Welcome back to the video course on fluid
mechanics. In fluid kinematics, in the last
lecture, we were discussing about the potential
flows. We can define the velocity potential
as: u is equal to del phi by del x, v is equal
to del phi by del y and w is equal to del
phi by del z. We have defined the consequence
rotationality of flow field for 3 dimensional
flows and the potential flow where we have
defined with respect to the rotational flow
fields. Further, we have discussed about the
potential flow, stream function and then we
have derived the Laplace equation which governs
the inviscid incompressible irrigational flow
fields.
In potential flows, we have seen how we can
define a problem and how the boundary conditions
are defined. As we discussed potential flows
which is the theories applicable for inviscid
incompressible irrotational flow fields given
by the Laplace equation and lines of constant
potential is equipotential; the stream function
is also defined and then the lines of constant
function is called stream line. We have also
seen some examples related to the potential
flows. Today, we will discuss about the potential
flow; we will see the basic potential flow;
then, super position of potential flow.
Potential flow is as we have seen it is governed
by the Laplace equation. Potential flow solutions
are always approximate since most of the fluid,
the assumption in potential flow is irrotational,
the viscosity is neglected and hence we are
assuming as frictionless. Potential flow solutions
are always approximates since fluid is assumed
to be frictionless. Exact solutions are got
from the potential flow theory and represents
approximate solutions to real flow problems.
Even we can derive the exact solutions for
potential flows but as far as real fluid flow
is concerned this is an approximation for
the reality or the real fluid problem. The
exact solution obtained in by using potential
flow theory gives only or represents approximate
solutions to the real flow problem. So, the
potential flow which we have seen here is
that we are assuming the flow as potential
but the reality is different; the solutions
which are derived for the potential flow are
just approximation for the real fluid flow.
We have already seen the Laplace equation
in the Cartesian coordinate system and now
for potential flow in the cylindrical coordinate
system, we can describe this del phi as shown
in this slide here: del phi is equal to del
phi by del r er plus 1 by r del phi by del
theta etheta plus del phi by del z ez. In
the cylindrical coordinate system, we are
defining in terms of r, theta and z. We can
use the unit vector er etheta and ez and then
we can represent the radial velocity as del
phi by del r and then the tangential velocity
vtheta we can represent as 1 by r del phi
by del theta and vz is represented as del
phi by del z.
In the cylindrical coordinate system, the
potential flow is represented with respect
to r, theta and z. This can be represented
in terms, here the radial velocity. So, in
a cylindrical coordinate system the parameters
are described as r, theta and z. The velocity
or the parameters can be described in terms,
as we have seen for the Cartesian coordinate
system, here represents the velocity in xyz
as uvw.
Correspondingly, in cylindrical coordinate
system we can represent the radial velocity
vr, the tangential velocity vtheta and the
velocity vz which are defined here as vr is
equal to del phi by del r with respect to
the potential function; then, the tangential
velocity is represent as 1 by r del phi by
del theta and the vz is represented as del
phi by del z. Finally, the velocity can be
represented as vr er; the unit vectors plus
vtheta e theta plus vz ez and the Laplace
equation in the cylindrical coordinate system
can be represented as 1 by r del r of r del
phi by del r plus 1 by r square del square
phi by del theta square plus del square phi
by del z square is equal to 0. So this equation
represents the Laplace equation cylindrical
coordinate system. Some type of problems where
we will be using theta in z coordinate system
or the cylindrical coordinate system we can
use the equation in this particular form.
Now, we have defined the potential function;
we have defined the stream function here.
So here with respect to the stream function.
We can represent the velocity for irrotational
function flow. We have already seen the stream
function can be defined as u is equal to del
psi by del y and the velocity in y direction
is minus del psi by del x, where phi is the
the stream function and then the velocities
here are represented in terms of x and y direction.
This is x this is y and psi is the stream
function. When we come as 2 dimensional flow
with respect to stream function we can write
u is equal to del psi by del y and v is equal
to minus del psi by del x which we have derived
earlier. From the condition of irrotationality
we can write del v by del x is equal to del
v by del y. From the condition of irrotationality
which we have seen earlier by substituting
by omegaz is equal to 0, we have show that
del v by del x is equal to del u by del y.
