Good morning. I welcome you all to this session.
In last class, we were discussing, we have
just started the discussion on continuity
equation, and we derive the continuity equation
in a Cartesian frame of reference. What is
a continuity equation? If we recall, it is
an equation relating to the velocity and density
field in a flow of fluid, which is deduced
from the principle of conservation of mass
applied to a control volume that means, continuity
equation basically signifies the principle
of conservation of mass.
Now, if we recall the equation, you please
see here that in a Cartesian frame of reference,
if you recall the equation, let this is the
Cartesian frame that we discussed x, y, z.
The equation of continuity was like this del
rho del t plus del del x of rho u, u is the
x component velocity, v is the y component
and w is the z component. So, typically this
equation was derived from the principle of
conservation of mass applied to a control
volume with respect to a Cartesian coordinate
system, rectangular Cartesian coordinate system.
This is precisely the equations where u rho
is the density and u v w are the velocity
components.
Now, one thing we can tell from this equation.
That in a flow field, if we get a description
of u v w as a function of x y z and t as a
function of x y z and t, v is also such functions
of x y z and t, w is also such function of
x y z and t and rho is also such functions.
That means, these dependent variables are
hydrodynamic parameters. For example, three
velocity components and density are expressed
as a function of independent variables like
the space coordinates and time. You know these
are the independent variables.
Then, this function must satisfy this equation.
Otherwise, the flow is impossible. Sometimes
we check whether a velocity field and density
field describe a physical flow or possible
flow or not. If they do not satisfy this equation,
means they do not satisfy the conservation
of mass. That means, this is an impossible
situation. This sets of functions can never
represent the velocity field and density field
of the fluid flow. It is most important for
the use of continuity equation. You should
know that.
Now, you see that this continuity equation
can be defined or described with respect to
different coordinate systems depending upon
the geometry of the flow. Now, before doing
that, we can just see that this equation expressed
in Cartesian coordinate system can be written
in a vector form like that, del rho del t
plus this term can be written as, if you recall
your preliminary knowledge in vector, so,
this term is divergence of rho v. rho is a
scalar and v is a vector. So, rho v is a vector.
You know the divergence operator is a vector
operator, which is I del del x. For a Cartesian
coordinate system, this tends like that. I
j k are the unit vector in x y and z direction.
So, we can write the operator del, which is
a vector operator k, just if you brush up
your preliminary knowledge in vector. So,
this operator being used with a dot product,
that is a scalar product with this vector,
where v bar that is the velocity field. You
know this is a vector. So, this has got distinct
three scalar components v and w along x y
and z directions. So, if you multiply these
two vectors, that is del dot rho v dot scalar
multiplication, we simply get this. So, that
you know del u del x plus del v del y plus
del w del z.
So therefore, one way of converting any equation
from one coordinate system to other coordinate
system is to see the equation in one coordinate
system and to find out its vector form, general
vector form. So, if you can do that, then
we can tell that this is the vector form and
we can expand this vector form in different
coordinate system. So that, we can get the
equation in different coordinate system.
So, before coming to that, I should describe
one thing, which probably we described earlier
also. If the flow is incompressible, in the
last class also, if the flow is incompressible
means incompressible flow. If the flow is
incompressible, rho is constant. That means,
density does not change in the flow field
neither with time nor with space coordinates.
What does it mean that del rho del t 0? That
means, in this case, del in this form del
rho del t 0.
Moreover, rho will come out from the differentials
and in this operations, in this vector form
rho will come out. So, ultimately the equations
will be divergence of the velocity vector
in vector form is 0. This is the equation
of continuity for incompressible flow. In
case of Cartesian coordinate, the expansion
of this will be del u del x plus del v del
y plus del w. That means, we can write this
in vector form and then, expand in case of
a Cartesian coordinate del dot v will be del
u del x plus del v del y plus del w del z.
Because v is a vector defined by like that
and del is the or we can straight forward
get it from the equation with respect to Cartesian
coordinate. That is will be 0 and rho will
come out and the common factor which cannot
b 0, so, del u del x plus del v del y plus
del w del z is 0.
