Welcome back to the video course on fluid mechanics. In this lecture, today we will
discuss one of the important topics on laminar and turbulent flow. The main objectives of
what we are going to study in this chapter: we will be checking how the forces due to
momentum changes on the fluid and viscous forces and how to compare? The main effect
of viscosity will be how the flow parameter is changing? What are the changes that take
place? How the fluid will be behaving with respect to the viscosity? How the system will
be the laminar or turbulent? We will be checking that.
We will be looking into various types of flow. We will be checking one of the important aspects
called Reynolds experiments. Where Reynold was born? Where did he utilize these experiments
to classify the flow as laminar turbulent and the various cases? Then, we will be discussing
the laminar flow between parallel plates; we will be discussing laminar flow in pipes.
Also, we will be discussing the turbulent flow in pipes. So, these are the main objectives
which we will be studying or which will be discussing with respect to this particular
chapter on laminar and turbulent flow. If you see the nature, we can see that the
flow can be most the time either laminar in nature or it can be turbulent in nature.
So, here we can see this. If we consider a river we can see that some parts of the river,
the flow is called comment; it is flowing in a very slow pase and we can see that the
flow is in laminar type or it is in flow is in layers but you can see some parts where
we can see some gates are there with respect to this bridge here. We can see that some
disturbances are created and then we can see that the laminar type of flow in this river
is transforming to turbulent flow or we can see whenever a boat is moving just like here,
that is, just behind the boat the flow is totally turbulent or totally disturbed. That
means here the same river at some parts the flow may be laminar and in some parts the
flow may be turbulent. Similarly, this is the case as far as open channel is concerned
or river is concerned; you can see the same in a pipe flow also. So in this photographic
we can see that whenever we open the pipe very slowly, when the valve is opened slowly
we can see that flow will be coming just like in layer type or laminar type and when you
are opening more, when the discharge is more, when the velocity is more we can see that
lot of mixing takes place and the flow become turbulent. We will see this again how the
flow can be laminar or turbulent in an experiment which we conducted in our hydraulics lab
So here we can see one of the experiments which we did in our hydraulics lab where some
pump test is done. Here, we can see one open channel which is coming. The flow is initially
laminar; then, there are some pumps and due to the pumping effect the flow velocity of
the discharge variation takes place and the velocity is increased. You can see that the
laminar flow at some part of the channel becomes turbulent. You can see that here, the flow
becomes very much disrupt. Then the laminar flow is transformed in to turbulent flow.
Like this we can see many places where initially the flow may be laminar; sometimes it may
be turbulent.
With respect to this, we will be discussing further the important parameters which decide
the flow is laminar or whether the flow is turbulent. How we can classify whether the
flow is laminar or turbulent and then we will be discussing further theories related to
the laminar and turbulent flow. Now, the same experiment which we saw in the video show
we can see here
Here, this is laminar flow and here due to the pumping effect each second becoming turbulent.
Here, some places is laminar, some place it is in transitional stage. You can see the
flow becomes turbulent. Now, with respect to the various theories on laminar and turbulent
flow we can see that most of the times we are dealing with real fluids. Real fluids
are concerned varies of course viscosity of the fluid is very important parameter and
viscous effects play a major role.
You can see that due to the viscous effect, as shown in this slide here, we can see there
will be no slip condition on the boundaries and then we can see there the flow before
becoming fully developed due to the boundary layer effects and viscosity effects. Some
parts initially the flow can be laminar and then after sometimes depending upon the flow
condition it can become turbulent. So, viscous effect is very important in fluid mechanics
in the case of real fluids.
With respect to this we can see that due to the viscosity or the viscous effects the fluid
tends to stick to the solid surface. In the previous slide, you can see due to the viscous
effects or viscosity the fluids tending to stick to this solid surface in the both sides.
Due to this sticking effect or no slip condition the boundary layer is created. The flow pattern
can change whenever flowing fluid is interacting with the solid surface. Similarly, we can
see the stresses also plays major role as far as various flow is concerned. When we
are classifying the flow into laminar or turbulent the stresses are also very important; shear
stress, normal stress and within the fluid how it will play is also very important. While
discussing this laminar turbulent flow, another important aspect is the Newton’s law of
viscosity. We have already seen in our introductory lectures that Newton’s law of viscosity
says that the shear stress is proportional to the velocity gradient. So, tow is proportional
to du by dy. With respect to the dynamic coefficient of viscosity we can write tow is equal to
mu du by dy. This is the Newton’s law of viscosity; the shear stress is equal to dynamic
coefficient viscosity multiplied by the velocity gradient.
