Overview: Mechanical Properties of Fluids

Any substance that can flow is a fluid.

• They do not oppose deformation; they get permanently deformed.
• They have the ability to flow.
• They can take the shape of the container. 

The branch of physics which deals with the properties of fluids at rest is called hydrostatics.

The normal force (F) exerted by a fluid at rest per unit surface area (A) of contact is called the pressure (P) of the fluid.

P = \[\frac {F}{A}\]

SI unit: Pascal
Dimension: [L^-1M¹T²]

Atmospheric pressure is the pressure exerted by the weight of the column of air above a unit area.

p = p₀ + pgh

Where:

• p_0​ = Atmospheric pressure
• ρ = Density of liquid
• g = Acceleration due to gravity
• h = Depth

Pascal's law states that the pressure applied at any point of an enclosed fluid at rest is transmitted equally and undiminished to every point of the fluid and also on the walls of the container, provided the effect of gravity is neglected. 

• Pascal’s law states that pressure applied at any point of an enclosed fluid at rest is transmitted equally and undiminished to every part of the fluid and to the walls of the container.
• It applies only to fluids at rest (static conditions).
• The transmitted pressure acts equally in all directions.
• Mathematically, pressure remains constant throughout the fluid:
  \[\frac {F_1}{A_1}\] = \[\frac {F_2}{A_2}\]​​
• It is the working principle of hydraulic devices such as hydraulic lifts and hydraulic brakes.

Any two mo.lecules attract each other. This force between molecules is called intermolecular force.

The force of attraction between the molecules of the same substance is called cohesive force or force of cohesion.

The force of attraction between the molecules of different substances is called the adhesive force or force of adhesion.

The maximum distance from a molecule up to which the molecular force is effective is called the range of molecular force.

An imaginary sphere with a molecule at its center and radius equal to the molecular range is called the sphere of influence of the molecule.

The surface layer of a liquid with thickness equal to the range of intermolecular force is called the surface film.

Surface tension T is defined as the tangential force acting per unit length on both sides of an imaginary line drawn· on the free surface of liquid.

Mathematically. T = \[\frac {F}{L}\]
Sl unit: N/m
Dimension:  [L⁰M¹T^-2]

The extra energy of the molecules in the surface layer is called the surface energy of the liquid.

The angle of contact is the angle between the tangent drawn to the free surface of the liquid and the solid surface at the point of contact, measured within the liquid.

• Soluble impurities, such as common salt, increase the surface tension of water, whereas detergents and phenol decrease it.
• Insoluble impurities reduce surface tension by decreasing cohesive forces, affecting droplet shape and meniscus formation.
• In most liquids, surface tension decreases with increasing temperature and vanishes at the critical temperature.

• For a plane surface, pressure just below the surface equals atmospheric pressure.
• For a convex surface, the pressure inside the liquid is greater than outside.
• For a concave surface, pressure inside the liquid is less than outside.

The phenomenon of rise or fall of a liquid in a capillary tube when dipped in the liquid is called capillarity.

h = \[\frac {2T cosθ}{rρg}\]

• Capillary rise occurs when liquid wets the tube (e.g., water in a glass).
• Capillary fall occurs when a liquid does not wet the tube (e.g., mercury in a glass tube).
• Rise or fall depends on surface tension and angle of contact.
• The narrower the tube, the greater the rise or fall.
• Capillarity plays an important role in natural processes like water rising in plants and oil rising in a lamp wick.

The branch of Physics which deals with the study of properties of fluids in motion is called hydrodynamics.

• In steady flow, properties such as pressure and velocity at a point remain constant over time.
• A flow line is the path followed by a particle in a moving fluid.
• A streamline is a curve whose tangent at any point gives the direction of fluid velocity.
• A flow tube is a bundle of streamlines; fluid does not cross its boundaries in steady flow.
• Laminar flow is smooth and orderly, while turbulent flow is irregular and chaotic.

The velocity beyond which a streamline flow becomes turbulent is called the critical velocity.

\[\mathbf{v}_{\mathrm{c}}=\frac{R_{\mathrm{n}}\eta}{\rho d}\]

where,
v_c = critical velocity of the fluid
R_n= Reynolds number
η = coefficient of viscosity
ρ = density of fluid
d = diameter of tube 

Viscosity is that property of fluid, by virtue of which, the relative motion between different layers of a fluid experience a dragging force.
SI unit = N s /m²

The rote of change of veiocity (dv) with distance (dx) measured from a stationary layer is coiled velocity gradient (dv/dx).

The coefficient of viscosity can be defined as the viscous force per unit area per unit velocity gradient.

The law states that, "The viscous force (F_v) acting on a small sphere falling through a viscous medium is directly proportional to the radius of the sphere (r), its velocity (v) through the fluid, and the coefficient of viscosity (η) of the fluid". 

\[\eta=\frac{2}{9}\frac{r^{2}\left(\rho-\sigma\right)g}{\mathrm{v}}\]

The continuity equation soys that the volume rote of flow of on incompressible fluid for a steady flow is the some throughout the flow.
A₁v₁ = A₂v₂ or, Av = constant

This is Bernoulli's equation. It states that the work done per unit volume of a fluid by the surrounding fluid is equal to the sum of the changes in kinetic and potential energies per unit volume that occur during the flow.

Mathematically.
p + \[\frac {1}{2}\]ρv² + ρgh = constant

• Speed of efflux: Liquid flowing out of a hole at depth hhh moves at the same speed as a body falling freely through a height hhh.
• Venturimeter: When a fluid passes through a narrow section, its speed increases and pressure decreases.
• Aeroplane lift: Faster airflow above the wings creates lower pressure, producing an upward lift force.
• Atomiser: High-speed air creates low pressure, causing liquid to rise and spray as fine droplets.
• Storm effect on roofs: Fast wind over a roof lowers pressure above it, and higher pressure below can lift the roof.