This is the equation of continuity for incompressible
flow. What will be the continuity equation
for a steady flow? For a steady flow, please
tell what is the continuity equation for a
steady flow? Which term will be omitted for
a steady flow? Steady flow means that parameters
will not change with time. So, which term
of these two terms del rho del t will be 0?
That means, for a steady flow, this is the
equation of continuity. That means, in Cartesian
coordinate, del del x of rho u plus del del
y of rho; v is a special case for a Cartesian
coordinate system del del z of rho.
Now, one interesting fact you must remember
that this is the continuity equation for a
steady flow of both compressible and incompressible.
So, if we take an additional constant of incompressible
flow along with the steadiness, then rho comes
constant. Then, we get from either of the
two equation divergence v is 0. So, for a
steady flow, this is the continuity equation.
Now, for an incompressible flow you see the
continuity equation is divergence v 0. So
therefore, from an incompressible flow, continuity
equation it is very difficult to infer whether
the flow is steady or unsteady, because even
if the flow is unsteady, but incompressible,
the equation remains same which is the fact,
what is the, why, what is the fact because
the derivative of rho with time only appears.
So, whenever the flow becomes incompressible,
so, it automatically goes because rho has
to be constant for an incompressible flow.
Even if it is unsteady, rho cannot depend
on time. Other parameters will depend on time.
Since the time derivative of no other parameters
appear in the continuity equation, therefore
it is very difficult to judge from the continuity
equation of an incompressible flow, whether
the flow is steady or not. For example, whenever
the flow is incompressible state becomes divergence
v 0; and that means, this is 0.
So, even if the flow is steady, there is no
scope of any further simplification or modification
of this equation because time derivative of
no variable appears in this equation. So therefore,
the continuity equation for both steady and
unsteady incompressible flow is given by divergence
v 0. But for a compressible flow, there is
a change between steady or unsteady equations
in the form of continuity equation. This is
why? Because this is the compressible unsteady
flow and this is the compressible part. That
means, the unsteady flow, compressible unsteady
flow, a steady flow and this is the compressible
unsteady flow.
So, a compressible flow, if it is unsteady,
term will be there. If a compressible flow,
if it is steady, then this term will not be
there. So therefore, we can distinguish the
steady flow or unsteady flow, if the flow
is not incompressible from the continuity
equation. But if the flow is incompressible,
whether the flow is steady or unsteady, the
continuity equation will always be divergence
v is 0. This part is cleared?
Now, next I come to the different coordinate
system. Now let us, first thing that is equation
in general for a incompressible unsteady flow,
this is the divergence of rho v. If the question
comes, what is the cylindrical coordinate?
What is the continuity equation in cylindrical
coordinate system? Continuity equation in
cylindrical coordinate system. So, one way,
for example what is cylindrical coordinate
system? Let us concentrate the cylindrical
coordinate system. Let this x and this y and
this z.
In a cylindrical coordinate system, instead
of x y, we define in x y plane, the point
by a radial location r and Azimuthal coordinate
theta and the z will be same. That is, in
the z direction, this has come from the concept
of a cylinder, geometry of a cylinder. So
that, at any point will be described by a
radius radial vector or radial coordinate
r azimuthal theta and this z. You know these
things. That means, r theta z instead of x
y. These are the coordinates in a cylindrical
polar coordinate system and not simply cylindrical
coordinate system.
In that case, one way of, did you see it?
Mathematically, without going for any other
physical complications, simply expand this
term. But before expanding this term, one
has to know the mathematical expression for
this del in different coordinate system. Probably,
you know, if we again brush up your preliminary
knowledge in vector, this del in Cartesian
coordinate as I have written earlier , that
represents del del y plus k del del z, where
i j k are the unit vector along x y and z
direction.
Similarly, in cylindrical coordinate system,
this will be del del r plus 1 by r. Probably
you know this thing, j 1 upon r del del theta
plus k del 0, where i j k at the unit vectors
along r, along this theta direction j and
along the z direction remains as it is. So,
if you define i j k, the unit vectors along
the coordinate directions r theta and j, this
is the operator. One knows this thing. For
him, it is easy to find out del dot rho v
because v in Cartesian coordinate system,
if you represent v r, let this is v r as the
radial component of velocity and v theta;
that means, this direction, the azimuthal
direction is the v theta tangential component
of velocity or azimuthal component of velocity.
v z is the z component, which we are using
w in case of Cartesian coordinate system.