This Newton’s law of viscosity is also very importance when we discuss the laminar and
turbulent flow. Now, we are discussing the laminar turbulent flow. With respect to this
it will be interesting to see the earlier classification which we have discussed in
the introductory lectures, say, the types of fluid flow.
We discussed in earlier lectures the fluid flow can be classified with respect to the
Rheological consideration like liquids and gases; Dilational consideration like which
is compressible or incompressible; then temporal variations with respect to time, how the fluid
is whether it is transient or steady state; and, spatial dimensions whether one dimensions
or two dimensions, that is, whether the flow which we are considering is one dimensions,
two dimensions or three dimensions. Also, the motion characteristics we have seen that
whether it is a laminar or turbulent. These types of fluid flow we have discussed earlier.
With respect to this, we are going to discuss in detail about the laminar and turbulent
flow; the various aspects as what kind of fluid we are dealing; what kind of the motion
characteristics or whether the spatial dimensions which we are considering or whether the temporal
variations. All these are very importance when we discuss whether the fluid is laminar
or turbulent.
Accordingly, this chat we have seen earlier in our introductive lecture. We have seen
that fluid flow can be classified as whether gas or liquid; with respect to the dilatational
characteristics whether the fluid flow is compressible or incompressible; or with respect
to temporal variations whether it is transient or steady stage; with respect to the viscous
effect whether it is viscosity considerate or in viscous or non-viscous fluid which we
are considering and also the rotational flow or irrigational flow. Especially, in turbulent
flow rotational effect is very important and then also the dimensions which we consider
whether one dimension, two dimensions or three dimensions. This type of fluid flow with respect
to what we are discussing now laminar or turbulent flow, many of these parameters are very important.
Before going in more details, the laminar and turbulent flow types of fluid flow also
can be classified according to the geometry of flow field.
Most of the flow with respect to the geometry, we can classify whether the fluid has the
internal flow or external flow. Now, internal flow means it involves flow in a boundary
region so just like in a pipe flow. We can see there is a pipe flow; the flow
is taking place in this way. We can see here the internal flow is in a pipe is the reason
is we can see that the flow is confined within this boundary of the pipe. That is why we
say that it is internal flow.
The approach which we will be analyzing in this kind of flow will be different just what
we are discussing so called external flow. The pipe flow is an example of the internal
flow. Now, other classification based upon the geometry is the external flow. So, here
the fluids flow in an unbounded region. We are considering the fluid flow in an unbounded
region so it can be just like in a flow pattern around a body immersed in the fluid.
For example, a building is concerned and if this is the building which we consider, with
respect to this if the wind comes what we can see with respect to surrounding the building.
This is very important for a many engineering design. These kinds of flow where the flow
happen with respect to the surrounding of a particular body or particular structure
or this kind of flow where the fluid flow is in an unbounded region. That is called
external flow. This is also very important when we are discussing the laminar and turbulent
flow. The real fluid motion significantly influenced by the boundary. Either it can
be an internal flow or it can be an external flow. As in the case a pipe flow we can see
that the solid surface where the fluid is having a no slip conditions, the boundary
is very important. So the no slip condition and the boundary layer formation are very
important as far as flow is concerned. Whether it is internal flow or the external flow,
where surrounding a building or whenever a boat is moving in a river we can say that
with respect to that surrounding the boat or surrounding the ship what is happening
that is very important. The formation of the boundary layer is also very important. With
respect to this laminar and turbulent flow which we will be discussing today whether
it is external flow or internal flow, that is also very important.
Now, as far as internal flow which we discussed is concerned, we can see that here is a pipe
flow. The boundary effects are likely to extend throughout the entire flow as we can see here
in this slide. So, we say here a fully developed velocity profile is formed and then after
some time the boundary effects is still there but still it has overcome. Then, the boundary
effects are likely to extend through the entire flow. This is again the internal flow.
