Fluid Mechanics: Fluids are a special category of matter which allows the constituent atoms or molecules of it to move. In simpler words, a fluid is a type of matter which can flow. Generally, these are either a gas or a liquid. In this article, we will learn more about fluid and their behaviour. We will discuss various properties and phenomena such as pressure, Buoyancy, flotation, Bernoulli’s theorem, viscosity, surface tension.

In this article, we have provided information regarding the properties of fluid, laws and theorems related to fluid mechanics. Students will also find sample questions to practice the concepts. Students can access practice questions and mock tests based on Fluid Mechanics on the Embibe app. Continue reading to know more about Fluid Mechanics.

Fluid Mechanics: Fluid Statics and Dynamics

When we discuss the properties of the fluid, when it is at rest with respect to the container it is kept in, it comes under fluid statics. Example: variation of pressure with depth, Pascal’s law, buoyancy.

When we discuss the properties of the fluid, when it is in motion with respect to the container it is kept. Example: viscosity, capillary action, Stoke’s law, Bernoulli’s theorem, etc.

Pressure and Pascal’s Law

Pressure is defined as thrust per unit area or force per unit area in the direction perpendicular to the area. Its SI unit is Pascals. 

The force from the direction perpendicular is also known as Thrust. Thus, pressure can also be defined as thrust per unit area.

\(P=\frac{T}{A}\)

Where,

\(T\) is Thrust

\(A\) is the area

Fig. 1

Fig 1: The pressure is applied at every point on the surface of the balloon, therefore the balloon is stretched.

Pascal’s law for fluid pressure states that any change in pressure at a particular point is transmitted at all points in the fluid, and the magnitude of the pressure change is equal to that change in pressure at the initial point.

Density and Relative Density

Density is a physical quantity that gives the amount of mass contained per unit volume of the material. That is, mass per unit volume of a body is known as the density of the body.

Relative density is the ratio of the density of a body to the density of given reference material.

Archimedes’ Principle and Buoyancy

When a body is wholly or partially submerged in a fluid, it displaces some amount of water, and the effective weight of the body decreases due to an upward force is known as buoyant force. Archimedes’ principle gives the magnitude of the buoyant force.

According to Archimedes’ principle in Fluid Mechanics, the buoyant force is equal to the weight of the fluid displaced by the body.

Fig. 2

In Fig. 2, the weight of the immersed body decreases due to the buoyant force, which is equal to the weight of the water dispersed.

Thus, the buoyant force is given by,

\(F=\rho_{F} V g\)

Where,

\(\rho_{F}\) is the density of the fluid.

\(V\) is the volume of water displaced.

\(g\) is the acceleration due to gravity.

Law of Floatation

Laws of floatation states that if the density of the submerged body is less than the density of the liquid, the body will float. Suppose the density of the material of the body is equal to the density of the liquid. In that case, the body floats fully submerged in liquid, but if the density of the body is more than the density of the liquid, the body will sink.

If \(W\) is the weight of the body and w is the buoyant force, then

1. If \(W>w\), then the body will sink to the bottom of the liquid.
2. If \(W<w\), then the body will float partially submerged in the liquid.
3. If \(W=w\), then the body will float in liquid if its whole volume is just immersed in the liquid.

Streamline flow

In the case of fluids, a streamline flow is one in which the fluids flow in separate layers with no mixing or disruption occurring between the layers at any point. In a streamlined flow, the velocity of each fluid particle remains constant over time.
Because of the lack of turbulent velocity fluctuations, the fluid will flow without any lateral mixing at low fluid velocities. The fluid particles tend to follow a specific order in which the movement or motion of fluid particles is parallel to the pipe wall. The movement occurs in such a way that the adjacent layers of fluid slide smoothly past each other.

Bernoulli’s Theorem

Bernoulli’s theorem is an extension of the law of conservation of energy for a fluid in motion. According to this law, the sum of kinetic energy, potential energy, and pressure energy remains constant for a fluid flowing in streamline.

Fig. 3

\(\frac{1}{2} \rho v^{2}+\rho g h+P= \text {constant}\)

Where,

\(\rho\) is the density of the fluid.

\(v\) is the velocity of the fluid.

\(h\) is the height of the fluid.

Viscosity and Viscous Force

Viscosity is defined as the opposition offered by the fluid to any relative motion between the different layers of a fluid or with the container. It is similar to friction which opposes the relative motion between the different objects in contact. The viscous force is given by,

Fig. 4

\(F=\mu A \frac{v_{\max }}{\Delta h}\)

Where,

\(\mu\) is the viscosity of the fluid.

\(A\) is the area under consideration.

\(v_{\max }\) is the velocity of the top layer under consideration.

\(\Delta h\) is the height of the fluid.

\(\frac{v_{\max }}{\Delta h}\) is the velocity gradient of the fluid.

Stoke’s Law and Terminal velocity

Stoke’s law defines the viscous force experienced by a spherical body when it moves in a fluid.

Fig. 5

In Fig. 5, the viscous force experienced by the spherical body is given by,

\(F=6 \pi r \eta v\)

Where,

\(r\) is the radius of the spherical ball.

\(\eta\) is the coefficient of viscosity.

\(v\) is the velocity of the ball.

From the above equation, we can say that the viscous force on the spherical ball will increase with velocity. Therefore at a particular velocity, the net force on the sphere will be zero, and that particular velocity is known as Terminal velocity. Terminal velocity of the sphere is given by,

\(V_{\text {term }}=\frac{2 r^{2} g\left(\rho_{s}-\rho_{f}\right)}{9 \eta}\)

Where,

\(\rho_{f}\) density of the fluid.

