FLUID MECHANICS INTRODUCTION CLASS NOTES FOR ENGINEERS

 

FLUID MECHANICS

INTRODUCTION CLASS NOTES

FOR ENGINEERS

Fluid Mechanics is a science which deals all

aspects of  fluids. It studies the fluids

 in motion or static. Fluid is a

common name to liquids, vapors and

gases. Branch which deals with the fluids is

called fluid mechanics.

  1. Definition of a fluid

Fluid is a common name for a liquid, vapor and a gas.

 (a) Liquids

Liquids have definite volume. These do not have definite shape. These do not fill the container fully. These are in-compressible and heavy compared to a gas.

 (b) Vapor

When liquid on heating changes phase at its boiling point, it is a vapor. It remains a vapor as long as degree of super  heat is ≤ to 500. No gas laws are applicable. Equations for vapors are highly complex. Therefore, use tables and charts to save time.

(c) Gases

Gases do not have definite volume & definite shape. It is, because, these fill the vessel completely. These are compressible and light. Therefore study of liquid flow and gas flow differ distinctly.

2. Properties of a fluid

Properties Playing Prominent Role in the Study of Fluid Mechanics:
Weight plays a predominant role in the fluid statics.
Density and viscosity play predominant roles in fluid dynamics.
Vapor pressure plays predominant role when dealing with Vacuum.
Surface tension plays an important role for both fluid at rest and in motion in SMALL PASSAGES like Capillary tubes.
Principles of thermodynamics play predominant roles while dealing with the compressible fluids (gases and vapors).

Three types of fluids

  • Liquids are in-compressible. There is no effect of pressure and temperature. Examples of liquid are water, milk, kerosene oil, petrol, fatty oils etc.

  • Vapor is compressible. Pressure and temperature affect vapors. It is studied only with the help of tabular data and charts.  Gas laws are not applicable to vapors. Examples of vapors are steam and refrigerants.

  • Gases are compressible. Pressure and temperature affect these. It is studied with gas laws and universal equation. Examples of gases are air, oxygen, nitrogen, hydrogen etc.

EXAMPLES OF MOVING FLUIDS

  1. Water flowing in pipe lines at home or factory

  2. Refrigerant flowing in a refrigerator or air conditioner

  3. Water flowing in water turbines and pumps

  4. Steam flowing in pipes to steam turbines

  5. Gas flowing to gas turbines

  6. Blood flowing in the human body

3. Fluid mechanics

There are two main branches of fluid mechanics.

 Fluid Statics: Which deals with fluids at rest. The resultant force is zero.

The resultant moment is zero.

 Fluid Dynamics: Which deals with fluids in motion. The resultant force

= mass x Acceleration= (Newton’s second law for accelerating bodies)

  Branches of Fluid Dynamics 

(a)    Fluid Kinematics

 Which deals with geometry of motion without the forces acting on the fluid. Geometry of motion means  velocity, acceleration and displacement of a fluid.

(b)   Fluid Kinetics

 Which deals with velocity, acceleration and displacement of a fluid along

with forces acting on the fluid.

4. Practical applications of fluid mechanics

Practical applications of fluid mechanics are as follows:

(i)  Mechanical engineering,

(ii) Chemical engineering,

(iii) Civil engineering,

(iv) Weather forecasting,

(v) Renewable energy systems,

(vi) Computer engineering

(vii) Pharmaceutical industry.

5. Fluid flow

It is mass transfer process  which helps to calculate the size of a pump and  a pipeline for transferring fluids. Fluid flow  effects  rates of heat transfer, mass transfer and  reaction rates .

6. Laws Used in Fluid Mechanics

Fundamental laws governing the fluid flow are

(i) Conservation of Mass

(ii) Conservation of Energy

(iii)   Conservation of Linear Momentum

(iv) Newton’s Laws of motion

(v) First and Second Laws of Thermodynamics.

7. Equations in fluid mechanics

The three conservation laws will form the basis for developing Continuity Equation, Bernoulli’s Equation, and the Momentum Equation. Firstly  derive these equations for the Ideal fluids. Then use these for the Real Fluids. A solid body gets distortion under a shear stress. It comes to original shape on release of the stress. However a fluid will continue to deform as long as the shearing stress is acting. It does not come back to the original position on the release of shear stress.