As far as external flow is concerned we can see that the frictional effects are confined
to the boundary layer next to the body; they drag or lift is more important. Here, when
we will discussed, for example, whenever a balloon is just lifting through the air you
can see that with respect to lift effect or the drag effect what happens. That is also
very important as far as the type of fluid flow is concerned. In the case of an external
flow as far as the laminar and turbulent flow is concerned we have to see that whenever
we discuss these kinds of problems, we have to see whether the flow is external flow or
internal flow.
Now, for the pipe of free flowing water if we inject a dye into the middle of a stream,
you can see just if we inject for example, a dye to the middle of the pipe flow, you
can see that with respect to the velocity of the flow in the pipe so initially we can
see that the fluid velocity is small.
Then you can see that the dye which we are injecting it will be just moving like a filament
of dye in layers. We can say now this is laminar flow where the flow is in layers and then
there is no disturbance takes place. We can see that the velocity of the flow increases
then we can see that after some time here in the second figure, we can see that there
is slight disturbance starts. Earlier, the flow was in layers and then slight disturbance
started and finally we can see when the velocity increase we can see that the dye is totally
mixing within the fluid. Then due to a lot of disturbance we can see the fluids are intermixing
and then we can see that they will be diluted totally mixing with the fluid inside the pipe.
We can see initially when the velocity is small we are injecting small mass of filament
of dye. We can see it is laminar and when the velocity is increased we can see it becomes
transitional flow and then finally lot of mixing takes place and we can see that it
becomes turbulent flow. In a similar we can see for the flow sometime can be laminar but
in the flow parameters like velocity changes then it becomes transitional. Further, we
can see that it will become turbulent. This is the way depending upon various flow parameters
the fluid flow changes from laminar to turbulent. With respect to what we discussed we can say
what is a laminar flow? The laminar flow we can see that the fluid moves in layers or
laminas. We can define the laminar flow as fluid moves in layers or laminas, one layer
gliding smoothly over an adjacent with only molecule interchange of momentum. With respect
to the first figure we can see the fluid moves in layers or laminas; one layer is gliding
smoothly over an adjacent with only molecule interchange of momentum takes place.
This is the definition of a laminar flow but then we can see that as I mentioned when the
velocity or other flow parameter changes then small kinds of mixing takes place instead
of smooth gliding. We can see that there is mixing starts and then the flow is transitional.
As far as turbulent flow is concerned, we can see here there are erratic motion fluid
particles with a violent transverse interchange of momentum. The turbulent flow we can define
as erratic motion of fluid particles with a violent transverse interchange of momentum.
So, here we can see whenever we injected the dye, when the velocity is increased or other
flow parameters changed, we can see that there is total mixing of the dye takes place within
the fluid and then the flow become total turbulent.
We can define the turbulent flow as wherever there is erratic motion of fluid particles
with a violent transverse interchange of momentum. Now, the question here is if a fluid flow
is there how we can define whether it is at laminar stage or whether it is at transitional
stage or whether it is in turbulent stage? This is a very important question. Most of
the time we will be dealing with pipe flow or we will be dealing with open channel flow
or river flow or many kinds of flow. It is very important to see that whether the fluid
flow is laminar or whether the flow is transitional stage or it is turbulent. The reason is that
most of theory whatever we are applying to laminar flow may not be applicable to turbulent
flow. So, many flow parameters have to be changed. The theory will be changing when
the flow is changing from laminar to turbulent.
In all the cases, it is in most of the fluid flow cases it is very important to define
whether the existing flow is in laminar stage or whether it is just in the transitional
stage or it is turbulent stage. A large number of scientists and engineers worked on this
topic and then in the 19th century, say, in 80s one famous scientist was born. Reynolds
defined the number called Reynolds number. After this, Reynolds introduced this dimensional
number called Reynolds number. With respect to this Reynolds number we can define whether
the concern flow is laminar or turbulent. This Reynolds number is to define whether
the flow is laminar turbulent. It shows the nature of flow, whether the flow is laminar
turbulent and then it says relative position along a scale. With respect to the dimensionless
number so called Reynolds number, defined by Osborne Reynolds, we can put scale whether
with respect to this scale: if the Reynolds number is 500 or 1000 or 2000, then it is
laminar as in the case of pipe flow; when it exceeds above 1000 it become turbulent.