If you substitute for u and v from these equations
here as shown then we will get del square
psi by del x square plus del square psi by
del y square is equal to 0s. This is another
form of the Laplace equation derived from
the condition of irrotationality and the definition
of the stream function. Finally, we get a
del square psi by del x square plus del square
psi by del y square is equal to 0. This is
again Laplace equation in terms of stream
function.
Depending up on the problem if we can represent
the flow field in terms of potential function
or if we can represent the flow field in terms
of stream function we can write either of
these two equations: if you represent phi
then we can write del square phi is equal
to the 0 for the potential flow problems;
if we can represent the flow field as a stream
function then we can use del square psi is
equal to 0. Both equations are valid for theorems
depending up on potential flow problems depending
up on whether you are using the potential
function or stream function. The stream function
similar to what we have seen in the case of
a potential function, we can represent the
stream function in terms of cylindrical coordinate
system, rtheta and z.
Since z coordinate system is not coming in
the case of stream function or since we are
considering two dimensions so vtheta is equal
to minus del psi by del r; this is the definition
of the tangential velocity with respect to
the stream functions psi. So vtheta is equal
to minus del psi by del r and vr is equal
to 1 by r del psi by del theta. In cylindrical
coordinate system rtheta with respect to stream
function we can write the tangential velocity
as minus del psi by del r and the radial velocity
vr is equal to 1 by r del psi by del theta
and change in value of the stream function
is related to the volume rate of flow.
That means if you draw the stream function
with respect to stream line, if we consider
a flow field like this and then if we can
draw the stream lines like this psi1, psi2
and psi3 extra, the stream in change in value
thing function from one position to another
that represent actually, the volume of ů.
This is psi1 and psi2, then the volume rate
of flow q is equal to between psi2 and psi1.
One we can write the volume rate of flow is
equal to psi2 minus psi1, that is, q1. Similarly,
between this psi2 and psi3 we can write q2
is equal to psi3 minus the stream functions
psi3 minus psi2. Stream function q is the
volume rate of flow can be represented as
the difference between this stream function.
This potential flow with respect to the stream
function we can use to calculate the volume
rate of flow between the fluid flows which
we will be considering. Now, with respect
to this stream function let us consider a
small example.
We can see an example for a fluid flow. With
fluid flow, radial velocity is given as vr
is equal to lambda by r square cos theta and
if vtheta the tangential velocity is equal
to 0, theta is equal to 0. Determine this
two stream functions psi and vtheta for the
fluid flow?
The problem is the data given are in terms
of the cylindrical coordinate system r and
theta; the radial velocity is already given
vr is equal to lambda by r square cos theta
where lambda is constant and then a condition
is given vtheta is equal to 0 and theta is
equal to 0 we have determine the stream functions
psi and then the tangential velocity vtheta
for the fluid flow.
Already given vr is the radial velocity equal
to lambda by r square cos theta. This with
respect to over definition here vr is equal
to 1 by r del psi by del theta, this lambda
by r square cos theta is equal to vr that
is equal to 1 by r del psi by del theta or
we can write del theta is equal to lambda
by r cos theta. Now, we got an expression
for psi with respect to theta as del psi by
del theta equal to lambda by r cos theta.
To get the stream function we can just integrate
this with respect to theta. If we integrate
theta del psi by del theta we get psi equal
to lambda by r sin theta f r. So, this stream
function is obtained as psi is equal to lambda
by r sin theta that is equal to fr. If you
want to determine the tangential velocity
vtheta definition is minus del psi by del
r. Here, we have already derived the stream
function so we can just differentiate with
respect to r to get the tangential velocity.
We differentiate the expression with respect
to r we get vtheta is equal to lambda by r
square sin theta plus f dash r which is constant.
One condition is given that the tangential
velocity vtheta is equal to 0 at theta is
equal to 0. We can apply this condition here.
So, vtheta is equal to 0 so we get fr, f dash
r is equal to 0. Finally, we can obtain vtheta
is equal to lambda by r square sin theta which
is the expression asked in the question.