FLUID MECHANICS ANALYSIS

NOTE: Why we consider the motion of a fluid particle in a fluid motion?

There is a basic difference between the motion of a solid and a fluid. A solid body is compact and moves as one element. There is no relative motion between the particles of a solid body. Thus, we study the motion of the entire body. Therefore, there is no necessity to study the motion of any particle of a solid body. But in fluids, we consider the motion of individual particles. Because, there is relative motion between various fluid particles. Prototypes help in the study of fluid mechanics. Further, prototypes are small size working objects for pumps, turbines, submarines, airplanes and dams. These prototypes should have geometric, dimensional, kinematic and dynamic similarities. Use empirical equations using dimensionless numbers in their study.

STUDY APPROACHES

There are two approaches for studying fluid mechanics.

1. Lagrangian Method
It considers the movement of a single FLUID PARTICLE. It studies the path taken by the fluid particle, changes in velocity, acceleration, density, pressure etc. of the fluid particle. The resulting equations are tedious, complex and cumbersome. These are difficult to solve. SO THIS METHOD IS NOT IN USE.

Eulerian Method
This considers the path, velocity, acceleration, density, pressure etc. at a SPACE POINT in the FLUID FLOW. This method makes the study simple and resulting equations are simple and easy to understand. FLUID DYNAMICS commonly use this method for analysis.

Represent velocity components as follows:

u = f(x, y, z, t)

v = f(x, y, z, t)

w = f(x, y, z, t)
Partial derivatives determine the acceleration components.

Three parts of fluid mechanics

Fluid statics

It deals with forces applied by fluids at rest. This fluid force is pressure. A fluid exerts pressure. It is normal force per unit area. It is the static pressure of the fluid. There are Some laws in fluid statics. These laws are

Pascal’s law

It states that an enclosed fluid transmit pressure equally in all directions.

Newton’s Law

It is for a static fluid.

Thus, ΣF = ma = 0 for a static fluid.

ΣFx = ΣFy = ΣFz = 0. It deals with equilibrium of forces on a fluid.

Buoyancy Law (Archimedes’ principle)

States that a body completely or partially submerged in a fluid (liquid or gas).

AT REST, an upward buoyant force acts.

The magnitude of which is equal to the weight of the fluid displaced by the body.

(b)Fluid Kinematics

Deals with fluids in motion without considering forces or energy acting. In this, we study displacement, velocity and acceleration. The other name of kinematics is geometry of motion. In this, we study ideal fluid, real fluid, in-compressible and compressible fluid. Thus, we also study various types of flows. For example,

(i) Laminar flow

(ii) Turbulent flow

(iii) Steady flow

(iv) Unsteady flow

(v) Uniform flow

(vi) Non-uniform flow

(vii) Rotational flow

(viii) Ir-rotational flow

(ix) One, two and three dimensional flows

There are two ways to study fluid mechanics. Out of Lagrangian and Eulerian methods, Eulerian method is relatively easy to apply. It concerns velocity field and continuity equation.

© Fluid Dynamics deals with fluids in motion with forces/energy acting on the moving fluids.

   

FLUID MOTION-KINEMATICS

 Fluid motion equations use two approaches.
(i) Integral approach–Not dealt here.
(ii) Differential approach

Differential approach uses Control Volume method

A control volume is a finite region having OPEN boundaries. There is mass transfer, momentum transfer and energy transfer across an open boundary. Control volume method considers the followings aspects:
a. Makes a balance between fluid flowing in and fluid flowing out
b. Find the force, or torque on a body
c. Help to find the total energy exchange between the fluid and the body

Control-volume approach uses three basic principles to achieve the aforesaid objectives.