For example, in open channel flow it is less than or equal to 500 whether it is laminar;
when it exceeds 600 or 1000 it become turbulent.
So, this Reynolds number is a scale relative which gives the relative position whether
the flow is laminar or turbulent. Now, Osborne Reynolds defines this Reynolds number based
upon the large number of experiments with respect to fluid movement. He conducted experiment
called Reynolds experiment and he conduct this experiment in 1880’s, end of 19th century
and then he define this so called a Reynolds number.
This Reynolds experiment we can see here. What Osborne Reynolds did is there is a small
tank where water is filled and then a pipe is connected with respect to that tank; there
is a valve attached at the end of the pipe; he introduces a small tank inside this large
tank where some dye has been put and then through a small nozzle he introduced the dye
to the pipe where the pipe starts like in this figure. And then what he did? With respect
to valve opening, he started slowly to open this valve and you can see as shown here.
The flow will be with respect to the dye and then the fluid movement here we can see that
the fluid movement is now with respect to dye; we can see it is laminar in nature.
With respect to what we have seen here, with respect to this color we can identify. It
is initially whenever the valve opening is small the velocity is small and discharge
is also small. Then, we can see the flow is with respect to dye. We can visualize the
fluid is moving in layers; they are in laminas.
When the valve is opened more we can see that this fluid which is moving in laminas or in
layers some disturbance starts. We can see that there is some oscillation for the fluid
and further when the valve is opened we can see that the velocity increases and then with
respect to dye movement we can visualize that there is mixing taking place; interchange
of masses takes place; there is erratic motions taking place. Finally, as we have seen here
the flow becomes turbulent. So, that is what Osborne Reynolds did in 1880s. Then, with
respect to a large number of experiments as mentioned here he defined so called Reynolds
number which we have seen.
After many experiments Reynolds showed that a particular ratio rho VD by mu, where rho
is the mass density of the fluid, V is the average velocity, D is the diameter of the
pipe and mu is the dynamic coefficient of viscosity. So, this ratio rho VD by mu helps
to predict the change in flow type. He defined this number as Reynolds number; this number
is dimensionless number. So, Osborne Reynolds defined this number as Reynolds number and
actually this number is the ratio of inertial forces to viscous forces. He puts this Reynolds
number is equal to inertial force divided by viscous force. For pipe flow, we can define
as rho VD by mu, where rho is the mass density of the fluid, V is the average of velocity,
D is the diameter of the pipe and mu is the coefficient of dynamic viscosity. Similar
way we can define this Reynolds number for open channel also. With respect to his experiments
he put this Reynolds number as a scale to define whether the concern fluid flow is laminar
or whether it is in the case of transitional stage or whether it is in the case of turbulent
stage.
Now, with respect to this Reynolds number we can say for example if we consider pipe
flow. As i mentioned whether the pipe flow can be pipe flow or open channel flow or river
flow like that, if we consider for pipe flow we can show that generally with respect to
various conditions we can say that when the Reynolds number is less than or equal to 2000
in this range of 2000 or less than 2000, we can see that the flow is laminar. A range
of between 2000 and 4000, we can see that the mixing starts and then the disturbance
start and then a transitional state occurs for pipe flow. Beyond 4000 when the Reynolds
number is greater than 4000, we can see that the flow become turbulent. With respect to
for pipe flow and with respect to Reynolds number we can say, whenever the Reynolds number
is less than 2000 the flow is laminar; between 2000 to 4000 we can say the flow is at a transitional
stage and beyond 4000, the flow is turbulent.
Through experiments we can show when the flow is laminar with respect to the Reynolds number
we can see that whenever the flow is laminar the velocity will be low which will be low
velocity; in transitional flow we can see the medium velocity and when the flow become
turbulent the velocity is high or high velocity flow. This is the case of pipe flow.
Similar way if we do experiments in open channel with respect to various conditions generally
we can say that the flow will be laminar when the Reynolds number is less than or equal
to 500; from 500 to 2000 it may be at a transitional stage but this can vary slightly and also
beyond 2000 we can see for open channel flow it can be turbulent in nature.
