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Fluid Dynamics¶

Fluids¶

Liquids and gases can deform their shape and can flow and are thereby classified as fluids.
The study of fluids at rest is called hydrostatics.
Fluid dynamics deals with fluids in motion.

Density¶

The density of a substance is defined as its mass per unit volume. $$ \text{Density}(\rho)=\frac{\text{Mass}(M)}{\text{Volume}(V)} $$
In $M.K.S.$ system the unit of density is $kgm^{-3}$.

Pressure¶

Pressure at a point due to fluid is the normal force exerted by the fluid on unit area containing the point. $$ \text{Pressure}(P)=\frac{\text{Force}(F)}{\text{Area}(A)} $$
Pressure is a scalar quantity. Pressure acts normal to a surface and it is always compressive in nature, therefore, only its magnitude is required for its complete description.
The SI unit of pressure is $Nm^{-2}$ and is also called $Pascal (Pa)$.
The other common units of pressure are the $atmospheric$ and $bar$. $$ 1 atm = 1.013 \times 10^{5}Pa \Rightarrow 1 bar = 10^{5}Pa $$

Thrust¶

The normal force exerted by a fluid on any surface in contact with it is called thrust. $$ \text{Thrust}=\text{Pressure}\times\text{Area} $$

Pascal's Law¶

The pressure exerted by a liquid at a point is the same in all directions. This principle is used in a hydraulic jack or a lift where a heavy load can be lifted up by a small force. $$ \frac{F_1}{F_2}=\frac{A_1}{A_2} $$

Archimedes Principle¶

A body that is partially or entirely submerged in a fluid, feels an upward force equal in magnitude to the weight of the displaced fluid.

Buoyancy:¶

The upward thrust which any liquid or gas exerts upon a body partly or fully submerged in it, is called its buoyancy.
The point at which the buoyancy of a liquid acts on a body is called the centre of buoyancy or the centre of floatation. The magnitude of this upward buoyant force (B) is given by Archimedes principle. i.e. $B = V\rho g$

Laws of Floatation:¶

When $w > w'$, i.e., the weight of the body is greater than the weight of the liquid displaced by it, the body sinks.
When $w = w'$, i.e. when the weight of the body is equal to the weight of the liquid displaced by it, the body floats being wholly immersed anywhere in the liquid.
When $w < w$, i.e., then the weight of the body is less than the weight of liquid displaced by it, the body floats on the surface of the liquid being partly immersed in the liquid.

Viscous Drag and Stokes Law¶

Viscosity is the measure of the force required to slide one layer of a liquid over another.
SI unit of coefficient of viscosity is $Kgm^{-1}s^{-1}$ and dimension $ML^{-1}T^{-1}$.
Substance which flows easily like water, etc. has lesser coefficient of viscosity.
The force experienced by an object while moving in a viscous medium is called drag force.
The drag force in a medium depends on the profile of the object and viscosity of the same object and nature of medium. Mathematically, drag force can be related with above stated factors using Stoke's law $$ F_D=6\pi\eta rv $$ The Stoke's law holds well at low speeds only for spherical bodies.

Terminal Velocity¶

Terminal Velocity Graph

When a spherical object falling gains a constant speed in a medium, then, the net force acting on it is zero and the corresponding speed is called as terminal velocity as

\[ v_t=\left(\frac{g}{6\pi\eta r}\right)m\space\text{OR}\space v_t=\left(\frac{2\rho g}{9\eta}\right)r^2 \]

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When an object is moving in a fluid at considerably higher speed, then the drag force is no more proportional to the speed.

Graph of Terminal Velocity¶

Fluid Flow¶

In a fluid, if every particle that passes a particular point, moves along exactly the same path as followed by particles which passed that point earlier then flow is called streamline or laminar.
Irregular or unsteady flow of the fluid is called turbulent flow.

Characteristics of Ideal Fluid¶

It is non viscous i.e. $\eta = 0$.
It is incompressible $\rho = \text{constant}$.
Its motion is steady.

Equation of Continuity¶

This equation is obtained by using law of conservation of mass of flowing of ideal fluid, and is written as: $$ \rho_1 A_1 v_1=\rho_2 A_2 v_2\space OR\space\rho Av=\text{constant}=\frac{\text{Mass}}{\text{Volume}}= \text{mass flow rate of fluid} $$

\[ A_1 v_1=A_2 v_2\space OR\space Av=\text{constant}=\frac{\text{Volume}}{\text{Time}}=\text{volume flow rate of fluid} \]

Equation of Continuity Figure

The flow of ideal fluid is assumed to be streamline flow.

Bernoulli's Equation¶

Bernoulli's equation is derived from conservation of mechanical energy.

Bernoulli's equation is $P+\frac{1}{2}\rho v_2 +\rho gh = \text{constant}.$
Fluid have three types of energies:
Potential energy = $mgh$ or potential energy per unit volume = $\rho gh$
Kinetic energy =$\frac{1}{2}mv^{2}$ or kinetic energy per unit volume =$\frac{1}{2}\rho v^{2}.$
Pressure energy = $PV$ or pressure energy per unit volume = $P$

Applications¶

Speed of Efflux¶

Speed of efflux is determined by Torricelli's theorem $v=\sqrt{2g(h_1 -h_2)}$ where $(h_1 -h_2)$ is height of fluid level from the orifice. This theorem explains streamline flow of ideal fluid under the action of gravity only.
Speed of efflux depends upon the height through which fluid falls under the action of gravity.
Pressure of fluid increases if the speed of fluid decreases and vice versa, according to the relation $P+\frac{1}{2}\rho v^{2}=\text{constant}.$
An aero plane lifts due to the difference of pressure of air on its wings.
Swing in cricket and tennis ball is also due to difference of pressure on its two sides.
There is a danger that a person standing near a fast moving train to fall towards it.
Venturimeter is a device used to measure the speed of liquid flow.
Venturimeter works according to venturi relation which is $P_1 -P_2 =\frac{1}{2}\rho v^{2}_2$ .
Mixing of petrol with air in carburetor is according to the Bernoulli's principle.