Module 2 of fluid machines - Kinematics of fluids

INTRODUCTION
 Fluids have a tendency to move or flow even if there exists a very small shear stress.
 The study of velocity and acceleration of a flowing fluid and the description and visualization
of the fluid motion are dealt with fluid kinematics.
Methods of describing fluid motion
 Two Methods:
1. Eulerian Method:
 concerned with field of flow.
 The properties (pressure, velocity, density etc.) are expressed as function of space and time.
F = f(x, y, z, t)
2. Lagrangian Method:
 follows an individual particle( or given mass) flowing through the flow field and thus
determines the fluid properties of individual particles or masses as a function of time like
p(t), v(t) etc.
 A fluid consists of large number of particles, therefore it is convenient to use the
Eulerian method for describing the fluid motion.

System and control volume
 SYSTEM :
 It refers to a fixed, identifiable quantity of mass in space:
 Three types of system: open system, closed system, isolated system.
 SURROUNDING:
 All other matter around the system is called surrounding
 BOUNDARY:
 The system boundaries forming a closed surface separates the system from
surrounding.
 The system boundary may be fixed or movable but there is no mass transfer occurs
across the system boundaries.
Example:
Consider a piston cylinder assembly:
• If the assembly is heated from out side, the
gas expands and the piston moves.
• Alternatively the gas in the cylinder can be
compressed by moving the piston.
• Although heat and work have transferred
through the boundary but there is no mass
transfer across the boundary.

 In case of fluid flow in pipes, channels, nozzles etc. it is difficult to focus attention
on fixed identifiable quantity of mass.
 It is convenient to focus attention on a volume, in a flow field, through which the
fluid flows. This volume is called control volume.
 Control volume:
 May be defined as an arbitrary volume in a flow field through which the fluid
flows
 The geometric boundary of the control volume is called control surface, that
may be real or imaginary or may beat rest or in motion.
Example:
Control volume for fluid flow in a pipe:
• The inside surface of pipe forms real
control surface.
• The vertical portions of control volume
are imaginary.
Points to remember:
• The size and shape of control volume can
be chosen arbitrarily.
• The location of control surface affects the
computational procedure

Visual description of fluid pattern
 While dealing with fluid flow problems, it is often advantageous to obtain a
visual representation of a flow field.
 The pattern of flow can be visualized in many different ways.
 Photographs
 Colored dyes
 flakes
 Hydrogen bubbles etc.
Path lines are the trajectories that individual fluid particles follow. These can be thought of as "recording" the
path of a fluid element in the flow over a certain period. The direction the path takes will be determined by the
streamlines of the fluid at each moment in time.
Timelines are the lines formed by a set of fluid particles that were marked at a previous instant in time,
creating a line or a curve that is displaced in time as the particles move.
Streamlines are a family of curves that are
instantaneously tangent to the velocity vector
of the flow. These show the direction a
massless fluid element will travel in at any
point in time.
There can be no flow across a streamline
Streak lines are the locus of points of all the
fluid particles that have passed continuously
through a particular spatial point in the past.
Dye steadily injected into the fluid at a fixed
point extends along a streak line.

 Stream tube:
 A typical set of neighboring streamlines that forms a passage through which
the fluid flows is called stream tube.
 No flow is possible across a stream tube except through its ends.
 The stream tube need not be a solid and are fluid surface.
 Stream filament:
 It is a stream tube with its cross section sufficiently small so that variation of
velocity over it may be considered negligible.

Velocity field

Acceleration of fluid particle in a velocity field

Streamline patterns and corresponding convective accelerations.

