Fluid Mechanics Concept Page - 12

Definition

Incompressible Fluid and Example of Incompressible Fluid Flow

Incompressible Fluid: The fluid whose density doesn't vary in any sort of flow  is considered as incompressible fluid.

Incompressible flow does not imply that the fluid itself is incompressible.
Example of incompressible fluid flow:
The stream of water flowing at high speed from a garden hose pipe. Which tends to spread like a fountain when held vertically up, but tends to narrow down when held vertically down. The reason being volume flow rate of fluid remains constant.

Definition

Conditions of steady flow and turbulent flow

Conditions of steady flow: When all the time derivatives of a flow field vanish, the flow is considered steady flow. Steady-state flow refers to the condition where the fluid properties at a point in the system do not change over time.

Conditions of turbulent flow:A flow is considered to be turbulent if the flow is characterized by recirculation, eddies, and apparent randomness. The fluid properties at a point in the system may change over time in a turbulent flow.

Definition

Streamline and its properties

Streamlines: Flow of a fluid in which its velocity at any point is constant or varies in a regular manner. It can be represented by streamlines Also called viscous flow.

Properties of streamline:
1. Streamlines never cross each other.
2. The velocity vector of the particle is tangent to the streamline at every point.

Definition

Continuity Equation

Continuity Equation: When a fluid is in motion, it must move in such a way that mass is conserved. To see how mass conservation places restrictions on the velocity field, consider the steady flow of fluid through a duct (that is, the inlet and outlet flows do not vary with time).

Example: An incompressible liquid flows through a horizontal tube L,M$L,M$ and N$N$ as shown in the figure. Then what is the velocity ′V′${}^{\prime }{V}^{\prime }$ of the liquid through the tube N$N$?

Solution:From the equation of continuity for an incompressible flow:
∑A1V1=0$\sum A_1{V}_1=0$
⇒2A×4=A2×4+AV$\Rightarrow 2A\times 4={\displaystyle \frac{A}{2}}\times 4+AV$
⇒8−2=V$\Rightarrow 8-2=V$
⇒V=6m/s$\Rightarrow V=6m/s$

Example

Problems based on continuity equation

Example: An incompressible liquid is continuously flowing through a cylindrical
pipe whose radius is 2 R at a point A. The radius at point B, in the direction of flow, is R. If the velocity of liquid at a point A is V then what be its velocity at point B?

Solution:Lets say, the velocity at point B is V′$V^{\prime }$, then from the equation of continuity, we have
a1v1=a2v2⇒π(2R)2×V=π(R)2×V′⇒V′=4V$$\begin{gathered} a_1{v}_1=a_2{v}_2 \\ \Rightarrow \pi (2R{)}^2\times V=\pi (R{)}^2\times V^{\prime } \\ \Rightarrow V^{\prime }=4V \end{gathered}$$

Example

Real life examples of viscosity

1) Consider two cases. The first one is water is poured into a beaker and the second one is honey poured into a beaker.
We can see that water flows faster than honey. This is because honey is more viscous than water.
2) Consider a river flowing. The water near the ground seems steady. On the other hand the water at the top moves faster. This is due to visous forces on different layers of the water. 

Definition

Reynold's Number and its use to identify nature of flow

The Reynolds number is an experimental number used in fluid flow to predict the flow velocity at which turbulence will occur. It is described as the ratio of inertial forces to viscous forces. For flow through a tube it is defined by the relationship: The parameters are viscosity , density and radius r.

R=2vρrη$R={\displaystyle \frac{2v\rho r}{\eta }}$

Reynolds' Number	Type of flow
<1000	Laminar
1000-2000	Transient
>2000	Turbulent

Definition

Terminal Velocity

When an object falls through a fluid, it attains a constant velocity through its subsequent motion. This happens because the net force on the body due to gravity and fluid becomes zero.

This constant velocity is termed as terminal velocity.

Definition

Understanding terminal velocity

Suppose a spherical object is dropped inside a container containing a certain fluid at time t=0 as shown in the figure(i).
Let us study the subsequent motion of the body.

1. Initially when the body is dropped, there are two forces acting on the body: a) force due to gravity Fg$F_g$ and b) buoyancy force FB$F_B$ as shown in the image (i). The drag force is zero as the velocity of the body is zero. Since the density of the body is more than density of the fluid, the net force will be downwards. Hence the body will accelerate downwards. 
2. As the body acquires some velocity, a drag force starts acting on the fluid upwards as shown in the image (ii). But still the net downward force would be greater as shown in the figure (ii).
3. As body further moves down the velocity of the body increases and hence drag force will also increase. (Since drag force is directly proportional to velocity)
4. A moment comes when the net upward force will be equal to net downward force as shown in the image(iii). Then the body moves with a constant speed as net acceleration on the body is zero.
5. The speed thus acquired by the body is called terminal velocity.

Example

Variation of pressure in a fluid in a rotating frame

Example: A U-shaped tube contains a liquid of density ρ$\rho$ and it is rotated about the left dotted line as shown in the figure. Find the difference in the levels of the liquid column and variation in pressure.

Solution:dF=dmxω2$dF=dmx{\omega }^2$
and (dp)A=(ρAdx)xω2$(dp)A=(\rho Adx)x{\omega }^2$
∫P2P1dp=ρω2∫L0ωdx${\int }_{P_1}^{P_2}dp=\rho {\omega }^2{\int }_0^L\omega dx$
P2−P1=ρω2L22$P_2-P_1={\displaystyle \frac{\rho {\omega }^2{L}^2}{2}}$
or ⇒ρgH=ρω2L22$\Rightarrow \rho gH={\displaystyle \frac{\rho {\omega }^2{L}^2}{2}}$
H=ω2L22g

BookMarks

Page 1  Page 2  Page 3  Page 4  Page 5  Page 6  Page 7  Page 8  Page 9  Page 10
Page 11  Page 12  Page 13  Page 14  Page 15  Page 16  Page 17  Page 18