Now, we are using v z; that means, the velocity
field can be expressed as its components v
theta plus k v z. So, one can find out now,
after knowing this and these two velocity
fields, rho is a scalar function that divergence
of rho v. Well, any questions please? So,
this will be this multiplied with this, that
means, del v r del r. I am sorry. This will
be multiplied with rho. So, del rho v r del
r plus rho v r by r plus 1 upon r del del
theta of rho v theta plus del del j of rho
v. Just simple mathematics. That means, what
I do? I expand this term, this second term
in a cylindrical polar coordinate or simply
cylindrical coordinate system. If I do so,
then I can write the continuity equation,
another additional term; that means, is del
rho del t plus, again I repeat the same thing,
del del r of rho v r plus rho v r by r plus
1 by r del del theta rho v theta plus del
del z rho v z. That means, what I do? del
rho del t, the equation plus divergence of
rho v 0. This is the general vector form.
That means, I have expanded this in cylindrical
coordinate system. That means, therefore,
I can tell precisely this is the continuity
equation in cylindrical coordinate system,
where v r v theta v z are the respective velocity
components corresponding to that coordinate
system. So, this is one way of reducing the
continuity equation in different coordinate
system. We can do it for spherical coordinate
system also. That I left as an exercise to
you. By expanding this term, this is a purely
mathematical exercise. But one can again find
out these from physical concept or from fundamentals.
What I told in the last class is that, if
we have to derive the continuity equation
from the fundamentals, means application of
conservation equation to a control volume,
then the first term is that, you will have
to take the control volume appropriate to
a coordinate system. Now, look into this figure.
Now, if we want to derive the continuity equation
in a cylindrical polar coordinate system,
which is defined by a radial location R azimuthal,
this is the azimuthal theta and Z, then we
will have to consider as volume, which is
parallel in case of a Cartesian coordinate
system. Now, the control volume will be like
this, where the different planes of the control
volume will be parallel to the coordinate
planes.
So, how we will choose it? This one direction
will be r d theta. If this is r and this radius
vector is at an angle d theta, so, this angle
is d theta. Another length will be d r and
this will be definitely d z. That means, a
control volume of dimensions d r d theta r
d theta d z, this type of a fluid element
or a slide of a fluid element have to be considered.
Now, let us give some name, otherwise it is
difficult to A B C D and this one is E F G
H. Now, A B C D E F G H represents the control
volume.
Now, you see we have to represent the mass
flux in different directions. So, there is
velocity in r direction. So therefore, there
is a mass influx. Now, let us consider, due
to the r direction velocity v r, so, velocities
are in r direction, theta direction v theta
, this is the v theta, theta direction, this
is the v r in r direction. This is as usual
v z. Here, we represent v z not as w, v z
direction velocity.
Now, if we concentrate in the similar fashion,
the mass flux in the radial direction which
is across the surface A B C D. This is A B
C D. That means, the r surface which is perpendicular
to the radial direction. So mass influx, if
we consider as m dot r and the mass efflux
from the r plane; that means, this plane E
F G H, there are two r planes; that means,
planes perpendicular to radial direction.
One is A B C D and another is E F G H.
So, because of the existence of the velocity
vector, we are in its usual positive direction.
Mass will come into the control volume across
A B C D surface and mass will leave the control
volume across E F G H surface. So, this will
be m dot r plus d d r of m dot r d r because
this mass flux has changed by this amount
because of a change in d r. In similar fashion,
we write what is the expression of m dot r,
that is the mass flux across this surface.
It is the volume flux times the area. So,
volume flux will be density. Sorry, density
will come for the mass. Let us write the volume
flux will be the velocity times the area.
What is the area of this surface r d theta
and d z? That means, already I get this d
z. Again it is duplication. Does not matter
r d theta d z and rho is multiplied to keep
m dot r. So therefore, what is this m dot
r d r m dot r d r? That means, let us write
it only d r m dot r d r. This will be equal
to d d r of rho v r r. So, d r d theta d z,
I take out.