Therefore, ………………(1)
………………(2)
………………(3)
So, the total acceleration is obtained by combining the acceleration given by the above three equations
………………(4)
………………(5)

Mass conservation(continuity Equation along
a stream tube
• The principle of mass conservation is valid for a flowing fluid.
• In a fixed region of flow constituting a control volume, mass of fluid entering the volume per unit time
must be equal to the mass of fluid leaving the control volume per unit time plus (or minus) the increase
(or decrease) of mass of fluid in the control volume per unit time.
• But for steady flow there can not be any change in mass within the control volume
Consider a stream tube of varying cross section

Differential equation of mass conservation
………………………….……………………(1)
………………………….……………………(2)
………………………….……………………(3)

Now total rate of mass flow to the control volume
………….……………………(4)
…………………………(5)
This equation is called the continuity equation in Cartesian coordinates.
In vector form
…………………………(6)
In cylindrical polar coordinates the continuity equation can be written as
…………………………(7)

…………….………….(8)
…………….………….(9)
…………….………….(10)

Consider a transparent plane parallel and at a unit distance from the board.

Module 2 of fluid machines - Kinematics of fluids

The partial derivative of the stream function with respect to any direction gives the
velocity component at 90o
anticlockwise to that direction.
The stream function is only defined for two dimensional in compressible flow, for which the continuity
equation can be written as
Putting the values of u and v from equation 1 and 2, we will get
This shows that the stream function always satisfies the continuity equation.
In polar coordinates
…………….(3)
…………….(4)

Fluid displacements
An infinitesimal element of fluid may move in fluid such
that the elements undergoes following basic
displacements.
1. Translation:
The elements moves bodily without being rotated or
deformed
2. Rotation:
Planes( top, base and sides of element) and medians as
well as diagonal of the element may change as a result of
their rotation about any one (or all three) of the
coordinate axes.

4. Linear deformation:
Shapes of the elements gets changed without change
in orientation of the element. The planes ( top, base
and sides of the element) as well as the median lines
of the elements are displaced while remaining parallel
to their original position.
3. Angular deformation:
It involves a distortion of the fluid element in which planes
that were initially perpendicular to each other are no longer
so.

Fluid rotation
• Counter clock wise rotation is considered to be positive.
• If the fluid particles in a flow region have rotation about any axis the flow is called
rotational flow or vortex flow.
• If the fluid particles in a flow region do not have rotation about any axis the flow is called
irrotational flow
………………………(A)

Consider the motion of a fluid particle in xy-plane
Derivation of expression for Rotation
The angular velocity of the line OA can be written as
and

Now, the rotation of fluid element about z-axis a per definition of rotation can be written as
Similarly the rotation of fluid element
about x and y axis can be found out as
………………………….(1)
………………………….(2)
………………………….(3)
Hence from equation A
………………………….(B)
In vector form
………………………….(C)

Another measure of rotation of fluid element is vorticity which is equal to twice the rotation
………………………….(4)
………………………….(5)
………………………….(6)
………………………….(D)

Angular deformation
here
And,
So,

Circulation
For closed curve OACB
Or,
Or,
This means circulation per unit area is equal to the vorticity or twice the rotation

The circulation around the closed curve ABPCA is given by:
Or,
Where,
vs is the component of velocity along ds
Or,
Or,

Or,
Substituting the value of u, v and w in continuity equation we will get
Now vorticity,
Since,

Flow Nets
• To check the squareness of a grid one can join
the diagonals of all the squares, the diagonals
also should result into a grid of squares

Combination of flow patterns
The magnitude or velocity components of the
resultant motion are given by the algebraic sum
of those for the constituent motion.