So have found the stream function as lambda
by r sin theta and we got the tangential velocity
v component vtheta is equal to lambda by r
square sin theta. As shown in this problem
by using these kinds of the polar coordinate
system or cylindrical coordinate system we
can solve this kinds of problem in terms of
the theta, the radial velocity or in times
of vr- the radial velocity or the tangential
velocity vtheta and with respect to r psi
we can determine various functions like vtheta
vr are distinct function. This is about the
representation of the potential flow with
respect to the stream function.
As I mentioned we can represent the streamlines
are the lines of constant psi. We can draw
like this in the figure shown here, Psi1 psi2
psi3, these lines are constant and hence these
lines are called stream lines. We can see
that there is no flow across a stream line,
flow is in the direction of the stream line
once the stream line with respect to stream
lines concept there cannot be a flow across
a stream line. So, flow across this 
stream line is impossible. Since at every
point stream line is tangent to the velocity
we can say that there is no flow across a
stream line. This means that a streamline,
we can consider as a solid boundary so that
flow cannot cross across a streamline. The
streamline concept is very useful to solve
many of the fluid mechanics problems especially
when we can approximate it is next and as
potential flows. The stream lines there cannot
be of any flow across a streamline and then
the streamline can be considered as a solid
boundary where we can consider as a boundary
between two various flow problems. So, streamline
is defined here.
With respect to this streamline and also for
the plane of irrotational flow we have defined
the potential function and stream function.
We have already seen that both satisfy the
Laplace equation. So, in Laplace equation
which we have derived del square phi is equal
to 0, del square psi is equal to 0. So, both
del square phi is equal to 0 function and
both potential function satisfies the for
plane irrotational flow satisfy the Laplace
equation.
Now as per our definition if we use the definition
of the stream function and potential function
we can write its dy by dx along psi is equal
to constant and is equal to v by u, that is,
the lines of constant psi or which we have
discussed the streamlines and then dy by dx
along the potential psi is constant that means
equal to, as per definition, that is equal
to minus u by v, where u is the velocity in
the x direction and u the velocity in the
y direction. These lines are constant where
the potential psi is constant where these
lines are called a equipotential lines. With
respect to the concept of the potential velocity
potential and the stream function we can draw
the stream lines where dy by dx along psi
is equal to constant which is equal to v by
u which are called the stream lines. Then
we can draw lines where the potentials are
constant or dy by dx along phi is equal to
constant as minus v by u. These are lines
of constant potential which are called a equipotential
lines.
This equipotential lines are when we draw
with respect to fluid flow we can see that
these equipotential lines are orthogonal to
the streamline. So if I draw here you can
see that these are the streamlines here. If
you plot the potential lines we can see that
these are orthogonal to the 90 degree angle.
So this is phi1, phi2 and phi3, like that
phi4. We can draw the potential equipotential
lines and then the streamlines already drawn
here. We can see equipotential lines are orthogonal
to the streamlines at all point where they
intersect or equipotential lines intersect
with respect to the streamlines at 90 degree
or they are orthogonal since the product of
slope is minus 1. When we draw the streamlines
and then when we draw the equipotential lines
these lines together consists a family of
streamlines and equipotential lines called
flow net. This flow net concept is very useful
in many of the fluid flow problems. So, flow
net consists of family of streamlines and
equipotential lines.
We have seen how to draw streamlines with
respect to the direction of the fluid flow
and we have also draw the equipotential lines;
together they are orthogonal or the product
of slope is minus 1 and then the family occurs
with respect to streamlines and equipotential
lines from the flow net.
So, here you can see the flow net. Flow net
consists of streamlines and equipotential
lines; the streamlines are drawn here just
like this red color and yellow color and equipotential
lines phi is constant is also drawn. So they
form the flow net. This flow net concept is
very useful in many of the flow net.
This flow net we can use for visualization
of flow patterns as shown here. It can be
also used to get graphical solution for fluid
flow problems. They can also determine the
velocity and discharge can be obtained from
the flow net since velocity is inversely proportional
to streamline spacing. Here this figure shows
a flow net for a fluid flow in a bend so there
is a flow through a bend.