\(\rho_{s}\) density of the sphere.

\(g\) is the acceleration due to gravity.

\(r\) is the radius sphere.

\(\eta\) is the coefficient of viscosity.

Poiseuille’s Equation

Hagen-Poiseuille’s equation or Poiseuille’s equation gives the pressure difference between the two ends of a cylindrical tube when a Newtonian fluid is flowing through it.

\(\Delta P=\frac{8 \pi \mu L Q}{A^{2}}\)

Where,

\(A\) is the area of the tube.

\(\mu\) is the coefficient of viscosity.

\(L\) is the distance between the two ends of the pipe.

\(Q\) is the volumetric flow rate of the fluid.

Reynold’s Number

Reynolds number helps us to predict the type of flow of a liquid (laminar or turbulent). It is given by,

\(R_{e}=\frac{{ \rho vd }}{\eta}\)

Where,

\(\rho\) is the density of the liquid.

\(v\) is the velocity of the liquid.

\(d\) is the diameter of the pipe.

\(\eta\) is the coefficient of viscosity of the liquid.

At low Reynolds number, the fluid flow is primarily laminar whereas, at higher Reynolds number, the flow is dominated by turbulent flow.

Surface Tension

The molecules of a matter attract each other with a force known as a cohesive force. 

Fig. 6

For a molecule in the bulk of the fluid, this force is balanced as there are molecules present in every direction. But for the molecules present in the top layer of the fluid, the net force is in a downward direction, due to which the top layer behaves like a stretched membrane. The surface tension has a unit of Newton per meter.

Fig. 7

Surface tension exerts force linearly; that is, if a thin, light twig is kept on the surface and is floating without getting immersed, then the surface tension exerts the force along the length of the twig.

Due to surface tension, the water tends to take the shape of the sphere to minimize the surface area. The excess pressure inside a bubble and water droplets is also due to the surface tension of the liquid.

The excess pressure inside the bubble is given by,

\(\frac{4S}{r}\)

Excess pressure inside a droplet is given by,

\(\frac{2S}{r}\)

Where,

\(S\) is the surface tension of the surface.

\(r\) is the radius of the bubble or sphere.

Surface tension is the property of the surface, and for a bubble, the number of the surface is twice the droplet.

Angle of Contact

The force between the molecule of two different materials is known as adhesive force. When the water is kept in the container, the molecule near the wall of the container experience both adhesive and cohesive force, and when these forces are unbalanced, then the surface near the walls of the container rises of dips depending on the adhesive force.

So, the angle made by the wall and the tangent to the surface at the point of contact to the walls of the container is known as the angle of contact.

Fig. 8

In Fig. 8, we can see that the contact angle can be less than \(90^{\circ}\), equal to \(90^{\circ}\) or greater than \(90^{\circ}\).

If the diameter of the container is sufficiently small, then the surface becomes spherical. Due to this, the pressure just above the surface change, and therefore, to compensate for this change, the water rises in the capillary tube. This phenomenon is known as capillary action.

The height of the capillary rise is given by,

\(h=\frac{2 T \cos (\theta)}{\rho g r}\)

Where,

\(T\) is the surface tension.

\(\theta\) is the contact angle.

\(\rho\) is the density of the liquid.

\(g\) is the acceleration due to gravity.

\(r\) is the radius of the capillary tube.

Thus, the height to which a liquid will rise in a capillary tube is directly proportional to the surface tension of the liquid.

Sample Questions on Fluid Mechanics

Q.1. How does the waterproofing agent work?
Ans:
The molecular force of attraction between the same type of molecules is known as a cohesive force, while the force of attraction between the two molecules of the different substances is called adhesive force.
For an object to become wet, it must retain water molecules when it comes in contact with the water. For this to happen, the force between the object and the molecules of water must be more than that of the force between the two water molecules.
Thus, for wetting, the adhesive force should be less than the cohesive force, and for the given condition, the contact angle is acute.
Therefore, a waterproofing agent will change the contact angle from acute to obtuse.

Q.2. Two soap bubbles have radii in the ratio of 2:1, then the ratio of the excess pressures inside them is
Ans: Given,
The ratio of the radii of the two soap bubbles is \(2:1\),
Let the radius of the first soap bubble be \(R_{1}\) then the radius of the second soap bubble will be \(R_{2}=\frac{R_{1}}{2}\).
We know that the excess pressure inside a soap bubble is given by,
\(\Delta P=\frac{4 T}{R}\)
Where the \(T\) is the surface tension, and the \(R\) is the radius of the bubble.
Thus, the excess pressure is inversely proportional to the radius,
\(\frac{\Delta P_{1}}{\Delta P_{2}}=\frac{R_{2}}{R_{1}}=\frac{1}{2}\)

Fluid Mechanics Summary

In this article, we briefly discussed the different mechanical properties such as pressure, pascal’s law, Archimedes principle, Buoyancy, etc.
Fluid mechanics is divided into two categories, namely fluid statics, and fluid dynamics. In fluid statics, the fluid is at rest with respect to the walls of the container, whereas in fluid dynamics, the liquid can be at rest or in motion with respect to the walls of the container.
Capillary action, excess pressure inside a bubble and droplet occurs due to surface tension which is due to the cohesive and adhesive forces between the molecules.
Bernoulli’s equation is an extension of the conservation of energy for fluids. 
Stoke’s law and terminal velocity are based on viscosity and viscous force, which is similar to friction and frictional force.

FAQs on Fluid Mechanics

Here are some of the most frequently asked questions on Fluid Mechanics:

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