The Acceleration Field of a Fluid
 Velocity is a vector function of position and time and thus
has three components u, v, and w, each a scalar field in
itself.
 This is the most important variable in fluid mechanics:
Knowledge of the velocity vector field is nearly
equivalent to solving a fluid flow problem.
 The acceleration vector field a of the flow is derived
from Newton’s second law by computing the total time
derivative of the velocity vector:
36

 Since each scalar component (u, v , w) is a function of
the four variables (x, y, z, t), we use the chain rule to
obtain each scalar time derivative. For example,
 But, by definition, dx/dt is the local velocity
component u, and dy/dt =v , and dz/dt = w.
 The total time derivative of u may thus be written as
follows, with exactly similar expressions for the time
derivatives of v and w:
37

 Summing these into a vector, we obtain the total
acceleration
:
38

 The term δV/δt is called the local acceleration, which
vanishes if the flow is steady-that is, independent of
time.
 The three terms in parentheses are called the convective
acceleration, which arises when the particle moves
through regions of spatially varying velocity, as in a nozzle
or diffuser.
 The gradient operator is given by:
39

Example 1. Acceleration field
Given the eulerian velocity vector field
find the total acceleration of a particle.
Solution step 2: In a similar manner, the convective acceleration
terms, are
40

Solution step 2: In a similar manner, the
convective acceleration terms, are
41

Example 2. Acceleration field
 An idealized velocity field is given by the formula
 Is this flow field steady or unsteady? Is it two- or three
dimensional? At the point (x, y, z) = (1, 1, 0), compute
the acceleration vector.
Solution
 The flow is unsteady because time t appears explicitly in
the components.
 The flow is three-dimensional because all three velocity
components are nonzero.
 Evaluate, by differentiation, the acceleration vector at (x, y,
z)
= (−1, +1, 0).
42

Example 2.
Acceleration field
43

Exercise 1
 The velocity in a certain two-dimensional flow field is
given by the equation
where the velocity is in m/s when x, y, and t are in meter and
seconds, respectively.
1. Determine expressions for the local and convective
components of acceleration in the x and y directions.
2. What is the magnitude and direction of the velocity and
the acceleration at the point x = y = 2 m at the time t = 0?
44

The Differential Equation of Mass Conservation
 Conservation of mass, often called the continuity relation,
states that the fluid mass cannot change.
 We apply this concept to a very small region. All the basic
differential equations can be derived by considering
either an elemental control volume or an elemental
system.
 We choose an infinitesimal fixed control volume (dx, dy,
dz), as in shown in fig below, and use basic control volume
relations.
 The flow through each side of the element is approximately
one-dimensional, and so the appropriate mass
conservation relation to use here is
45

 The element is so small that the volume integral
simply reduces to a differential term:
The Differential Equation of Mass
Conservation
46

 The mass flow terms occur on all six faces, three inlets and
three outlets.
 Using the field or continuum concept where all fluid
properties are considered to be uniformly varying
functions of time and position, such as ρ= ρ (x, y, z, t).
 Thus, if T is the temperature on the left face of the
element,
the right face will have a slightly different temperature
 For mass conservation, if ρu is known on the left face, the
value of this product on the right face is
47

 Introducing these terms into the main relation
 Simplifying gives
The Differential Equation of Mass
Conservation
48

 enables us to rewrite the equation of continuity in
a compact form
 so that the compact form of the continuity relation
is
 The vector gradient operator
49

Incompressible Flow
 A special case that affords great simplification is
incompressible flow, where the density changes are
regardless of whether the
negligible. Then
flow is steady or unsteady,
 The result
 is valid for steady or unsteady incompressible flow. The
two coordinate forms are
50

 The criterion for incompressible flow is
 where Ma = V/a is the dimensionless Mach number of
the flow.
 For air at standard conditions, a flow can thus be
considered incompressible if the velocity is less than
about 100 m/s.
51

Example 3
 Consider the steady, two-dimensional velocity field given by
 Verify that this flow field is incompressible.
Solution
 Analysis. The flow is two-dimensional, implying no z component of
velocity and no variation of u or v with z.
 The components of velocity in the x and y directions respectively are
 To check if the flow is incompressible, we see if the
incompressible continuity equation is satisfied:
 So we see that the incompressible continuity equation is indeed
satisfied. Hence the flow field is incompressible.
24