Now, you see the net mass efflux. Because
of the mass flux across the r surfaces or
r planes; that means, the plane perpendicular
to r, because of these two planes A B C D
and E F G H is this minus this. So, net mass
efflux, I am not writing. I just tell you
net mass efflux from the control volume. Because
of the mass flux is across the r planes, that
is planes perpendicular to r direction; that
means, these two planes A B C D E F G H is
equal to this minus this. That means, m d
d r of m dot d r. That means this quantity.
This is the net mass efflux, because of the
flux across the r planes.
This we can write by taking 1 by r multiply
and then v r, little rearrangement d theta
d r d z. What is r d theta d r d z? Volume.
Very good. So, 1 by r d d r of rho v r r into
d v. In the similar way, we can find out for
mass fluxes across the theta.
That means, m dot theta. That means, theta
direction. That means, the mass flux due to
the planes perpendicular to theta directions;
that means, now we are interested for mass
flux across the planes B F G H and A E H D.
Two parallel theta planes perpendicular to
theta direction.
Now, because of the existence of the which
is the component of the velocity in its usual
positive direction, the mass flux will enter
the control volume across B F G H. Let us
define that m dot theta. Because of the same
reason, the mass flux will go out of the control
volume across this plane or phase A E H D,
which we should write m dot theta. Why because
this is change at this location, because of
a change in r d theta.
So, you can write in terms of d theta, d d
theta because in the theta direction, angular
direction we are doing it m dot theta d theta.
So, what is m dot theta? Please tell me. m
dot theta now, we will be mass into the volume
flow, that is v theta times the d r into d
z. Very good. d r into d z. All right. Now,
what will be this m dot d d theta of m dot
theta d theta? This will be equal to del del
theta is rho v theta d r d z d theta. So,
this can be written as 1 by r del del theta
of rho v theta into 1 by r r d theta d r d
z. That means, this is d v, where d v is the
elemental volume of this control volume. Again,
this is the quantity which represents the
net mass efflux from the control volume due
to the mass fluxes across these two planes.
That is theta planes B F G C and A E H D.
So, this represents simply the net mass flux,
from the mass efflux, from the control volume
due to the mass flux across these two parallel
planes. That is, the planes perpendicular
to theta direction. Similar way, if we see
or if we investigate the mass fluxes through
planes perpendicular to z direction. What
are the planes? A B F E and D C G H. That
means, bottom and top plane in this drawing.
So, due to the existence of the positive direction
velocity v z, in usual positive direction
of the coordinate axis, the mass will come
into the control volume across the phase A
B F E. Similarly, the mass will leave the
control volume across the phase D C G H. Simply
we can write now rho, this will be v z times
r d theta d r. Simply this will be m dot z
plus d d z of m dot z d z as usual. That means,
there is a change in the mass flow because
of a vertical displacement of d z. So, d d
z of m dot z d z can be written as, rather
I can write d d z that is del del z. I am
writing in this fashion though I am writing
d d, but ultimately this is a partial differential.
That is why I am changing from d to del. Does
not matter.
So, this will be equal to del del z. You can
write all these in terms of del, because in
the conception there is no problem. But in
a mathematical notation, this is not a total
differential concept because these are all
partial differential. Because all the quantities
vary with both r theta z and time also. So,
this is an instantaneous picture. So that,
when we make differentiation with j this should
be del only.
However, for the conception there is no problem.
So, del del z of what we can rho v z and r
d theta d r d z will automatically make d
v. This is nothing but the amount, which corresponds
to the net efflux to the control volume due
to the fluxes is across the z planes. That
means, across two planes A B A V and D C G
H. So therefore, we get this is the net efflux
from the control volume due to the mass fluxes
across r planes. Similarly, this is the net
efflux from the control volume due to the
mass fluxes across theta planes and this is
the mass efflux from the control volume due
to mass fluxes across z plane.
So, net mass efflux from the 
control volume will be sum of this. That means,
del del r, if you want to have a look, 1 upon
r del del r rho v r. So, I now write with
this 1 by r del del r rho v r into r. You
can have a look; plus d v, we will take common,
plus this 1 by r del del theta of rho v theta
plus del del z of rho v z into d v plus. If
you recall this is the net mass efflux. So,
continuity equation will be what? Continuity
equation therefore, if you recall the continuity
equation in its statement form that the net
rate of mass efflux from the control volume
plus the rate of change of mass within the
control volume.
