.

Fluid Dynamics:

• NCERT Reference: Class 12, Chapter 13: Properties of Matter

Streamlines and Streamtubes: Streamlines are imaginary lines that show the direction of fluid flow at any instant, while streamtubes are bundles of streamlines. Equation of Continuity: It states that the mass of fluid entering a section of a pipe per unit time is equal to the mass of fluid leaving the section per unit time. Bernoulli’s Equation: This equation describes the conservation of energy in fluid flow and relates pressure, velocity, and height within a fluid.

• Applications include hydraulic systems such as dams, water supply systems, and car brakes. Venturimeter and Pitot Tube: Devices to measure fluid flow velocity based on Bernoulli’s principle.

Fluid Flow:

• NCERT Reference: Class 12, Chapter 13: Properties of Matter

Laminar and Turbulent Flow: Laminar flow occurs when fluid layers flow smoothly without any disturbances, while turbulent flow is chaotic and characterized by swirls and eddies. Reynolds Number: It is a dimensionless quantity that determines the type of flow in a pipe. Laminar Flow Equations: Hagen-Poiseuille equation gives the relationship between pressure drop, flow rate, fluid viscosity, and pipe radius for laminar flow in a cylindrical pipe. Turbulent Flow Equations: Darcy-Weisbach equation expresses the pressure drop in a pipe due to frictional losses for turbulent flow.

Fluid Statics:

• NCERT Reference: Class 11, Chapter 11: Properties of Matter

Pascal’s Law: Pressure applied to a confined fluid is transmitted equally to every point of the fluid. Hydraulic machines, such as hydraulic presses, utilize this principle. Hydrostatic Pressure: The pressure exerted by a fluid at rest due to gravity. It increases with depth. Buoyancy and Equilibrium of Fluids: Buoyancy force opposes the weight of an object immersed in a fluid. When the net force acting on the object is zero, it is in equilibrium. Metacentre and Stability of Floating Bodies: Metacentre is a point related to the stability of floating bodies. A floating body is stable if its metacentre is above its centre of gravity.

Applications:

Fluid mechanics has numerous applications across various fields, including:

• Hydraulic machines: Pumps, turbines, and hydraulic presses use fluid power.
• Fluid power systems: Used in vehicles, construction machinery, and industrial automation.
• Aerodynamics: Designing aircraft wings, rockets, and cars that efficiently move through the air.
• Oceanography: Studying ocean currents, waves, and underwater vehicles.