ME 303 FLUID MECHANICS I

Prof. Dr. Haşmet Türkoğlu

Çankaya University
Faculty of Engineering
Mechanical Engineering Department

Fall, 2023
INTRODUCTION
Fluid mechanics: Subject that deals with the study of the behavior of a fluid at rest
and in motion.

Fluid Mechanics in Engineering

Many engineering applications involve fluid in motion or at rest.

Examples: Home and city water supply systems, transportation of oil and natural gas
in pipelines, flow of blood in vessels, air flow over an aircraft, motion of a ship in
water, pumps, turbines and many others.

Design and operation of all such devices require a good understanding of fluid
behavior when it is stationary or in motion, and its interaction with the surface in
contact with.

To characterize fluid behavior, velocity, pressure, temperature, density, etc.

distributions are determined.

To characterize the interaction between fluid and the surface in contact with it, force,
energy, torque, etc. acting on surfaces due to flow are determined.
1 Introduction 2
 Definition of a Fluid

Consider imaginary chunks of both a solid and a fluid. Chunks are fixed along one
edge, and a shear force is applied at the opposite edge.

A short time after application of the force, the solid assumes a deformed shape
which can be measured by the angle 1. If we maintain this force and examine the
solid at later times, we find that deformation is exactly the same, that is 2= 3 =1.
On application of a shear force, a solid assumes a certain deformed shape and
retains that shape as long as the same force is applied.

Consider the response of the fluid to the applied shear force. A short time after
application of the force, a fluid assumes a deformed shape, as indicated by the angle
1. At a later time, the deformation is greater, 2>1, in fact the fluid continues to
deform as long as the force is applied.

A fluid is a substance that deforms continuously under the action of an applied shear
force. The process of continuous deformation is called flow.

This has the implication that if a fluid is at rest there are NO shearing forces acting
on it, and forces must be acting perpendicular to the fluid surface.

1 Introduction 3
Scope of Fluid Mechanics

As pointed out above, many engineering applications involve fluids in motion or at

rest. We cannot consider all these specific problems of fluid mechanics. Instead, the
purpose of this course is to introduce the basic laws and associated physical concepts
that provide the basis or starting point in the analysis of any problem in fluid
mechanics.

Basic Laws
Analysis of any fluid mechanics problems begins, either directly or indirectly with the
basic laws governing the fluid motion. The basic laws, which are applicable to any fluid
flow are,

1. Conservation of mass
2. Newton’s second law of motion
3. Moment of momentum
4. The first law of thermodynamics
5. The second law of thermodynamics

It should be emphasized that not all the basic laws are required to solve every
problem.

Constitutive Equations
In some problems, it is necessary to bring into the analysis additional relations, in the
form of equation of state or constitutive equations;

Equation of state: 𝑝𝑣 = 𝑅𝑇
du
Newton’s law of viscosity:  yx   dy

Constitutive relations are not generally valid for all fluids and for all flows.

1 Introduction 4
 METHOD OF ANALYSIS

The basic laws can be applied to a control volume or to a system. The first step in
solving a problem is to define the system that is going to be analyzed.

System
A system is defined as a fixed, identifiable
quantity of mass.

Gas The boundaries of a system may be fixed or

moveable; however, there is no mass transfer
across the system boundaries; i.e. the amount
of mass in the system is fixed.

Control Volume

A control volume is an arbitrary volume in space through which fluid flows.

Control surface

flow C

Mathematical Formulation

Differential vs Integral Approach

The basic laws that we apply in fluid mechanics problems can be formulated in
differential and integral forms. The solution of differential equations provides a
means of determining the detailed (point by point) behavior of the fluid.

1 Introduction 5
 The Systems of Dimensions

a) Mass [M], length [L], time [t], temperature [T]

b) Force [F], length [L], time [t], temperature [T]
c) Force [F], mass [M], length [L], [t], time [T]

These are primary dimensions in different systems of dimensions. Dimensions of

other quantities are known as secondary dimension and derived from the primary
dimensions.