The streamlines are plotted here like this
and then equipotential lines are also plotted.
If we can draw the streamlines and then equipotential
lines finally get the flow net we can get
a visualization of the pattern as per the
fluid flow is concerned especially for potential
flows or the flows which we can the approximate
as potential flow. Many of the problems like
flow through an earth dam we can use this
concept.
For example, you consider an earth dam like
this. In this earth dam, let us assume that
it is impermeable. Here, if the head or the
potential is 10 meter and here is 2 meter
you can see that there is fluid surface and
then with respect to this for the earth dam
problem. We can draw the streamlines for this
and then correspondingly we can plot the equipotential
lines which give the flow net. The flow net
for an earth dam is drawn here. With respect
to this, once the flow net is drawn we can
use this flow net pattern to calculate the
velocities or discharge since the velocity
is inversely proportional to this streamline
spacing so this concept of the streamlines
or the flow net with respect to streamline
and equipotential lines. There are large numbers
of applications like in earth dam so flow
through a bend, wherever the flow problems
can be approximated as a potential flow we
can use the concept of the flow net.
Now, we also consider here another problem
with respect to a flow beneath a concrete
dam. Here, in this slide we can see there
is a concrete dam; this is the dam position
and then we are considering a domain 60 meter
length, this is 0,0 then 60,0; the depth is
60 by 15 and here 0,20. We are now considering
there is a concrete dam on a permeable foundation
like this. You can see that here the flow
condition, the boundary condition here it
is 5 meter depth and here it is the downstream
end; it is 5 meter. There is a level difference,
here the boundary conditions and here the
potential if we consider with respect to this
line as datum, you can see that there is a
potential meter of 10 meter. So on the upstream
side, the flow comes and the potential function
phi is equal to 10 meter and downstream side
this is the water level. The downstream side
phi is equal to 5 meter and since we are considering
this boundary at the bottom as impermeable
we can say that del phi by del n that means
no flux can cross this boundary. So, del phi
by del n is equal to 0; here also del phi
by del n is equal to 0 and this side also
del phi by del n is equal to 0. Also we can
assume that stream functions psi is equal
to 0, psi is equal to 0 here and here also
psi is equal to 0. Then, similarly, here there
is a concrete bed and the stream function
here, let us assume psi is equal to 10 and
on this also del phi by del n is concrete
dam; then in upstream there is a concrete
bed; in downstream also there is a concrete
bed. So del phi by del n is equal to 0 on
this phase also. With respect to this problem
we can solve now. The equations of the del
square phi is equal to 0 we can solve and
del square psi is equal to 0. The Laplace
equation in terms of phi the potential function
and then the Laplace equation terms of stream
function also we can solve. In this particular
domain, the boundary conditions are given
here and then we can determine the phi and
psi at various points like this, at various
points we can find the potential function
and stream functions. Then we can interpolate
between two to get the stream lines and the
potential lines like in the figure.
So here we have drawn with respect to the
problem given here and the boundary conditions
and then the use the Laplace equation in terms
of phi and then use the Laplace equation in
terms of psi del square phi is equal to 0
and del square psi is equal to 0.With respect
to this boundary conditions we can get the
potential function and the stream function
in various locations and finally we can call
this stream lines like this.
You can see here these stream lines are plotted
like this and then the potential functions
are also plotted. So, a flow net is formed
here for the flow beneath the concrete dam
and this can be used to calculate how much
will be the discharge. These are the equipotential
lines and these are stream lines. So, finally,
we get a flow net for this particular problem.
This can be used to find the discharge or
the flow which can go through the pores media
from this place to this place and then finally
it will exist at this place. This can be used
to find the velocity; these are discharge
between with respect to the potential flow
equations and potential flow theories.
Potential flow theory has got lag in applications
as described in the earth dam here or the
flow beneath concrete dam as described here.