1. Principle of conservation of mass which gives the continuity equation
2. Principle of conservation of energy which gives the energy equation
3. Principle of conservation of linear momentum which gives equations for the dynamic forces exerted by flowing fluids

EQUATIONS OF FLUID MOTION

  1. CONTINUITY EQUATION

Obtain Continuity equation under steady state condition using conservation of mass
Total Mass flowing out = Total Mass flowing in
For an in-compressible flow, density is constant. Replace mass flow rates with volumetric flow rates. Thus between any two sections 1 and 2 along the pipe, the continuity equation becomes
Q=V1A1=V2A2 =constant

GENERAL CONTINUITY EQUATION FOR 3-D FLOW

∂ ρ/∂t + ∂(ρu)/ ∂x +∂(ρv)/ ∂y + ∂(ρw)/ ∂z =0

Continuity equation is applicable to all types of flow as given below:

(a) Steady and unsteady flows
(b) Uniform and non-uniform flows
(c) Compressible and in-compressible flows.

 

∂ ρ/∂t + ∂(ρu)/ ∂x +∂(ρv)/ ∂y + ∂(ρw)/ ∂z =0

Continuity equation is applicable to all types of flow as given below:

(a) Steady and unsteady flows
(b) Uniform and non-uniform flows
(c) Compressible and in-compressible flows.

For a steady flow

∂ ρ/∂t = 0

∴ The continuity equation becomes

∂(ρu)/ ∂x +∂(ρv)/ ∂y + ∂(ρw)/ ∂z=0

For an in-compressible fluid with steady flow, ρ= constant

∴ The continuity equation becomes

∂(u)/ ∂x +∂(v)/ ∂y + ∂(w)/ ∂z =0

For 2-D flow
w=0
∴ The continuity equation becomes
∂(u)/ ∂x +∂(v)/ ∂y =0

VELOCITY

Let V= Resultant velocity at any point in a fluid flow.
(u, v, w) are the velocity components in x, y and z directions.  These are functions of space coordinates and time.
u=u(x, y, z, t) ; v=v(x, y, z, t) ; w=w(x, y, z, t).

Resultant velocity vector

V=u i +v j + w k

|V|= (u2+v2+w2)1/2

ACCELERATION

Let ax, ay, az are the acceleration components in the x, y, z directions

ax= [du/ dt]= [∂u/ ∂x] [∂x/ ∂t]+ [∂u/ ∂y][ ∂y/ ∂t]+ [∂u/ ∂z][ ∂z/ ∂t]+∂u/ ∂t]

ax= [du/dt] =u[∂u/ ∂x]+v[∂u/ ∂y]+w[∂u/ ∂z]+[ ∂u/∂t]

Similarly,

ay= [dv/dt] = u[∂v/ ∂x]+ v[∂v/ ∂y]+w[∂v/ ∂z]+[ ∂v/∂t]

az= [dw/dt] = u[∂w/ ∂x]+ v[∂w/ ∂y]+w[∂w/ ∂z]+[ ∂w/ ∂t]

TYPES OF ACCELERATIONS

1.CONVECTIVE ACCELERATION COMPONENTS

It is the rate of change of velocity due to change of position of the fluid particles. Convective components of ax, ay, az are
ax= u[∂u/ ∂x]+v[∂u/ ∂y]+w[∂u/ ∂z]
ay= u[∂v/ ∂x]+ v[∂v/ ∂y]+w[∂v/ ∂z]
az= u[∂w/ ∂x]+ v[∂w/ ∂y]+w[∂w/ ∂z]

LOCAL ACCELERATION COMPONENTS

It is the rate of change of velocity with respect to time only. These are LOCAL COMPONENTS of acceleration. These are as given below:
ax=∂u/∂t
ay= ∂v/∂t

az= ∂w/ ∂t

In a steady flow, local acceleration components as well as total local acceleration is zero.

∴ [∂u/ ∂t]= [∂v/ ∂t]= [∂w/ ∂t]= 0

In an uniform flow, space derivatives are zero. Hence convective acceleration components are zero and hence total convective acceleration is zero.

ax= u[∂u/∂x]+v[∂u/∂y]+w[∂u/∂z]
ay=  u[∂v/∂x]+v[∂v/∂y]+w[∂v/∂z]
az=  u[∂w/∂x]+v[∂w/∂y]+w[∂w/∂z]

Acceleration Vector

a = ax i +ay j + az k;

|A|=[ax2+ay2+az2]1/2      

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