Example 4
 Consider the following steady, three-dimensional velocity
field in Cartesian coordinates:
where a, b, c, and d are constants. Under what conditions is
this flow field incompressible?
Solution
Condition for incompressibility:
 Thus to guarantee incompressibility, constants a and c must
satisfy the following relationship:
a = −3c
53

Example 5
 An idealized incompressible flow has the proposed three-
dimensional velocity distribution
 Find the appropriate form of the function f(y) which
satisfies the continuity relation.
 Solution: Simply substitute the given velocity components
into the incompressible continuity equation:
54

Example 6
 For a certain incompressible flow field it is suggested that the
velocity components are given by the equations
Is this a physically possible flow field? Explain.
55

Example 7
 For a certain incompressible, two-dimensional flow
field the velocity component in the y direction is given
by the equation
 Determine the velocity in the x direction so that
the continuity equation is satisfied.
56

Example 8
 The radial velocity component in an incompressible, two
dimensional flow field is
 Determine the corresponding tangential velocity
component, required to satisfy conservation of mass.
Solution.
 The continuity equation for incompressible steady flow
in cylindrical coordinates is given by
58

The Stream Function
 Consider the simple case of incompressible, two-dimensional
flow in the xy-plane.
 The continuity equation in Cartesian coordinates reduces to
(1)
 A clever variable transformation enables us to rewrite this
equation (Eq. 1) in terms of one dependent variable (ψ)
instead of two dependent variables (u and v).
 We define the stream function ψ as
(2)
60

 Substitution of Eq. 2 into Eq. 1 yields
 which is identically satisfied for any smooth function ψ(x, y).
What have we gained by this transformation?
 First, as already mentioned, a single variable (ψ) replaces two
variables (u and v)—once ψ is known, we can generate both u and v
via Eq. 2 and we are guaranteed that the solution satisfies continuity,
Eq. 1.
 Second, it turns out that the stream function has useful
physical significance . Namely, Curves of constant ψ are
streamlines of the flow.
The Stream Function
61

 This is easily proven by
considering a streamline in
the xy-plane
The Stream Function
Curves of constant stream function
represent streamlines of the flow
62

 Along a line of constant ψ we
have dψ = 0 so that
 and, therefore, along a line
of constant ψ
The Stream Function
 The change in the value of ψ as
we move from one point (x, y) to
a nearby point (x + dx, y + dy)
is given by the relationship:
63

 Along a streamline:
 where we have applied Eq. 2, the definition of ψ. Thus along
a
streamline:
 But for any smooth function ψ of two variables x and y, we
know by the chain rule of mathematics that the total change
of ψ from point (x, y) to another point (x + dx, y + dy) some
infinitesimal distance away is
The Stream Function
64

 Total change of ψ:
 By comparing the above two equations we see that dψ = 0
along a streamline;
The Stream Function
65

 and the velocity components,
and
can be related to
the stream function, through the
equations
The Stream Function
 In cylindrical coordinates the continuity equation for
incompressible, plane, two dimensional flow reduces to
66

The Differential Equation of Linear Momentum
 Using the same elemental control volume as in mass
conservation, for which the appropriate form of the
linear momentum relation is
67

 The momentum fluxes occur on all six faces, three
inlets and three outlets.
 Again the element is so small that the volume integral
simply reduces to a derivative term:
68

 A simplification occurs if we split up the term in
brackets as follows:
 The term in brackets on the right-hand side is seen to be
the equation of continuity, which vanishes identically
 Introducing these terms
69

 Thus now we have
 This equation points out that the net force on the control
volume must be of differential size and proportional to
the element volume.
 The long term in parentheses on the right-hand side is the
total acceleration of a particle that instantaneously
occupies the control volume:
70

 These forces are of two types, body forces and surface
forces.
 Body forces are due to external fields (gravity,
magnetism, electric potential) that act on the entire mass
within the element.
 The only body force we shall consider is gravity.
 The gravity force on the differential mass ρ dx dy dz
within
the control volume is
 The surface forces are due to the stresses on the sides of the
control surface. These stresses are the sum of hydrostatic
pressure plus viscous stresses τij that arise from motion with
velocity gradients
71

The Differential Equation of
Linear Momentum
72
Fig. Elemental Cartesian fixed
control volume showing the
surface forces in the x direction
only.

