Saturday, 26 May 2018

EULER EQUATION OF MOTION DERIVATION

We were discussing the basic concept of streamline and equipotential linedimensional homogeneityBuckingham pi theoremdifference between model and prototypebasic principle of similitude i.e. types of similarity, various forces acting on moving fluid and model laws or similarity laws in the subject of fluid mechanics, in our recent posts.

Now we will go ahead to understand the fundamentals and derivation of Euler’s equation of motion of a fluid, in the subject of fluid mechanics, with the help of this post. 

Euler’s Equation of motion

Euler’s equation of motion of an ideal fluid, for a steady flow along a stream line, is basically a relation between velocity, pressure and density of a moving fluid. Euler’s equation of motion is based on the basic concept of Newton’s second law of motion.

When fluid will be in motion, there will be following forces associated as mentioned here.
1. Pressure force
2. Gravity force
3. Friction force due to viscosity
4. Force due to turbulence force
5. Force due to compressibility

In Euler’s equation of motion, we will consider the forces due to gravity and pressure only. Other forces will be neglected.

Assumptions

Euler’s equation of motion is based on the following assumptions as mentioned here
1. The fluid is non-viscous. Frictional losses will be zero
2. The fluid is homogeneous and incompressible.
3. Fluid flow is steady, continuous and along the streamline.
4. Fluid flow velocity is uniform over the section
5. Only gravity force and pressure force will be under consideration.

Let us consider that fluid is flowing from point A to point B and we have considered here one very small cylindrical section of this fluid flow of length dS and cross-sectional area dA as displayed here in following figure.

Let us think about the forces acting on the cylindrical element

Pressure force PdA, in the direction of fluid flow
Pressure force [P + (P/∂S) dS] dA, in the opposite direction of fluid flow
Weight of fluid element (ρ g dA dS)
Image: Force on a fluid element
Let us consider that θ is the angle between the direction of fluid flow and the line of action of weight of the fluid element.

As we have mentioned above that Euler’s equation of motion is based on the basic concept of Newton’s second law of motion. Therefore, we can write here following equation as mentioned here
Net force over the fluid element in the direction of S = Mass of the fluid element x acceleration in the direction S.

Above equation is termed as Euler’s equation of motion. 

We will now derive the Bernoulli’s equation from Euler’s Equation of motions, in the subject of fluid mechanics, in our next post.

Do you have any suggestions? Please write in comment box.

Reference:

Fluid mechanics, By R. K. Bansal
Image Courtesy: Google

Also read

Continue Reading

Sunday, 20 May 2018

MODEL LAWS OR SIMILARITY LAWS

We were discussing the basic concept of streamline and equipotential linedimensional homogeneityBuckingham pi theoremdifference between model and prototype, basic principle of similitude i.e. types of similarity and various forces acting on moving fluid in the subject of fluid mechanics, in our recent posts.

Now we will go ahead to understand the basic concept of model laws or similarity laws in the subject of fluid mechanics with the help of this post.

Model laws or similarity laws

For the dynamic similarity between the model and the prototype, ratio of corresponding forces acting on corresponding points in the model and the prototype should be same.

 
Ratios of the forces are dimensionless numbers. Therefore we can say that for the dynamic similarity between the model and the prototype, dimensionless numbers should be equal for the model and the prototype.

However, it is quite difficult to satisfy the condition that all the dimensionless numbers should be equal for the model and the prototype.

However for practical problems, it is observed that one force will be most significant as compared to others and that force is considered as predominant force. Therefore for dynamic similarity, predominant force will be considered in practical problems.

Therefore, models are designed on the basis of ratio of force which is dominating in the phenomenon.
Hence, we can define the model laws or similarity laws as the law on which models are designed for the dynamic similarity.

There are following types of model laws

Reynold’s Model law
Froude Model law
Euler Model law
Weber Model law
Mach Model law

Reynold’s Model law

Reynold’s model law could be defined as a model law or similarity law where models are designed on the basis of Reynold’s numbers.

According to the Reynold’s model law, for the dynamic similarity between the model and the prototype, Reynold’s number should be equal for the model and the prototype.

In simple, we can say that Reynold’s number for the model must be equal to the Reynold’s number for the prototype.