So control volume, volume is d v and rho is
its density. So, it is the instantaneous mass
within the control volume. So, rate of change
of mass within the control volume will be
del del t of rho d v plus this quantity. That
means, I am not writing it again d v is equal
to 0. That means this quantity. So, d v will
come out of this del del t because control
volume by definition and d v is fixed. So
therefore, we can write del del t plus, this
in bracket; that means, this term, the entire
thing d v is equal to 0. d v cannot be 0.
It is valid for any volume, any finite volume
of the control volume. So, this part will
be 0.
So therefore, in the similar fashion, we can
write that finally, del rho del t plus, now
I write 1 by r del del r of rho v r into r
plus this probably. You can see that this
I am writing 1 by r del del theta 1 by r del
del theta of rho v theta plus del del z of
rho v z is equal to 0 . So, this is the precisely
the continuity equation. We can write it in
a different form, del rho del t. If we just
expand this, the differentiation v r r, so
we can get we can write del del r of rho v
r plus rho v r by r, by taking rho v r the
first function, and r is the second function.
If we differentiate it, this cooled level
thing, del del theta rho v theta plus.
So, this is the equation we also derive straight
from expanding the vector form of the continuity
equation. So, there is no need always of deriving
it from the fundamental. Just for your conception,
I show you. Because earlier, what we did was
we know this vector form of the continuity
equation. We simply expand this with the idea
or with the knowledge that the del operator
is defined in a cylindrical coordinate system
like this, so that, we can straight away this
in different coordinate systems.
For example, the expression in Cartesian coordinate
system will be del del x of rho u plus del
del y of rho v plus del del z of rho w. Similarly,
for a cylindrical coordinate system, this
will be the expression. But this can also
be derived again from the fundamental. That
means, taking a control volume appropriate
to a coordinate system. For example, they
are cylindrical coordinate system and applying
the law of conservation of mass, considering
all the mass fluxes coming in and coming out
from the control volume across different plane
surfaces, so that, I can or we can derive
the continuity equation.
Similar way, the continuity equation can be
derived in a spherical coordinate system.
Again we will make more complications, because
the control volume will be little complicated
by geometry either by using the fundamental
concept or by expanding these forms, which
is left as an exercise to you.
Now after this, I will go to a very important
concept in a fluid flow, which is the concept
of stream function. What is a stream function,
concept of stream function? Now, we know that
for an incompressible steady flow or we should
not always tell that is incompressible steady
even for an incompressible flow. That means,
even if it is steady or unsteady, the continuity
equation is such that, means, if I have an
incompressible flow. My velocity fields are
given as a function of space coordinates and
t, with scalar components u as a function
of x y z t, v as a function of, if it is a
Cartesian coordinate system. Or, if it is
a cylindrical coordinate, it will be a function
of r theta z. So, I am just describing in
generally one test and claims, that is flow
is incompressible one can tell him. Please
wait. I will check whether your velocity functions
explicitly given in this form satisfies this
particular equation or not. Divergence of
v 0, if it is satisfied, we will tell your
flow field is possible. If it does not satisfied,
then we can tell it is an erroneous flow field.
This velocity function can never define an
incompressible flow.
This is the concept. So therefore, if we come
again to mathematics, the divergence of the
velocity vector is 0 for an incompressible
flow. Let us consider a Cartesian system.
First, simply it is nothing but del u del
x plus del v del y is 0. Always we consider
a two dimensional flow stream. Concept of
stream function is associated only for a two
dimensional flow. You must know, this is not
for a three dimensional flow. In three dimensional
asymmetric flow only, which again reduces
the three dimensional flow in a two dimensional
flow. I will explain it afterwards. Then,
we can define this stream function. Usually
the stream function is defined only for two
dimensional flow.
So, for a two dimensional flow, the expansion
of this term for a Cartesian coordinate system
is like that. Or in other words, in a two
dimensional flow, defining u and v as a function
of x y and t, the continuity equation for
an incompressible flow at any instant is this.
Now, if I define a function psi, which is
a function of x y and t, at any instant t.