 Splitting into pressure plus viscous stresses
 where dv = dx dy dz.
 Similarly we can derive the y and z forces per unit volume
on the control surface
 The net surface force in the x direction is given by
73

 The net vector surface force can be written
as
The Differential Equation of
Linear Momentum
74

 is the viscous stress tensor acting on the element
 The surface force is thus the sum of the pressure
gradient vector and the divergence of the viscous stress
tensor
 In divergence form
75

 In words
 The basic differential momentum equation for an
infinitesimal element is thus
76

 This is the differential momentum equation in its full glory,
and it is valid for any fluid in any general motion,
particular fluids being characterized by particular viscous
stress terms.
 the component equations are
77

Newtonian Fluid: Navier-Stokes Equations
 For a newtonian fluid, the viscous stresses are
proportional to the element strain rates and the coefficient
of viscosity.
 where μ is the viscosity coefficient
 Substitution gives the differential momentum equation for a
newtonian fluid with constant density and viscosity:
78

 These are the incompressible flow Navier-Stokes
equations named after C. L. M. H. Navier (1785–1836) and
Sir George G. Stokes (1819–1903), who are credited with
their derivation.
Newtonian Fluid: Navier-
Stokes Equations
79

Inviscid Flow
 Shearing stresses develop in a moving fluid because of
the viscosity of the fluid.
 We know that for some common fluids, such as air and
water, the viscosity is small, therefore it seems reasonable
to assume that under some circumstances we may be able
to simply neglect the effect of viscosity (and thus
shearing stresses).
 Flow fields in which the shearing stresses are assumed to
be negligible are said to be inviscid, nonviscous, or
frictionless.
 For fluids in which there are no shearing stresses the
normal stress at a point is independent of direction—that
is σxx = σyy = σzz.
80

Euler’s Equations of Motion
 For an inviscid flow in which all the shearing stresses are
zero and the Euler’s equation of motion is written as
 In vector notation Euler’s equations can be expressed as
Inviscid Flow
81

Vorticity and Irrotationality
 The assumption of zero fluid angular velocity, or
irrotationality, is a very useful simplification.
 Here we show that angular velocity is associated with
the curl of the local velocity vector.
 The differential relations for deformation of a fluid
element can be derived by examining the Fig. below.
 Two fluid lines AB and BC, initially perpendicular at time t,
move and deform so that at t + dt they have slightly
different lengths A’B’ and B’C’ and are slightly off the
perpendicular by angles dα and dβ.
82

Vorticity and
Irrotationality
83

 But from the fig. dα and dβ are each directly related
to velocity derivatives in the limit of small dt:
 Substitution results
 We define the angular velocity ωz about the z axis as the
average rate of counterclockwise turning of the two lines:
84

is thus one-half the curl of
 The vector
the velocity vector
 A vector twice as large is called the vorticity
Vorticity and
Irrotationality
85

 Many flows have negligible or zero vorticity and are
called
irrotational.
 Example. For a certain two-dimensional flow field the
velocity is given by the equation
 Is this flow irrotational?
Solution.
 For the prescribed velocity field
86

Vorticity and
Irrotationality
87

Velocity Potential
 The velocity components of irrotational flow can be
expressed in terms of a scalar function (
ϕ x, y, z, t) as
 where ϕ is called the velocity potential.
 In vector form, it can be written as
 so that for an irrotational flow the velocity is expressible
as the gradient of a scalar function .
ϕ
 The velocity potential is a consequence of the
irrotationality of the flow field, whereas the stream
function is a consequence of conservation of mass
88