The exact range depends on various other flow conditions exist in the flow regime, various
other boundary conditions or the initial conditions. Depending upon this kind of various conditions
this range can vary. But, approximately this is the range put forward by various scientists
and engineers for open channel flow and pipe flow. With respect to the Reynolds experiments
which we have seen we can put forward certain interpretations.
First one is when viscous forces are dominant. We have seen that the Reynolds number is the
ratio of inertial force to viscous force. With respect to this, we can interpret when
the viscous force are dominant the flow is slow or the Reynolds number is low and then
they are sufficient enough to keep all fluid particles in line or flow is laminar. Even
though there is any tendency for any disturbances then here for this particular kind of flow,
wherever velocity is small or the slow flow and low Reynolds number we can see that the
disturbance will be dampened. Here, the viscous force is dominant and flow is laminar. Whenever
the fluid flow is faster or the velocity is larger the Reynolds number is larger then
we can see that inertial forces dominant over viscous forces and then finally the flow is
becoming turbulent. This is what is happening. Whether with respect
to the fluid flow velocity we can see that when the Reynolds number is lower or higher
and whether the viscous force is dominant or the inertial force is dominant. So, accordingly,
we can classify whether the concern flow which we are dealing is laminar in nature or turbulent
in nature. Also, while conducting these experiments even though we have seen that in certain range
the flow is laminar and beyond that it become transitional or turbulent.
When we conduct large number of experiments we can define a lower critical Reynolds number
where the Reynolds number above which flow changes from laminar to transitions or turbulent.
This is so called lower critical Reynolds number and then we can define, for example
the velocity of flow with respect to the valve movement with the velocity is reduced. In
a similar way when the flow is transforming from laminar to turbulent in the opposite
direction, when we reduce the valve opening then you can see that flow velocity will be
slowly and coming down and then there will be a transitional from turbulent to transitional
or turbulent to laminar.
There we can define a Reynolds number called upper critical Reynolds number. Here, the
Reynolds number below which turbulent flow changes to laminar flow. So this critical
Reynolds number we can have two ranges: one is lower critical Reynolds number and another
one is the upper critical Reynolds number. This critical Reynolds number which we have
seen is based upon which we classify the flow as laminar or turbulent. This varies depends
upon various flow conditions like what kind of fluid and then what are the boundaries
just as we have seen whether it is pipe flow or whether it is open channel.
It is difficult to put an exact range; when exact, the flow is transforming from laminar
to turbulent but this depends upon various other different flow conditions. Generally,
we prescribe a lower critical Reynolds number where the flow changes from laminar to turbulent
or transitional or and then upper critical Reynolds number where they are turbulent flow
changes to laminar flow.
Now also we can see whenever we conduct these kinds of experiments in the laboratory we
can see that this laminar flow which is we have seen is flowing in layers and then in
some stages we can see instability of laminar flow. Here, the inertial forces associated
with fluid mass try to amplify disturbances in the flow. Initially, the flow is laminar
but then there can be different kinds of disturbances like boundary friction or the shearing effect
or the other kinds of effect then we can see that the inertial forces with respect to the
fluid which is flowing. So, inertial forces associated with fluid mass try to amplify
disturbances in the flow.
Then there is instability of laminar flow. Viscous forces try to damp this disturbance
but the various forces which are again trying to amplify the disturbances are dominating
above the viscous forces. Then we can see that the laminar flow become unstable; instability
of laminar flow takes place and depending on the flow fluid characteristics like mass,
density, viscosity, velocity gradient proximity to boundary etc., the disturbances are dampened
or amplified.
For the given conditions, if the disturbances are dampened again the flow will be continuing
as laminar flow but if the disturbances are amplified further due to various forces like
shear force or various frictional forces then it may change from laminar to turbulent flow
or that the laminar flow become unstable or instability of laminar flow takes place. So
if the disturbances are dampened the flow is laminar; otherwise, that flow becomes turbulent.
Also, with respect to what we have discussed, we see that as far as turbulent flow is concerned
various important characteristics which we can observe in nature or observe in the laboratory
includes irregularity. That means here when flow become turbulent we can see that when
the flow is laminar we can see it is very regular; one layer is sliding over the other
one; it is very much regular.