As we know that Reynold’s number is basically the ratio of inertia force and viscous force, therefore a fluid flow situation where viscous forces are alone predominant, models will be designed on the basis of Reynold’s model law for the dynamic similarity between the model and the prototype.
Image: Reynold’s model law
Where,
Vm = Velocity of the fluid in the model
Lm = Length of the model
νm = Kinematic viscosity of the fluid in the model
VP = Velocity of the fluid in the prototype
LP = Length of the prototype
νP = Kinematic viscosity of the fluid in the prototype

Models based on the Reynold’s model law

Pipe flow
Resistance experienced by submarines, airplanes etc.

Froude Model law

Froude model law could be defined as a model law or similarity law where models are designed on the basis of Froude numbers.

According to the Froude model law, for the dynamic similarity between the model and the prototype, Froude number should be equal for the model and the prototype.

In simple, we can say that Froude number for the model must be equal to the Froude number for the prototype.

 
As we know that Froude number is basically the ratio of inertia force and gravity force, therefore a fluid flow situation where gravity forces are alone predominant, models will be designed on the basis of Froude model law for the dynamic similarity between the model and the prototype.
Image: Froude model law
Where,
Vm = Velocity of the fluid in the model
Lm = Length of the model
gm = Acceleration due to gravity at a place where model is tested
VP = Velocity of the fluid in the prototype
LP = Length of the prototype
gP = Acceleration due to gravity at a place where prototype is tested

Models based on the Froude model law

Free surface flows such as flow over spillways, weirs, sluices, channels etc,
Flow of jet from an orifice or from a nozzle,
Where waves are likely to be formed on surface
Where fluids of different densities flow over one another

Euler’s Model law

Euler’s model law could be defined as a model law or similarity law where models are designed on the basis of Euler’s numbers.

According to the Euler’s model law, for the dynamic similarity between the model and the prototype, Euler’s number should be equal for the model and the prototype.

In simple, we can say that Euler’s number for the model must be equal to the Euler’s number for the prototype.

As we know that Euler’s number is basically the ratio of pressure force and inertia force, therefore a fluid flow situation where pressure forces are alone predominant, models will be designed on the basis of Euler’s model law for the dynamic similarity between the model and the prototype.
Image: Euler’s model law
Where,
Vm = Velocity of the fluid in the model
Pm = Pressure of fluid in the model
ρm = Density of the fluid in the model
VP = Velocity of the fluid in the prototype
PP = Pressure of fluid in the prototype
ρP = Density of the fluid in the prototype

Models based on the Euler’s model law

Euler’s model law will be applicable for a fluid flow situation where flow is taking place in a closed pipe, in which case turbulence will be fully developed so that viscous forces will be negligible and gravity force and surface tension force will be absent.

Weber Model law

Weber model law could be defined as a model law or similarity law where models are designed on the basis of Weber numbers.

According to the Weber model law, for the dynamic similarity between the model and the prototype, Weber number should be equal for the model and the prototype.

In simple, we can say that Weber number for the model must be equal to the Weber number for the prototype.

As we know that Weber number is basically the ratio of inertia force and surface tension force, therefore a fluid flow situation where surface tension forces are alone predominant, models will be designed on the basis of Weber model law for the dynamic similarity between the model and the prototype.
Image: Weber model law
Where,
Vm = Velocity of the fluid in the model
σm = Surface tension force in the model
ρm = Density of the fluid in the model
Lm = Length of surface in the model
VP = Velocity of the fluid in the prototype
σP = Surface tension force in the prototype
ρP = Density of the fluid in the prototype
LP = Length of surface in the prototype

 

Models based on the Weber model law

Capillary rise in narrow passage
Capillary movement of water in soil
Capillary waves in channels
Flow over weirs for small heads

Mach Model law

Mach model law could be defined as a model law or similarity law where models are designed on the basis of Mach numbers.

According to the Mach model law, for the dynamic similarity between the model and the prototype, Mach number should be equal for the model and the prototype.  

In simple, we can say that Mach number for the model must be equal to the Mach number for the prototype.

As we know that Mach number is basically the ratio of inertia force and Elastic force, therefore a fluid flow situation where elastic forces are alone predominant, models will be designed on the basis of Mach model law for the dynamic similarity between the model and the prototype.
 
Image: Mach model law
Where,
Vm = Velocity of the fluid in the model
Km = Elastic stress for model
ρm = Density of the fluid in the model
VP = Velocity of the fluid in the prototype
KP = Elastic stress for prototype
ρP = Density of the fluid in the prototype

Models based on the Mach model law

Water hammer problems
Under water testing of torpedoes
Aerodynamic testing
Flow of aeroplane and projectile through air at supersonic speed

We will discuss another important topic i.e. Euler's Equation of motions in the subject of fluid mechanics in our next post.