Systems of Units
MLtT
SI (kg, m, s, K)
FLtT
British Gravitational (lbf, ft, s, oR)
FMLtT
English Engineering (lbf, lbm, ft, s, oR)

Preferred Systems of Units

System of International SI (kg, m, s, K) 1 N = 1 kg.m/s2 [ML/t2]

Metric system (gr, cm, s, K) 1 dyne = 1 gr.cm/s2 [ML/t2]

British Gravitational (BG) unit system (lbf, ft, s, oR)

The unit of the mass is slug and it is a secondary dimension which is derived from the
Newton’s secon law.
1 slug = 1 lbf.s2/ft [Ft2/L]

English Engineering (EE) unit system (lbf, lbm, ft, s, oR)

Both mass and force are primary dimensions. Newton’s secon law is written as
𝑚𝑎
F= where gc conversion factor, gc = 32.2 ft.lbm/(lbf.s2)
𝑔𝑐
1 slug = 32.2 lbm
1 Introduction 6
FUNDANENTAL CONCEPTS

Topics
- Fluid as a Continuum
- Velocity Field
- Stress Field
- Viscosity
- Surface Tension
- Description and Classification of Fluid Motions

1 Introduction 7
FUNDAMENTAL CONCEPTS

Fluid as a Continuum
All fluids are composed of molecules in constant motion. However, in most
engineering applications we are interested in the average or macroscopic effects of
many molecules. We thus treat a fluid as an infinitely divisible substance, a
continuum, and do not concern with the behavior of individual molecule.

For continuum model to be valid, the smallest sample of the matter of practical
interest must contain a large number of molecules so that meaningful averages can
be calculated.

The condition for the validity of continuum approach is that the distance between
the molecules (mean free path of the fluid) of the fluid should be much smaller than
the smallest characteristic length of the problem. Mathematically, l/Lc<<1

As a consequence of the continuum assumption, fluid properties and flow variables

can be expressed as continuous functions of position and time, i.e.
For Air:
l=6.8 x10-8 m, number of mol per cm3=2.7x1019 (at standart atmosphere)
l=0.1 m (at 100 km altitude)

 =  (x,y,z,t)
u = u (x,y,z,t)
T = T (x,y,z,t)
p = p (x,y,z,t)
…
The value of a fluid property at a point is defined as an average considering a
volume around that point.

1 Introduction 8
 Specific gravity: An alternative way of expressing the density of a substance (solid or
fluid) is to compare it with a reference density value (density of water at 4 oC).

SG=Density of substance/density of water

SG=/H2O

Specific weight: g=g

VELOCITY FIELD
Continuum assumption leads to description of all the fluid properties at every point in
the flow domain.

The fluid velocity at a point C is defined as the velocity of the center of gravity of
volume  surrounding the point C.

The velocity of fluid at any point in the flow field is a function of space and time, i.e.

𝑉=𝑉 𝑥, 𝑦, 𝑧, 𝑡
Velocity vector, can be written in terms of scalar components,

u is x-component of velocity
V  ui  vj  wk v is y-component of velocity
w is z-component of velocity
CLASSIFICATION OF FLOWS

1. Classification of flows based on the dependency on time, t

Steady Flow and Unsteady Flow

If properties at each point in a flow do not change with time, the flow is called steady
flow. Mathematically, for any property  for steady flows,

0     ( x, y , z )
t
V
0  V  V ( x, y , z )
t
If properties at each point in a flow changes with time, the flow is called unsteady flow.
Mathematically, for any property  for steady flows,

 V
0  0  V  V ( x, y , z , t )
t t

1 Introduction 9
2. Classification of flows based on the dependency on space
variables x, y and z

ONE- TWO- AND THREE-DIMENSIONAL FLOWS

A flow is classified as one-, two-, or three-dimensional depending on the

number of space coordinates required to specify the velocity field.

Examples:

1-D Flow
r R
x u

Velocity depends only on r, Hence, the flow is one-

dimensional.

2-D Flow

To be completed in class

3-D Flow

To be completed in class

1 Introduction 10
Uniform Flow

To simplify the analysis, sometimes velocity at a cross-section is assumed to be

constant over the cross-section. If velocity at a given cross section is assumed to be
uniform, flow is called uniform flow.

At a given cross section, velocity is

assumed to be the same at all points.

1 Introduction 11
Timelines, Pathlines, Streaklines, and Streamlines

Timelines, pathlines, streaklines and streamlines provide a visual representation of

a flow field (motion of fluid particles).