If it is an unsteady flow, the variable t
will come. Otherwise, it will be a function
of x and y only. That means, if in a two dimensional
flow field, I defined a function. Think mathematically
first. I defined a function, so that these
functions satisfies this condition u is equal
to del psi del y and v is equal to minus del
psi del x. That means this function is such
whose partial derivative with respect to y,
defines the x component of velocity at a particular
point. If it is a function of time, this will
be a function of time also. Similarly, its
x derivative with respect to x with a negative
sign defines the velocity, y component velocity
at that instant and then, this function is
defined as the stream function.
Now, question comes, why so arbitrarily we
are defining a function such that u becomes
del shy del y and v becomes del shy del. Mathematically
it is understandable. We will define a function
shy, which is a function of x y and t in such
a way, that u and v are defined in terms of
this function in this manner. Then, this function
is called this stream function. But what is
the significance of it?
Let us now see the mathematical significance.
If you defined this way, then if we put this
stream function in continuity equation; that
means, if the continuity equation is now substituted
in terms of the stream function, what we will
get first term? del square shy del x del y.
What we will get in the next term? It is minus
del square shy del y del x. Now, if shy is
a continuous function of x and y, you know
that this order change does not have any difference.
That means, del square shy del x del y is
del square shy del y del x. That means, if
you differentiate shy first, we takes and
then with y or first with y and then with
x, they will be equal if the function is continuous.
Again brushing up your school level mathematics.
So therefore, this is equal to 0. That means,
it is automatically satisfied. That means,
we do not get any extra equation as a continuity
equation, if the flow field is defined in
terms of stream function. Try to understand.
This is a real tough concept. This level,
only by reading books you may not understand
this. That means, instead of defining flow
field in terms of u v, if we define a flow
field in terms of stream function, it automatically
satisfies the continuity equation. Because
if we substitute the stream function, because
stream function is defined this way.
So that, if we substitute this continuity
equation get 0; that means, if we define this
flow field in terms of a stream function,
so, continuity equation is automatically satisfied.
That means, we do not have an extra equation
to satisfy the conservation of mass; that
means, the equation is automatically satisfied.
This is the mathematical implication of stream
function.
Let us see the physical implications of stream
function. Now, before recognizing the physical
implications of stream function, let us consider
one thing that stream function is a function
describing shy as a function of x y. Now,
a change in stream function. One interesting
thing now we like to express. Now, we write
a total change in stream function x y t. It
can be written in mathematical form del shy
del x del shy del t d t plus del shy del x.
This you know d x plus del shy del y d y in
a two dimensional flow.
Now, at any instant if you are interested
that any instant value or for a steady flow,
this term is not coming into picture. So therefore,
d shy is equal to; that means, the change
in the stream function at any instant or in
a simplified manner, we can think of a steady
flow, where the change in the stream function
can be defined like this. It is very simple.
It is no way connected to mechanic fluid.
If there is a function of x y, the change
in the functions shy d shy is the change due
to x and change due to y, del shy del x d
x plus del shy del y d y.
Now, what is the definition of stream function?
This function is not a very arbitrary function.
This is such a function that u is equal to
del shy del y or simply del shy del y is equal
to u and v is equal to minus del shy del x.
So therefore, here I can write minus v d x
plus u d y. What is the value? Can you tell
me along a streamline? Along a stream line,
what is the value of right-hand side? For
a streamline, the equation of streamline we
know. What is the equation of a streamline?
If you recollect, it is d y d x is v by u
because the tangent at that line, at any point
is the direction of the velocity vector. For
a two dimensional case, streamline is d y
d x v by u. So therefore, v d x minus u d
y or u d x minus v or u d y minus v d x is
0 along a streamline.
So therefore, along a streamline, if this
is a streamline this value is 0, which means,
d shy is equal to 0 along a streamline. That
means, the shy function along a streamline,
let us consider this is the x, this is the
y, this is the z and this is a streamline.
So, z concept does not come. It is only x
y concept. Let it is like this and there is
x and y and this is a streamline. So, shy
functions, this is a streamline and this is
a streamline. So, what we get? Along a streamline,
the values of shy are same. That means, shy
is equal to constant along a streamline. So,
this is proved.
