Stream-line and Turbulent Flow, Energy of a Liquid and Bernoulli's Theorem

Streamline and turbulent flow

When the flow of liquid is such that the velocity, v of every particle at any point of the fluid is constant then the flow is said to be steady or streamline flow. The path followed by a particle of the fluid in stream-line flow is called steady or streamline flow. The path followed by a particle of the fluid in stream-line flow is called steady or stream-line. It is a curve whose tangent at any point is in the direction of the liquid velocity at that point. The stream-lines never cross to each other in a stream-line flow of a liquid.

Laminar flow

If a liquid is flowing over a horizontal surface with a steady flow and moves in the form of layers of different velocities which do not mix with each other then the flow of liquid is called laminar flow. In general laminar flow is a streamline flow.

Turbulent flow

When a liquid moves with a velocity greater than its critical velocity, the motion of the particles of the liquid becomes disorderly or irregular. Such a flow is turbulent flow. The smoke from a cigarette after rising a short distance, the flow of water just behind boat or ship, the air flow behind a moving bus or train are some examples of turbulent flow.

Equation of continuity

Let us consider the steady flow of non-viscous liquid through a pipe of varying cross-sectional area as shown in the figure. Let a₁, v₁ and p₁ be the area of a cross-section of the tube, velocity of flow of the liquid density of the liquid respectively at point A of the tube. Similarly, a₂, v₂ and p₂ be corresponding values at the point B of the tube.

Volume of the liquid entering per second at A = a₁v₁

Mass of liquid entering per second at A = a₁v₁ρ₁

Similarly, mass of liquid leaving per second at B = a₂v₂ρ₂

If the liquid is incompressible then, the density remains the same and ρ₁ =ρ₂. Therefore

$$a_1v_1 = a_2v_2$$

$$\text {or,} \: av = \text {constant} $$

This is called equation of continuity. This equation states that if the area of a cross-section of the tube becomes larger the liquid's speed becomes smaller and vice-versa.

Energy of a Liquid

A liquid can posses three types of energies:

1. kinetic energy
   It is the energy possessed by a liquid by virtue of its motion. If v be the velocity of liquid and m be the mass of liquid, then
   \begin{align*} \text {KE of a liquid} &= \frac 12 mv^2 \\ \text {KE per unit mass, } E_k &= \frac {1/2 mv^2}{m} = \frac 12 v^2 \\ \end{align*}
2. Potential energy
   It is the energy possessed by a liquid by virtue of its height or position above the surface of earth.
   \begin{align*}\text {The potential energy of a liquid of mass m at the height} (h) &= mgh. \\ \text {P.E. per unit energy} E_{P.E}&= \frac {mgh}{m} = gh \\ \end{align*}
3. Pressure energy
   The energy possessed by a liquid by virtue of its pressure is called pressure energy.
   let us consider a piston fitted in the bottom of a container containing liquid as shown in the figure.
   Let A be the area of the cross-section of the piston and P be the pressure exerted by the liquid on it. Let dx be the displacement of the piston due to the force exerted by the liquid on it.
   The workdone by liquid is
   $$W = Fdx = PA dx = Pdv \dots (i)$$
   where dv is the volume of liquid. Equation (i) represents pressure energy of liquid of volume dv. The pressure energy per unit mass is \(E.P = \frac Wm = \frac {Pdv}{m} = \frac {P}{\rho } \dots (ii)\) whereρ is density of liquid.

Bernoulli's theorem

The total energy per unit mass of an ideal per unit mass of an ideal liquid (non-viscous and incompressible) remains constant through out its flow.

$$\text {i.e.} \frac {P} {\rho} + gh + \frac {V^2}{2} = \text {constant}$$

Let us consider an ideal liquid flowing through a pipe AB as shown in the figure and this can be done by applying an external force. Let P₁, a₁, v₁ be the pressure area of cross-section and velocity of a liquid at the end A of a pipe respectively.

Let P₂, a₂, v₂be their corresponding value at the end B. Here P₁ > P_2.The pipe AB has the non-uniform area of cross-section throughout its length.

Let ds be the distance travelled by the liquid in a time dt at end A. So, external force does work on the liquid at end A and its given by

$$W = F \times ds = P_1a_1v_1dt = P_1 \times \text {volume of a liquid entering at A in time dt} $$

$$= P_1 \times \frac {dm}{\rho}\dots (i) $$

where dm is the mass of liquid entering at the end A in time dt andρ be its density. The same mass odf liquid leaves the end B in time dt. so work done by the liquid at end B is

$$W = P_2 \frac {dm}{2} \dots (ii)$$

The net work done on the liquid by external force is

$$W = W_1 - W_2 = (\rho _1 - \rho _2) \frac {dm}{\rho} \dots (iii)$$

This work done on the liquid is stored in it in the form of its increase in kinetic energy and potential energy.

\begin{align*} \text {increase in K.E.}\Delta K.E. &= \frac 12 dm v_2^2 - \frac12 dmv_1^2 = \frac 12 dm(v_2^2 -V_1^2)^2 \dots (ii) \\ \text {increase in P.E.}\Delta P.E. &= dmgh_2 - dmgh_1 = dmg(h_2 - h_1) \dots (v) \\ \text {increase in K.E. and P.E. of liquid}\Delta E &= \frac 12 dm(v_2^2 -V_1^2)^2 + dmg (h_2 - h_1) \dots (vi)\\ \end{align*}

According to conservation of energy

\begin{align*} W &= \Delta E \\ \text {or,} \: (\rho _1 - \rho _2) \frac {dm} {\rho} &=\frac 12 dm (v_2^2 -V_1^2)^2 + dmg (h_2 - h_1) \\ \text {or,} \:\frac {P_1} {\rho} + gh_1 + \frac {v^1}{2} &=\frac {P_2} {\rho} + gh_2 + \frac {v^2}{2} \\ \therefore\frac {P} {\rho} + gh + \frac {V^2}{2} &= \text {constant} \\ \end{align*}

Applications of Bernoulli's Theorem

Atomiser or sprayer

When the rubber bulb B at the end of the pipe is squeezed, the air blows in the tube T with high speed. According to Bernoulli's theorem, when the air blows in the tube T, the pressure in it becomes less than the pressure in the vessel R. Due to this, the liquid rises up in the tube T and is pushed out with air through the nozzle N in the form of the spray.

Flowmeter-Venturimeter

It is a device used for measuring the rate of flow of liquid through pipes. Its working is based on Bernoulli's theorem.

Lift on an aeroplane

The shape of the aeroplane wings is slightly convex upward and concave downward. Therefore, the speed of air above the wings become more than the below the wings. So, the pressure below the wings is more than that above the wings. Due to this difference in pressure, a vertical lift acts on the aeroplane. When this lift is sufficient to overcome the gravitational pull on the aeroplane, it is lifted up.

Lift on an aeroplane wing.