Timeline:

If a number of adjacent fluid particles in a flow field are marked at a given instant,
they form a line in the fluid at that instant, this line is called a timeline.
Observation of the timeline at a later instant may provide information about the
flow field.

Pathline:

A pathline is the path or trajectory traced out by a moving fluid particle. A pathline
may be obtained by following a fluid particle (i.e. by use of dye) in the flow field.

1 Introduction 12
Streakline:

A line joining the fluid particles that pass through the same point in the flow field is
called the streakline.

Streaklines over an automobile in a wind

tunnel

Streamline:
Streamlines are lines drawn in the flow field so that at a given instant they are
tangent to the direction of flow at every point in the flow field. Streamlines are
tangent to the velocity vector at every point in the flow field.

In steady flows, pathlines, streaklines, and streamlines are identical

lines in the flow field.

1 Introduction 13
   
Example (Fox): A velocity given by V  axi  ayj , the units of velocity are m/s; and x
and y are given in meters; a=0.1 sec-1.
a) Determine the equation for the streamline passing through the point
(x0, y0, 0)=(2, 8, 0)
b) Determine the velocity of a particle at the point (2, 8, 0)
c) If the particle passing through the point (x0, y0, 0) is marked at time t0=0, determine
the location of the particle at time t=20 sec.
d) What is the velocity of the particle at t=20 sec.
e) Show that the equation of the pathline is the same as the equation of the
streamline.

To be completed in class

1 Introduction 14
1 Introduction 15
FORCES ACTING ON DIFFERENTIAL FLUID ELEMENT

In studying fluid mechanics, we need to understand what kinds of forces

act on a fluid particle.

Forces acting on a fluid element

- Surface forces
- Body forces

Surface forces include all forces acting on the boundaries of a medium

through direct contact.
- Pressure force
- Friction (viscous) force

Forces developed without physical contact and distributed over the

volume of the fluid are called body forces.
- Gravitational force
- Electromagnetic force

Gravitational body force acting on a fluid element of volume d is

and gravitational body force acting on per unit volume of a fluid element is
g

The concept of stress field provides a convenient means to describe forces

acting on boundaries of a fluid medium and transmitted through the
medium.

1 Introduction 16
STRESS FIELD

Consider an area around point C in a continuum. The force

acting on can be resolved into two components, one normal and the
other tangential to the area
n: normal unit vector of the area A
Fn: normal component
Ft: tangential component

Normal stress sn and shear stress n are defined as

Note: Subscript, n, indicates that the stresses are associated with a

particular surface whose normal vector is n.

Note that at a point C in a continuum, different surfaces can be drawn.

However, for purpose of analysis, we usually reference the area to some
coordinate system. In rectangular coordinate system, we might consider
the stress components acting on planes whose outward drawn normal
vectors are in x, y or z-directions.

1 Introduction 17
 Force and stress components on area element Ax can be shown as

Stress components shown in above figure is defined as

sxx =

To be completed in class

We have used a double subscript notation to label the stresses.

i,j
i: indicates plane on which stress acts (axis perpendicular
to plane on which stress acts)
j: direction in which stress acts

Consideration of an area element, Ay, would lead to the definition of

stresses syy, yx, yz, and use of area element Az would similarly lead to
the definitions of szz, zx, zy.

To be completed in class

1 Introduction 18
 y

syy

yz yx
xy
zy
xz sxx
szz zx x

An infinite number of planes can be passed through point C, resulting an infinite

number of stresses associated with that point. Fortunately, state of stress at a
point can be described completely by specifying the stresses acting on three
mutually perpendicular planes passing through the point. Hence, stress at a
point is specified by the nine components.

s xx  xy  xz 
 
 yx s yy  yz 
 zx  zy s zz 

The planes are named in terms of the coordinate axes. The planes are named
and denoted as positive or negative according to the direction of the outward
drawn normal to the plane. Thus, the top plane for example is a positive y-plane
and the back plane is a negative z-plane.

It is also necessary to adopt a sign convention for stresses. A stress component

is considered positive when the direction of the stress component and the
outward normal of plane on which it acts are both positive or both negative.

Thus, yx=2.4 N/m2 represents a shear stress on positive y-plane in positive x-

direction or shear stress on negative y-plane in negative x-direction.