So, this is one of the very important conclusions
that along a streamline d shy is 0; that means,
shy is equal to constant. So, shy is equal
to constant along a streamline. So therefore,
in a two dimensional flow field, we can define
streamlines by telling different values of
shy; shy 1, shy 2, shy 3 and shy 4, which
are the constant values because shy is constant
along a streamline. It does not change along
a streamline.
So, a streamline can be specified by giving
a constant value of shy stream function. From
this, we can proceed another step further
regarding the streamline, physical implication
of streamline. Let us consider again x y,
the two dimension and let us consider two
shy 1 and shy 2, the two streamlines, shy
1 and shy 2. Let us consider one point A on
a streamline, any other point A and B on two
streamline. Let us join. Now, let us make;
Let us consider a control volume, whose dimension
in this direction perpendicular to this plane
of the figure or plane of the paper is unit.
Let it be one unit and this length be d s,
that is the length joining two points on two
streamlines defined by stream functions shy
1 and shy 2.
Therefore, this length, this is dy and this
is dx obviously. Now, let us consider the
flow field is such that because of which there
is a flow coming into this control volume
across this surface of length d s and plane
unit dimension in this direction. If there
is a flow coming into the control volume and
if we consider the flow going out from these
two planes, then what is the flow coming out
from this? It is u d y volume flow. Now, I
have told that this concept of stream function,
we are discussing for an incompressible 2D
flow. It is always valid; incompressible 2D
flow. Because from the very beginning, we
have defined that the continuity equation,
which is getting satisfied is del u del x
plus del v del y 0. That means, basically
it is two dimensional and incompressible.
So, in case of incompressible flow, the volume
flow rate instead of mass product, we can
use because rho becomes a constant scale factor.
That means, the volume flow which is coming
to the control volume across these planes
must go out from these two planes. If we consider
through these two planes, the volume the fluid
is going out. So, this is the volume fluoride
across this phase and this is the volume fluoride
across this phase because the area of this
phase is d y into unit distance and area of
this phase is d x into this distance unit.
So therefore, we can write the d q. If I tell
the d q amount of volume flow, q is the nomenclature
for volume flow rate is crossing these planes,
joining a and b must be equal to u d y. Now,
this v d x in this frame of reference, where
y is positive in this way, then this will
be minus v d x because v is always negative
sign. So, minus v d x plus of minus v d x;
that means, u d y minus v d x.
Now, what is u d y minus v d x? This is the
difference d shy between the two points because
if I relate shy 1 shy 2 minus shy 1 as d shy,
it will be del shy del x as we have seen,
d x plus del shy del y dy. That means, u minus
v d x plus u d y. That means, this quantity,
this is equal to d shy; that means, this gives
the difference of stream function between
these two streamlines. So, this is the flow
rate by unit length in the perpendicular direction.
So, this gives the most important physical
conclusion, that the difference in stream
function between two adjacent streamlines,
between two any streamline gives the volume
flow rate within the streamlines per unit
in the normal direction.
There is no volume flow rate across a streamline.
That means, if we consider a control volume
like that, the volume is flowing like this.
So, this is given. So, if we consider two
streamlines and this consists of a stream
two; that means, bounded by two streamlines,
then we can tell the volume flow through this
channels made by two streamlines is given
by the difference of this stream functions
defining these two streamlines, per unit length
or per unit width in the normal direction.
Because if you see the unit, you see that
shy 2 minus shy 1 u d y minus v d x, this
unit is meter square per second. That means,
this is the volume flow rate per unit within
this direction.
So, it is cleared. Therefore, we can tell
that this stream function signifies this physical
role that difference between this stream function
gives the flow rate within the two streamlines.
Well, I think we can conclude here. So, streamlines
is like stream function is constant along
a streamline. Stream function is a function
of x y, such a way that it is derivative with
respect to x and y coordinate, defines the
velocity component in such a way that it automatically
satisfies the equation of continuity and incompressible
two dimensional flow. This is number one.
Number two is the streamline function remains
constant along a streamline. Number three
is the difference between the stream functions
between two streamlines gives the rate of
flow through these two streamlines along the
channel found by two streamlines per unit
width or length in the perpendicular direction.
Thank you.