Further this will be this aspect will be discussed
later. Now, after the flow net we will see
some of the basic potential flows. The potential
flow where the applications directly we can
apply Laplace equation and where the simple
flow surface and we can call some basic potential
flows like uniform flow, like a source since
and double x. Since the potential flows are
governed by the Laplace equation which is
a linear equation we can have number of particular
solution and with this particular solution
can be added to yield solution for complex
problem.
If we have a problem like where the potential
functions for example potential function phi1
phi2 phi3 extra are known then since the Laplace
equation which we are considered del square
phi is equal to 0 or del square psi is equal
to 0; this is the linear form of the linear
equation. We can superpose or we can add to
yield the solution of complex problem. If
phi1 phi2 phi3 extra the solution obtained
then for various problem we can superpose
the problems to the elementary flows like
uniform flow, like a flow with source and
sink and then we can superpose whether to
get complex flow problem. This concept we
can use to solve many of the complex problem
which can be kindly approximated with respect
to the potential theories. This we will discuss
before going to the complex problems. You
will see the basic potential flows.
As mentioned elementary flows are uniform
flow, source, sink and the vortex. These are
the elementary flows which we consider in
the potential flows theory, uniform flow source
and sink and vortex so each one of this discuss
in detail. Now we will discuss the basic potential
flows.
First, we will discuss the uniform flow. This
is the simplest plane flow described by either
stream function or velocity potential. Some
of the flows like the ground water flow without
the pumping or velocities very low then it
can be approximated as uniform flow sometimes
depending up on the flow conditions. The uniform
flow concept becomes most simple or simplest
plane flow where there is no complexity; there
are no sources or nothing.
In the uniform flow we can approximate using
potential flow theory. This we can either
represent using the stream function equation
or the potential equation and the stream lines
are straight and parallel as far as the potential
flow is concerned and the magnitude of the
velocities is constant.
Here, we can see that it can be horizontal
or implant but as you can see here the stream
lines are straight and parallel so that is
the peculiarity of uniform flow and magnitude
of velocity is constant. Here you can see
the psi is equal to psi1 or psi is equal to
psi2 and then the streamlines are psi is equal
to psi1 or psi is equal to psi2 or psi is
equal to psi3 or psi is equal to psi4 like
that. We can represent the uniform flow with
equipotential lines and streamlines are drawn
as shown in this uniform flow or it can be
implanted like with respect to an angle alpha.
Here also phi1 and phi2 represent the equipotential
lines and psi1 psi2 psi3 psi4 represent the
streamline.
With respect to this now we will use the potential
flow theory. As we have represented the velocity
in the x direction it can be represented as
u is equal to del phi by del x and the for
this particular problem del phi by del y is
equal to 0 for the case of the uniform flow.
So, this is the flow in one direction; we
can represent del phi by del x is equal to
u and then we can write phi is equal to ux
plus c, where c is an arbitrary constant and
this can be said to 0. So that for the uniform
flow we can write phi is equal to Ux, where
U is the velocity the x direction, phi is
equal to Ux plus c and c is an arbitrary constant
which we said to 0, phi is equal to Ux. So
that we can write now del psi by del y is
equal to u as per the definition of the stream
function; del psi by del y is equal to 0 and
also the uniform flow del psi by del x is
equal to 0. Since we assume that v is equal
to as per the definition of the uniform flow,
the stream lines are straight and parallel
and the magnitude of the velocity is constant.
So, with respect to this we can write that
phi is equal to Ux and then psi is equal to
Uy. So that del phi by del x is equal to U
and del psi by del y is equal to U. Finally,
we get the expression for the potential as
phi is equal to Ux plus c and the expression
for Ux plus c, we said c is equal to 0, we
get phi is equal to U into x and psi is equal
to U into y. This represents the uniform flow.
These results can be generalized to provide
the velocity potential and stream function
for a uniform flow at an angle alpha with
the x axis. As shown in the previous slide,
these lines are present: phi is equal to Ux
and psi is equal to Uy and if you consider
certain angle as shown here then we can represent
the psi and phi like this. For phi is equal
to U into x cos alpha plus y sin alpha and
psi is equal to U into y cos alpha minus y
x sin alpha So the results are not generalized,
provided the velocity potential and stream
function are uniform flow pattern angle alpha.