1 Introduction 19
VISCOSITY
We have learned that a fluid is a substance that undergoes continuous
deformation when subjected to a shear stress. This shear stress is function of rate
of deformation. For many common fluids, the shear stress is proportional to the
rate of deformation. The coefficient of proportionality, called viscosity, is a fluid
property.

To develop the defining equation for the relation between viscosity and the rate of
shear deformation, we consider a flow in x-y plane in which x-direction velocity
varies with y. du e
u δy
dy


Fluid y
element

y x u u
x

Fluid element at time t Fluid element at time t+t

Consider the fluid element in the figure. The top of the fluid element moves faster
than the bottom, so in time fluid element deforms.

We measure shear deformation by the angle , which can be related to the fluid
velocity.

e =

Hence, shear stress is

 Based on the relation between shear stress and rate
 yx

t
of deformation (flow), fluids are grouped as
- Newtonian fluid
or - Non-Newtonian fluid
du
 yx

dy
1 Introduction 20
Newtonian Fluid
Fluid in which coefficient of proportionality in the above expression is
constant and equal to the viscosity called Newtonian fluid.

Examples:

Newton’s law of viscosity:

du
 yx   dy
: dynamic viscosity (absolute viscosity)

Unit of 
F
:
L2  Ft
du 1  : (Dimension of dynamic viscosity)
: L2
dy t 

N sec kg
In SI system: : :
m2 m. sec

g kg
In metric system: poise  1 poise  0.1
cm. sec m sec

Kinematic Viscosity, 
  L2   m 2 
  ,  ,  
  t   sec 
cm2 m2
In metric system stoke 1 stoke  0.0001
sec sec

1 Introduction 21
Non-Newtonian Fluid
Not all fluids follow the Newton’s law of viscosity (stress-strain
relation). Such fluids are called non-Newtonain. Some fluids such as
ketchup, are ‘shear-thinning’; that is the coefficient of resistance
decreases with increasing strain rate (it all comes out of the bottle at
once). Others, such as a mixture of sand and water are ‘shear-
thickening’. Some fluids do not begin to flow until a finite stress has
been applied (toothpaste).

In these fluids, shear stress-deformation rate (shear strain) relation

may be represented by the power law model,
n
 du 
 yx  k   n: flow behavior index, k: consistency index
 dy 

If the above equation is written in the form

n 1
du du du
 yx dy
 k
dy

dy

n 1
du
k is referred to as the apparent viscosity.
dy

1 Introduction 22
Classification of Non-Newtonian Fluids
Based on the variation of apparent viscosity with rate of deformation, non-
Newtonian fluids are classified as follows:

Pseudoplastic Fluid (Shear thinning): Apparent viscosity decreases with increasing

deformation rate.
Example: Polymer solutions, ketchup

Dilatant (Shear thickening): Apparent viscosity increases with increasing

deformation rate.
Example: Sand suspension

Bingham plastic: Deformation (flow) does not begin until a finite stress is applied.
Example: toothpaste, drilling muds, clay suspensions

Based on the variation of apparent viscosity in time under constant shear stress,
non-Newtonian fluids are classified as follows:

Rheopectic fluid: Apparent viscosity increases with time under constant shear
stress.
Example: Plaster, cement

Thixotropic fluid: Apparent viscosity decreases with time under constant shear
stress.
Example: Paints

Viscoelastic fluid: Fluid which partially returns to original shape when the applied
stress is released. Example: Jell

1 Introduction 23
Dependency of Viscosity on Temperature

In liquids, resistance to shear force depends on the cohesion between molecules.

The distance between liquid molecules increases with increasing temperature, and
hence cohesion between molecules decreases. As a result of this, in liquids, viscosity
decreases with increasing temperature.

In gases, resistance to shear force depends on the momentum transfer between

molecules. With increasing temperature, motion of the gas molecules increases and
hence momentum transfer increases among the gas molecules, as a result viscosity
increases.

1 Introduction 24
Expample: Consider a fluid flowing on an inclined surface. Its velocity profile is
given by
y
 y  y 2 
u ( y )  U 2    
 Y  Y   Y
U

width, w
g
Find shear stress at y=0, Y/2 and Y

Solution: q x

To be completed in class

1 Introduction 25
COMPRESSIBILITY OF FLUIDS

Bulk Modulus

Pressure and density variations of liquids are related by the bulk compressibility
modulus or modulus of elasticity.