But you can see when the flow become turbulent, some of the important flow characteristics
of turbulent flow are: first one is irregular in nature, the flow is irregular; and the
second one is diffusivity, as far as diffusivity is concerned we can see the case of laminar
flow we that the molecular diffusivity is more important but as for as turbulent flow
is concerned the turbulent diffusivity is much more than the molecular diffusivity and
then the flow is changing from laminar to turbulent.
Some of the other important characteristics are higher Reynolds number as we have seen
and then in many cases we can see that the flow may be rotational in nature. In most
of the laminar flow the rotational effective is very minimal or it can be even rotational
in nature. But in most of the turbulent flow we can see that the motion will be rotational
nature and due to this most of the time we have to consider the fluid flow in three dimensions.
Since in one dimension or two dimensions it will be very difficult to see what is realistically
happening and how the disturbance takes place? Which direction it takes place. Most of the
time as far as turbulent flow is concerned we have to do three dimensional analysis.
As far as turbulent flow is concerned motion can be rotational and three dimensions and
then the fluctuations we can see here; the fluctuations can be wide spectrum. If you
just consider for example, if the flow is laminar but as far as turbulent flow is concerned
we can see that the whenever the flow becomes turbulent the fluctuations that takes place
will be wide spectrum . Another important flow characteristic in turbulent flow motion
is dissipative. Various energies like heat energy and all will be dissipated with respect
to the turbulent flow. With respect to the discussion so far we can say that the turbulent
in a flow is feature of flow but it is not a property. It is very difficult to say that
turbulent is not a fluid property but we can define a turbulent is a feature of the flow.
Whenever a flow is there we have to see that if it is if we turbulent flow it is the future
of the flow but it is not a fluid property. Further, we will be discussing various aspects
of the laminar flow and turbulent flow in detail in this chapter.
So initially we will be discussing the laminar flow and then we will be discussing the turbulent
flow. Now, we will discuss laminar flow in detail. Initially, we will see the various
flow characteristics as far as laminar flow between parallel plates and then will be discussing
the laminar flow in pipes. Further, we will be discussing the turbulent flow. Now first
one is laminar in compressible flow between parallel plates. Here we will be deriving
various relationships like the velocity variation and then the shear stress variation etc.,
as far as laminar incompressible flow between parallel plates. Let us consider the flow
of a viscous fluid between two very large parallel flat plates spaced at a distance
of 2h or equal to 2y apart. We consider here the laminar flow so the laminar flow what
we are considering here is we are considering two very large parallel plates like this so
now if we consider this sheet of paper as the parallel plates what we are doing here
is in between two parallel plates are placed like this and then two parallel flat plates.
In between if fluid flow is there laminar flow how the flow parameter varies so that
we are going to discuss here; the incompressible flow between parallel plates.
Since it is parallel flow we assume the flow is only one velocity component and then we
also assume the flow is steady state and fluid is incompressible. This kind of flow wherever
the flow between two large parallel plates this kinds of flow is general, you called
a plane poisecuillie flow. The flow between two parallel plates which is laminar in nature
at the parallel plates is placed at distance. This kind of flow between these parallel plates
is known as plane poiseuillie flow.
Now we will discuss in detail various flow parameters or the velocity variation takes
place. So here we can see the velocity profile for flow between parallel plates: one parallel
plate is here and another one just placed 2h above at distance 2h. Here the flow is
between these parallel plates. If we plot the velocity profile we can see that we are
discussing the real fluid. Due to the viscosity of the fluid we can see the no slip condition.
Here the velocity on this side on the upper plate where the contact between fluids takes
place the velocity will be 0; here the velocity will be 0 and then we can see since the fluid
is between two parallel plates. Generally, the velocity will be maximum at the center
and then you can see if you brought the velocity then you can see it will be parabolic in nature.
So starting was 0 both sides and then it will be maximum at the center. If you bring the
velocity we can see that it will parabolic in nature 0 on the both sides of the parallel
plates and then maximum at the center. Now, we want to get the velocity variation we want
to find out how the velocity variation and once the velocity variation is known we can
find out the shear stress or other fluid flow parameters.
Now, we are considering the flow which we consider it has two dimensions in nature.