Velocity Potential
 It is to be noted, however, that the velocity potential can
be defined for a general three-dimensional flow, whereas
the stream function is restricted to two-dimensional
flows.
 For an incompressible fluid we know from conservation of
mass that
 and therefore for incompressible, irrotational flow (with
) it follows that
89

Velocity
Potential
 This differential equation arises in
many different areas of engineering
and physics and is called
Laplace’s equation. Thus,
inviscid, incompressible,
irrotational flow fields are
governed by Laplace’s equation.
 This type of flow is commonly called
a potential flow.
 Potential flows are irrotational flows.
That is, the vorticity is zero
throughout. If vorticity is present
(e.g., boundary layer, wake), then the
flow cannot be described by Laplace’s
equation.
64

Velocity Potential
 For some problems it will be convenient to use cylindrical
coordinates, r,θ, and z. In this coordinate system the
gradient operator is
91

Example 1
 The two-dimensional flow of a nonviscous, incompressible fluid
in the vicinity of the corner of Fig. is described by the stream
function
 where ψ has units of m2/s when r is in meters. Assume the fluid
density is 103 kg/m3 and the x–y plane is horizontal that is, there
is no difference in elevation between points (1) and (2).
FIND
a) Determine, if possible, the corresponding velocity potential.
b) If the pressure at point (1) on the wall is 30 kPa, what is
the pressure at point (2)?
93

Example 1
Solution
 The radial and tangential velocity components can be
obtained from the stream function as
94

Source and Sink
 Consider a fluid flowing radially outward from a line
through the origin perpendicular to the x–y plane as is
shown in Fig. Let m be the volume rate of flow
emanating from the line (per unit length), and therefore to
satisfy conservation of mass
 or
99

 It follows that
 If m is positive, the flow is radially outward, and the flow
is considered to be a source flow. If m is negative, the flow
is toward the origin, and the flow is considered to be a sink
flow. The flowrate, m, is the strength of the source or sink.
Source and Sink
 A source or sink represents a purely radial flow.
 Since the flow is a purely radial flow, ,
the corresponding velocity potential can be
obtained by integrating the equations
100

 To yield
 The streamlines (lines of ψ = constant ) are radial lines, and
the equipotential lines (lines of = constant) are
ϕ
concentric circles centered at the origin.
Source and Sink
 The stream function for the source can be obtained
by integrating the relationships
101

Example 2
 A nonviscous, incompressible fluid flows between wedge-
shaped walls into a small opening as shown in Fig. The
velocity potential (in ft/s2), which approximately
describes this flow is
 Determine the volume rate of flow (per unit length) into
the
opening.
102

The negative sign indicates that the flow is toward the
opening,
that is, in the negative radial direction
103

Vortex
82
 We next consider a flow field in which the streamlines are
concentric circles—that is, we interchange the velocity
potential and stream function for the source. Thus, let
and
where K is a constant. In this
case the streamlines are
concentric circles with
and
This result indicates that the tangential velocity varies
inversely with the distance from the origin

 A mathematical concept commonly associated with
vortex motion is that of circulation. The circulation, Γ, is
defined as the line integral of the tangential component
of the velocity taken around a closed curve in the flow
field. In equation form, Γ, can be expressed as
83
Circulation
where the integral sign
means that the integration
is taken around a closed
curve, C, in the
counterclockwise
direction, and ds is a
differential length along
the curve

,
 This result indicates that for an irrotational flow the
circulation will generally be zero.
 However, for the free vortex with , the
circulation around the circular path of radius r is
 which shows that the circulation is nonzero.
 However, for irrotational flows the circulation around any path
that does not include a singular point will be zero.
84
Circulation
 For an irrotational flow

Circulation
 The velocity potential and stream function for the free
vortex are commonly expressed in terms of the circulation
as
and
107

Module 2 of fluid machines - Kinematics of fluids

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