𝑑𝑃
𝐸𝑣 = − 𝑜𝑟 𝐸𝑣 =dP/(d/)
𝑑∀/∀
Where dP is the differential change in pressure needed to create a differential change
dV in volume V (or differential change d in density ).

For water: Ev = 2.15×109 N/m2

Large values of bulk modulus indicate that the fluid is incompressible. That is it takes a
large pressure change to create a change in volume (or in density).

Compression and Expansion of Gases

When gases are compressed (or expanded) the relationship between pressure and
density (volume) depends on the nature of the process.

If the compression or expansion takes place under constant temperature conditions

(isothermal process), then from the ideal gas equation (equation of state) we
obtain:
𝑃
𝑃 = 𝑅𝑇 = constant
𝜌

If the compression or expansion is frictionless and no heat is exchanged with the

surroundings (isentropic process) then

𝑃
𝜌𝑘
= constant

Where k is the specific heat ratio, k=Cp/Cv

1 Introduction 26
Speed of Sound

The velocity at which small disturbances (the pressure wave of infinitesimal strength)
propagate in a medium ( fluid) is called the acoustic velocity or speed of sound.

The peed of sound is an important characteristic parameter for compressible flows.

The speed of sound is related to changes in pressure and density of the fluid medium
by the equation:

𝑑𝑃
c= 𝑑

𝐸𝑣
or in terms of bulk modulus c= 

𝑘𝑃
For gases undergoing an isentropic process, Ev = kP. Hence, c=


For ideal gases it follows 𝑐= 𝑘𝑅𝑇

For air at 20 C, speed of sound is c = 343 m/s.

For water at 20 C, Ev = 2.15×109 N/m2 and  = 998.2 kg/m3 hence speed of sound is
obtained as c = 1481 m/s.

If a fluid were truly incompressible (Ev = ), the speed of sound would be infinite.

1 Introduction 27
Surface Tension, s

At the interface between a liquid and a gas or between two immiscible liquids, forces
developed at the liquid surface which causes the surface to behave as if it were a «skin»
or «membrane» stretched over the liquid.

Surface tension is a property that results from the attractive forces between molecules.
The molecules at the interior of the fluid mass are surrounded by the same molecules
that are attracted to each other equally. However, molecules along the surface are
subjected to a net force toward the interior. Due to this unbalanced force, a tensile force
acts in plane of the surface along a line in the surface.

Surface tension has the unit of the force per unit length, i.e. N/m

Surface tension values for some fluid pairs:

Air – octane interface: s = 21.8 N/m

Air – water interface : s = 72.8 N/m
Air – mercury interface : s = 484 N/m
Water – mercury interface : s = 375 N/m
Water – octane interface : s = 50.8 N/m

Specific Gravity, SG

SG= density of Fluid/Density of Water = /H2O

1 Introduction 28
DESCRIPTION AND CLASSIFICATION OF FLUID MOTIONS

Since there is much overlap in the types of flow fields encountered, there
is no universally accepted classification scheme. One possible
classification,

Viscous and Inviscid Flows

There are many problems where viscosity plays no role on the motion
of fluid and hence an assumption that =0 will simplify the analysis,
and at the same time lead to meaningful results. In an inviscid flow,
the fluid viscosity, , is assumed to be zero. Fluids with zero viscosity
do not exist. All fluids possess a non-zero viscosity.

Example:

du
U U  0   yx  0 Flow is frictionless (inviscid )
dy

y du
 0   yx  0 Flow is visvous
dy

In any viscous flow, the flow in direct contact with a solid boundary
has the same velocity as the boundary itself. There is no slip at the
boundary.
1 Introduction 29
Laminar and Turbulent Flows

The laminar flow is characterized by smooth motion of fluid particles in

laminae or layers.

The turbulent flow is characterized by random, three-dimensional

motions of fluid particles superimposed on the mean motion.

In laminar flow there is no microscopic mixing of adjacent fluid layer.

https://www.youtube.com/watch?v=WG-YCpAGgQQ

https://www.youtube.com/watch?v=6OzAx1bPGD4

1 Introduction 30