For two dimensional flow we have already seen that the continuity equation we can write
del u by del x so if the velocity in x direction is u velocity in y direction is v then we
can write from the continuity equation as del u by del x plus del v by del y this is
equal to 0. So this is the continuity equation and now we can see that since the streamlines
curve is straight here is the stream if you bring the streamlines, the stream lines are
parallel so that we can say that it is parallel to the x direction. You can see that here
the velocity component v is equal to 0.
Finally, what we can see for this particular case there will be only one velocity component
in x direction the other velocity component v is equal to 0. From this continuity equation
we can write del v by del x is equal to del v by del y since v is equal to 0 that del
v by del x is equal to del v del y is equal to 0. If we put into this continuity equation
we can write del u by del x is equal to 0. That means, for this particular case laminar
incompressible flow between parallel plates velocity does not change with x; velocity
field, v is equal to 0 and the velocity u is changing with respect to y only as we can
see the flat here we can see the velocities varying with respect to y only. So u is a
function of y as written here.
Finally, with respect to this simplification we can see that the flow is one dimensional
and unique directional or one directional. This type of flow with the velocity profile
unchanging as the fluid moves downstream is so called fully developed flow. This type
of flow with the velocity profile unchanging we can see the velocity profile here which
is unchanging with respect to the as the fluid moves down downstream so we this fully developed
flow. We have already seen the velocity variation here and now we want to find out the velocity
distinguishes; we want to get an expression for the velocity variation so that we can
draw the velocity. Here we will derive the expression for velocity by using the first
principle or the Newton’s second law we can apply here. We have already discussed
about Newton’s second law in this application earlier.
By using Newton’s second law for a particle if we can say particular particle in the control
volume which we are dealing, we can write the mass of particle multiplied by the acceleration
of particle is equal to net pressure force on the particle plus net gravity force on
particle plus net stress force on particle. So, we can see from the Newton’s second
law for the particle which we consider the control volume we can write mass of the particle
multiplied by acceleration of particle that is equal to net pressure force on particle
plus net gravity force on particle plus net stress force on the particle. So here we use
the Newton’s second law to derive the equation. For the x direction, from the Newton’s second
law we can write mass into acceleration is equal to pressure force plus gravity force
plus shear force.
If we consider a small fluid element for which the mass is delta Mp and acceleration is ax
in x direction so that is equal to delta Fx, pressure plus delta Fx, gravity plus delta
Fx shear. Here, if we consider, the delta Fx is concerned so here we can write down
the expression for this by using here.
We will consider a control volume that means here the fluid from between parallel plates
is here so we consider a small fluid element and then we can write the pressure force we
write with respect to this fluid element on left side we can write p minus del p by del
x into delta x by 2delta x is the size of the fluid in x direction; delta y is the size
of the fluid in y direction and the plate is placed at distance of 2y; here the pressure
force is p minus del p by del x into del x by 2.
And other side of the fluid element p plus del p by del x into delta x by 2 and then
other side is concerned, we can write p plus del p del y into delta y by 2 and the opposite
side it will be p minus del p by del y into delta y by 2. Similarly, the shear stress
is concerned which is another important force shear force. So we can write on this side
tow plus delta tow by del y into delta y by 2. On this side, it is tow plus del tow by
del x into delta x by 2 and this side of the fluid element it will be tow minus del tow
by del y into del y by two and this side it will be tow minus del tow by del x into delta
x by 2.
Now we are considering this control volume and we are considering fluid element; all
the pressure force and shear force is concerned. Finally, we can write with respect to this
the delta Fx pressure we can write del p del x into delta x into delta y into, if we consider
the unit thickness, it is multiplied by 1.
Similarly, we can write with the gravity force is concerned delta Fx which is acting on the
fluid element will be rho g x into delta x into delta y into 1. This gives the gravity
force and as the shear force is concerned we can write with respect to the earlier figure
shown we can write del tow by del y into delta x into delta y.
Now by using Newton’s second law, we have written for the fluid particle of the fluid
element is concerned the mass of the fluid element or fluid particle multiplied by saturation
that is equal to the pressure force on the fluid particle plus gravity force on the fluid
particle plus shear force on the fluid particle. So we have derived the expression for the
pressure force on the fluid element p minus del p by del x into delta x into delta y and
gravity force, we got rho g x into delta x into delta y and shear force we got del tow
by del y into delta x into delta y.