Do you have any suggestions? Please write in comment box.

Reference:

Continue Reading

Sunday, 13 May 2018

TYPES OF FORCES ACTING IN MOVING FLUID

We were discussing the basic concept of streamline and equipotential linedimensional homogeneity, Buckingham pi theoremdifference between model and prototype and basic principle of similitude i.e. types of similarity in the subject of fluid mechanics, in our recent posts.

Now we will go ahead to understand the various forces acting on moving fluid in the field of fluid mechanics with the help of this post.

 
We will see another important topic in the field of fluid mechanics i.e. Types of forces acting in moving fluid with the help of this post.

There are various types of forces, as mentioned here, acting on moving fluid. Forces acting on a fluid mass may be any one or combinations of the several of the following forces.

Inertia force (Fi)

Inertia force is basically defined as the force which is the product of mass and acceleration of the flowing fluid. Inertia force will act on flowing fluid in a direction opposite to the direction of acceleration.

Inertia force will be displayed by Fi and will always exist in the fluid flow problems.

Viscous force (FV)

Viscous force is basically defined as the force which is the product of shear stress due to viscosity and surface area of the flowing fluid.

Viscous force will be displayed by FV and will always exist in the fluid flow problems when viscosity will be in very important role.

Gravity force (Fg)

Gravity force is basically defined as the force which is the product of mass and acceleration due to gravity of the flowing fluid.

Gravity force will be displayed by Fg and will always exist in case of open surface fluid flow problems.

Pressure force (FP)

Pressure force is basically defined as the force which is the product of pressure intensity and cross-sectional area of the flowing fluid.

Pressure force will be displayed by FP and will always exist in case of pipe flow problems.

 

Surface tension force (FS)

Surface tension force is basically defined as the force which is the product of surface tension and length of surface of the flowing fluid.

Surface tension force will be displayed by FS.

Elastic force (Fe)

Elastic force is basically defined as the force which is the product of elastic stress and area of the flowing fluid.

Elastic force will be displayed by Fe.

There will be one or two forces out of above mentioned forces, which will dominate the other forces. These dominating forces will govern the fluid flow problems.

We will see another important topic i.e. Dimensionless numbers and Model laws or similarity laws
in our next post.

Do you have any suggestions? Please write in comment box.

 

Reference:

Fluid mechanics, By R. K. Bansal
Image Courtesy: Google

Also read

Continue Reading

Sunday, 29 April 2018

SIMILITUDE AND SIMILARITY IN FLUID MECHANICS

We were discussing the basic concept of streamline and equipotential linedimensional homogeneity, Buckingham pi theorem and difference between model and prototype in the subject of fluid mechanics, in our recent posts.

Now we will go ahead to understand the basic principle of similitude i.e. types of similarity in the field of fluid mechanics with the help of this post.

 
Let us have a brief look over the basics of model and prototypes

In order to secure the information about the performance of any hydraulic structure such as dam or any hydraulic machine such as turbine, before going for construction and manufacturing of actual of structure or machine, models are prepared of the actual structure or machine and experiments are carried out on the models to secure the desired result.

Therefore we can define the model as the small scale replica of the actual structure or machine. Actual structure or machine will be termed as prototype.

Similitude – Types of similarities

Similitude is basically defined as the similarity between model and its prototype in each and every respect. It suggests us that model and prototype will have similar properties or we can say that similitude explains that model and prototype will be completely similar.

Three types of similarities must exist between model and prototype and these similarities are as mentioned here.

Geometric similarity
Kinemtaic similarity
Dynamic similarity

We will discuss each type of similarity one by one in detail. Let us first see here geometric similarity.

Geometric similarity

 
Geometric similarity is the similarity of shape. Geometric similarity is said to exist between model and prototype, if the ratio of all respective linear dimension in model and prototype are equal.
Ratio of dimension of model and corresponding dimension of prototype will be termed as scale ratio i.e. Lr.