This is called uniform flow and the uniform
flow represented with respect to the potential
function and then the stream function as phi
is equal to Ux and psi is equal to Uy as shown
here. phi is equal to Ux and psi is equal
to Uy for the horizontal type of flow like
this and for inclined type flow it is phi
is equal to U into x cos alpha where alpha
is the angle plus y sin alpha and psi is equal
to Y into y cos alpha minus x sin alpha, this
is about the uniform potential flow with respect
to the uniform flow which we have discussed.
This is the simplest plane flow. As discussed
here uniform flow is the simplest flow described
by either stream function and we have seen
the stream function and velocity potential.
Now, the second kind of the basic or elementary
potential flow which we represent is called
the source and sinks; the source and sink
is purely radial flow type. So the fluid flow
is radialy outward from the origin perpendicular
to the xy plane only we have to consider the
vr- the radial velocity here and psi is equal
to constant. This line represents psi is equal
to constant and this dash line represents
the phi is equal to constant.
If we consider the flow with respect to an
angle theta, we can derive various values
of the potential function and the stream function.
Let us consider m as the volume rate of flow
emanating from the line per unit length as
shown in this figure. If m is the volume rate
of flow emanating from the line per unit length,
if we use the conservation of mass we can
write this as 2phi r into vr is equal to m.
With respect to this figure, we can write
m is equal 2phi r into vr, r is shown here.
If we consider the radial distance which we
consider here 2phi r into vr is equal to m
where vr is the radial velocity. Finally,
for the considered source sink we can write
the volume rate of flow emanating from the
line per unit length vr is equal to m divided
by 2phi r. So with respect to this conservation
of mass we can write vr- radio velocity is
equal to m by 2phi r and for the potential
flow, the tangential velocity is equal to
0. As defined here it is purely radial flow;
there is no tangential component for the velocity.
So, the radial velocity is equal to m divided
by 2phi r and Vtheta is equal to 0. So that
we can write del phi by del r which is the
radial velocity component del phi by del r
is equal to m by 2phi r. Finally, we can get
an expression for phi. The potential function,
phi is defined as phi is equal to m divided
by 2phi by r. The integration of expression
del phi by del r is equal to m by 2phi by
r. We get an expression for the velocity potential
phi as 2phi by r. So this represent as far
the source flow is concerned as shown in this
figure the potential function phi is represented
as m is equal to m divided by 2phi natural
log r, where m is the volume rate of flow
emanating from the line per unit length as
shown in this figure. So, if volume rate of
flow, 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.
This is either with respect to the domain
which we are considering what is coming and
what is going out; with respect to that particular
point we can define if m is positive then
it can be the source flow and if m is negative
it can be the sink flow. So the flow rate,
m is called the strength of the source or
sinks. We have considered the source or sink.
As shown here whether it can be coming to
the domain or going out of the domain. It
can be a sink or a source depending up on
the case whether m is negative or positive,
where m is defined as the volume rate of flow
emanating from the line per unit length. This
m is called with respect to the potential
function phi is equal to m by 2phi natural
of r. Here, this m is called the strength
of the source or sink. With respect to stream
function, we can define this in polar coordinate
system in cylindrical coordinate system define
theta is equal to the tangential velocity.
vtheta is equal to minus del psi by del r
and the radial velocity vr is equal to 1 by
r del psi by del theta.
With respect to this for the source or sink
it is purely radial flow we can that vtheta
is equal to 0 so that here del psi by del
r is equal to 0 or now find the V r is equal
to one by r del psi by del theta so this is
the radial velocity which we have already
seen so this is equal to m by 2phi r so which
will give the stream function is equal to
m by 2 phi theta.
So finally, we can derive here del psi by
del theta. This we can integrate that we will
get psi is equal to m by 2 phi theta, this
gives the stream function. For a source or
sink type flow which is described here, source
or sink which are purely radial flow we have
derived the velocity potential as m by 2phi
natural log r and then the stream function
psi is equal to m divided by 2phi into theta,
where m is the strength of the source or sink
which we considered. Now we got the stream
function and the potential function.
As far as source and sink is concerned which
we represent, the stream lines are just radial
lines.