Now, the acceleration of the particle as we have already seen earlier we can write ax
is equal to del u by del t plus du into del u by del x plus v into del u by del y. This
gives the local acceleration plus converting acceleration. So ax is equal to del u by del
t plus u into del u by del x plus v into del u by del y.
You can see here we are considering the fluid steady state, this is 0 and del u by del x
is 0 and then v is 0. Finally, ax is equal to 0. The Newton’s law reduces to the force
is equal to the pressure force plus gravity force plus shear force on the fluid is equal
to 0. Finally, we get the equation as minus del p del x plus rho g x plus del tow by del
y is equal to 0.
This is the expression which we get for the fluid element which we consider. Now, in the
earlier equation if you introduce here this rho g x is concerned, introduce z coordinate
pointing upward opposite the gravity vector. We can write minus del p by del x minus rho
g del z by del x plus del tow by del y is equal to 0 and here you can see that if the
pressure p and the acceleration gravity g are constant. We can combined these two terms
and write del by del x p plus gamma z plus del tow by del y is equal to 0. We can finally
get the expression as del by minus del by del x of p plus gamma z plus del tow by del
y is equal to 0.
Similar way what we have seen in the earlier equation is the x direction. So, similar way
we can write the y direction: minus del by del y of p plus gamma z plus del tow by del
x is equal to 0. Here you can see since the velocity profile do not change in x direction
we can write del tow by del x is equal to 0 and hence we can write del by del y of p
plus gamma z is equal to 0 with respect to this earlier equation.
So that we can say the pressure distribution in the y direction is hydrostatic and the
term p plus gamma z is a function of x only. Now this earlier equation number 3, which
is written the x direction we can finally write as minus d by dx of p plus gamma z plus
del tow by del y is equal to 0 equation number 5.
Earlier, we have seen the Newton’s law of viscosity as tow is equal to mu into del u
by del y. If you substitute for this tow here finally we get minus d by dx of p plus gamma
z plus mu into del square u by del y squared is equal to 0. This is the final expression
which we get connecting the pressure variation and the velocity variation. Here, you can
see u is only the function of y. So that we can write del square u by del y square can
be written as derivate d square u by dy square.
Therefore, finally we can write the expression as mu into d square u by dy square is equal
to d by dx into p plus gamma z and if you put this p delta is equal to d by dx of p
plus gamma z, finally we can write d square u by dy square is equal to p delta by mu,
where mu is the coefficient dynamic viscosity and now our aim is to get an expression for
velocity. So this expression we can integrate two times and that write u is equal to p delta
by 2mu y squared plus C1 y plus C2.
Now, we will introduce the boundary condition. Here we can see the boundary conditions on
the plate. Due to the no slip condition we can write at y is equal to plus or minus y
the velocity is 0 since due to no slip conditions the velocity is 0 from which we will get the
constant of integration C1 is equal to 0 and finally we will get C2 is equal to minus p
delta y squared by 2mu. So we get C2 is equal to minus p y squared by 2mu and C1 is equal
to 0. Finally, we get an expression 4u- the velocity variation is equal to minus p delta
y squared by 2mu into 1 minus y by Y whole squared, where as we have seen the distance
between the plate is 2y.
Finally, we get an expression for the velocity variation and then we can see that with respect
to velocity variation maximum at the center. So y is equal to 0, so that umax is equal
to minus p delta y squared by 2mu and then also we can find out the average velocity
v bar is equal to 1 by 2y. We can integrate between minus y to plus y u d y. That would
give the average velocity as minus 2 by 3 p delta y squared by 2mu that is equal to
2 by 3 umax.
And then if we introduce the term g and p bar which we have already seen with respect
to that the final expression for velocity can be obtained as u is equal to minus y squared
by 2mu d by dp plus gamma z by dx into 1 minus y by Y whole squared or umax also can be written
as y squared by 2mu dp plus gamma z by dx and the average velocity we can show that
it will be two third of the maximum velocity that is equal to minus y squared by 3 mu into
d of p plus gamma z by dx.
So, like this we can derive the relationship for velocity and from the velocity we can
go for the other parameters like shear stress, variation or pressure distribution. Further
we will be discussing on this laminar flow in the next lecture.