Let us assume the following linear dimension in model and prototype.
Lm = Length of model, LP = Length of prototype
Bm = Breadth of model, BP = Breadth prototype
Dm = Diameter of model, DP = Diameter of prototype
Am = Area of model, AP = Area of prototype
Vm = Volume of model, VP = Volume of prototype

Kinematic Similarity

The Kinemetic similarity is said to exist between model and prototype, if the ratios of velocity and acceleration at a point in model and at the respective point in the prototype are the same.
We must note it here that the direction of velocity and acceleration in the model and prototype must be identical.
Vm = Velocity of fluid at a point in model, VP = Velocity of fluid at respective point in prototype
am = Acceleration of fluid at a point in model, aP = Acceleration of fluid at respective point in prototype

Dynamic Similarity

The dynamic similarity is said to exist between model and prototype, if the ratios of corresponding forces acting at the corresponding points are the same.

We must note it here that the direction of forces at the corresponding points in the model and prototype must be same.
Fm = Force at a point in model, FP = Force at respective point in prototype

 
We will see another important topic in the field of fluid mechanics i.e. Types of forces acting in moving fluid with the help of our next post.

Do you have any suggestions? Please write in comment box.

Reference:

Fluid mechanics, By R. K. Bansal
Image Courtesy: Google 

Also read

Continue Reading

Sunday, 15 April 2018

DIFFERENTIATE BETWEEN MODEL AND PROTOTYPE

We were discussing the basic concept of streamline and equipotential linedimensional homogeneity and Buckingham pi theorem in the subject of fluid mechanics, in our recent posts.

Now we will go ahead to understand the basic differentiate between model and prototype with the help of this post.

Model, prototype and dimensional analysis

 
In order to secure the information about the performance of any hydraulic structure such as dam or any hydraulic machine such as turbine, before going for construction and manufacturing of actual of structure or machine, models are prepared of the actual structure or machine and experiments are carried out on the models to secure the desired result.

If result obtained from the experiment carried out on the models of actual structure or machine is not achieved as desired, respective modification will be carried out to secure the desired result.

Let us consider one example of an aircraft. If we want to design and manufacture one aircraft, first we must have the fundamental information about the aircraft like drag forces etc. 

Various experiments will be carried out on the aircraft in order to secure all the desired result. If we want to conduct the experiments on actual aircraft, it will not feasible as it will be very expensive and not safe also.

So what we will do?

We will prepare one model of the actual aircraft. Model of the actual aircraft will be in smaller size. We will conduct the experiments over this model of actual aircraft and we will analyse the result obtained from the experiment.

If result obtained from the experiment is not as per requirement, we will do some modification in the design and other parameters of the model of the actual aircraft.

Once we will secure the desired result from experiments, we will send this model to production cell and construction and manufacturing of the actual aircraft will be commenced on the basis of the prepared model of the actual aircraft.

Therefore we can define the model as the small scale replica of the actual structure or machine. Actual structure or machine will be termed as prototype.

We have considered above the case of designing and manufacturing of an aircraft. We have discussed above that the model of actual aircraft will be in smaller size as compared to the actual aircraft. But, it is not always necessary that model will be in smaller size as compared to the actual structure or machine.

In some cases, model of the machine might be in larger size as compared to the actual machine.

 
Let us think the case of designing and manufacturing of watch. As we know very well that parts used in manufacturing of watch will be of very small size and it will be very difficult to conduct the experiments and analyse the results to predict the performance of the watch.

Therefore in this case, model will be prepared of larger size as compared to the actual size of the watch. After preparing the correct model, manufacturing of the respective set of watches will be commenced on the basis of the model.

Therefore, model could be of smaller size or larger size as compared to the actual machine.
We can define the model analysis as the study of the models of actual machines.

Advantage of the dimensional and model analysis

Let us see here few advantages of the dimensional and model analysis

With the help of dimensional and model analysis, we can easily secure the information about the performance of actual machine or structure before going for manufacturing of actual machine or structure.

With the help of dimensional and model analysis, we can also consider the alternative design of actual machine or structure and we can select most economical and safe design.

With the help of dimensional and model analysis, we will secure the relationship between variables influencing a flow problem in terms of dimensionless parameters and this relationship will help in conducting the experiments on the model.

 
We will see another important topic in the field of fluid mechanics i.e. similitude and similarity with the help of our next post.

Do you have any suggestions? Please write in comment box.

Reference:

Fluid mechanics, By R. K. Bansal
Image Courtesy: Google 

Also read

Continue Reading

Archivo del blog

Copyright © HKDIVEDI.COM | Powered by Blogger | Designed by Dapinder