The source the stream lines are radial lines
and the equipotential lines are concentric
circles centered at the origin as shown in
figure. So point source or sink is a point
of singularity in the flow field. Since the
radial velocity, r is defined like this is
the origin so from origin r tends to 0 the
sources strength is put on the origin. So,
when r tends to 0 we can see that the radial
velocity Vr radial velocity v will be tending
to become infinity as per the definition.
We can see singularity may occur at the point
source of singularity so Vr is defined as
m by 2phi r, when r tends to 0 as shown here
you can see that Vr becomes Vr tending to
infinity. Then source or sink point source
become a singularity, the point of singularity
where r is tending to 0. Vr becomes infinity
which is called a singular point or sink is
we can represent as a singularity. So this
concept of source and sink on applications
especially we consider the point media flow
so sometimes there are not much complexity.
For the flow in media is homogeneous, isotopic
cases we can consider. If there is a recharge
well or there is a plumbing well application
of the potential theory we can approximate
and then we will try to solve the problem
with respect to the pores media flow. These
are some of the applications which will be
discussing later. With respect to these now
point sources is singularity as discussed
and now the third one is so called the vortex
flow.
The vortex flow: it is the flow in which the
streamlines are concentric circle. First we
discussed the uniform flow; second one, we
discussed the source and sink and third type
of basic or elementary potential flow is called
the vortex flow. The vortex flow is the flow
in which the streamlines are concentric circles
and the velocity along each streamline is
inversely proportional to the distance from
the center. This kind of flow is called a
vortex flow and the vortex motion can be either
rotational or irrotational.
This represents the rotational type which
we have discussed here; the velocity, the
flow, streamlines are concentric circles and
velocity along each streamline is inversely
proportional to the distance as shown in this
figure. So, phi constant is the dotted lines
here and these lines represent the psi constant
lines.
Then we can define the potential function
with respect to as phi is equal to K theta,
where K is a constant, theta is show in this
figure. So phi is equal to K theta and psi
can be represented as psi is equal to minus
K natural log r. Finally, in this case, vortex
flows Vr is equal to the radial velocity component
is equal to 0 and vtheta is defined as 1 by
r del phi by del theta which is equal to minus
del psi by del r. This is equal to vtheta
is equal to minus K by r. For vortex flow
we define phi is equal to K theta; the potential
function for vortex flow is defined as phi
is equal to K theta and psi is equal to minus
K natural log r, where K is the constant.
If the fluid were rotating as a rigid body
has already shown earlier vtheta is equal
to K1r, where K1 is a constant.
This type of vortex motion is rotation and
cannot be discussed with respect to velocity
potential; the vortex flow can be either rotational
or irrotational.
We call the rotational vertex as forced vortex,
for example, the motion of a fluid or the
motion of a liquid contained in a tank that
is rotated about its axis with angular velocity
omega. This is called a rotational vortex.
Then irrotational vortex, free vortex, for
example the swirling motion of the water as
it drains from a bathtub. This represents
the irrigational vortex.
The irrotational vortex is called the free
vortex and for example swirling motion of
the water as it drain from a bathtub is called
irrotational vortex and rotational vortex
is called the forced vortex so the motion
of the fluid contained in a tank that is rotated
about its axis with angular velocity omega
so this is called rotational vortex and also
we can have combined vortex. The combined
vortex is the forced vortex as a central core
and a velocity distribution corresponding
to that of a free vortex outside the core.
We can write vtheta is equal to omega into
r, where r is less than are equal to r0 and
vtheta is equal to K by r as represented earlier;
vtheta is the tangential velocity and K is
the constant which we considered. The vortex
flow can be rotational vortex which is called
forced vortex or it can be irrotational vortex
called free vortex or we can also have combined
vortex which is called forced vortex as defined.
So this is about the vortex flows.
Further, we will be discussing about the circulation
with respect to the vortex flows and then
we will be discussing the doublet; then the
combination of all the elementary basic flows
to represent complex flow system which can
be in certain places we can apply for the
real fluid flow problem. So, this will be
discussing in the next lecture.
