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 

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Saturday, 14 April 2018

STATE BUCKINGHAM PI THEOREM

We were discussing the basic concept of streamline and equipotential line and dimensional homogeneity in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to understand the basic concept of Buckingham π theorem with the help of this post. 

Buckingham π theorem

 
In an equation, if the variables are more than the numbers of fundamental dimensions i.e. M, L and T. The Rayleigh’s method of dimensional analysis will be more laborious and this problem was resolved by one theorem or concept and that theorem, as stated below, was termed as Buckingham π theorem. 

According to Buckingham π theorem, if there are n variables (Independent and dependent variables) in a physical phenomenon and if these variables contain m fundamental dimensions i.e. M, L and T. Then the variables are arranged in to (n-m) dimensionless terms and each term will be termed as π term. 

Let X1, X2, X3….Xn are the variables involved in a physical problem. Let us think that X1 is the dependent variable and X2, X3….Xn are the independent variables on which X1 will be dependent.

We can also say that X1 will be a function of X2, X3….Xn and mathematically we can write as mentioned here. 

X1 = f(X2, X3….Xn)

Example

Let us discuss one example here to understand the concept of Buckingham π theorem.


 
The power required by an agitator in a tank is a function of following variables as mentioned here. 
Diameter of the agitator (D) 
Number of the rotations of the impeller per unit time (N) 
Viscosity of liquid (µ
Density of liquid (ρ) 

We will secure here one relation between power required by agitator and above mentioned four variables by using the concept of Buckingham π theorem. 

There are total five variables here. Power (P) is dependent variable and rest four variables (D, N, µ and ρ) are independent variables. Power (P) will be dependent over the above mentioned four variables. 

Number of variables = 5 
Number of fundamental dimensions = 3 
Number of dimensionless groups = 5-3 = 2 

We will select here the variables so as to represent the dimensions, let us select N, D and ρ.

N = [T-1]

T = [N-1]

D = [L]

L = [D]

ρ = [ML-3]

M = ρ [L3] = ρ [D3]

For the other variables,

Dimension of the power P will be [ML2T-3]

Therefore P M-1L-2T3 will be dimensionless

Therefore the П1 term will be given as mentioned here

П= P M-1L-2T3
П= P ρ-1 D-3 D-2N-3
П= P ρ-1 D-5 N-3
П= P / (ρ D5 N3)

Dimension of the viscosity μ will be [ML-1T-1]

Therefore μ will be μ [M-1LT] will be dimensionless term

Therefore the П2 term will be given as mentioned here

П= μ [M-1LT] = μ ρ-1 D-3DN-1

П= μ / (ρ D2N) 

So, we have determined the relation between the variables with the help of Buckingham π theorem.


 
We will see another important topic in the field of fluid mechanics i.e. differentiate between model and prototype with the help of our next post. 

Do you have any suggestions? Please write in comment box.
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Saturday, 31 March 2018

DIMENSIONAL HOMOGENEITY PRINCIPLE

We were discussing the basic concept of streamline and equipotential line, in the subject of fluid mechanics, in our recent posts.

Now we will go ahead to understand the basic concept of dimensional homogeneity with the help of this post.

Dimensional homogeneity

Dimensional homogeneity suggests that the dimensions of each term in an equation on both sides will be equal.

Dimension of LHS = Dimension of RHS

 
Therefore an equation will be termed as dimensionally homogeneous equation or dimensionally consistent, if dimensions of each term of an equation on both sides are identical.

We have already studied that there are three fundamental dimensions i.e. Length (L), Mass (M) and Time (T). For a dimensionally homogeneous equation, the powers of fundamental dimensions (L, M and T) for each term of equation on both sides will be same.

Explanation with an example

Let us consider one equation and we will see here the dimensions of each term of equation on both sides. We will analyze the equation to secure the information that the given equation is dimensionally homogeneous equation or not.

Example -1

Let us find the below equation as mentioned here
d = Final distance
d0 = Initial position
v0 = Initial Velocity
t = Time of travel
a = Acceleration

Example -2

 
Let us consider the equation as mentioned here
V = Final Velocity
h = Height of fall
g = Acceleration due to gravity


We will see another important topic in the field of fluid mechanics i.e. Buckingham π theorem 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

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Thursday, 29 March 2018

DIFFERENCE BETWEEN STREAMLINE AND EQUIPOTENTIAL LINE

We were discussing the basic concept of Types of fluid flowDischarge or flow rateContinuity equation in three dimensionsvelocity potential function and stream function  and total acceleration, in the subject of fluid mechanics, in our recent posts.

 

Now we will go ahead to understand the basic difference between streamline and equipotential line, in the field of fluid mechanics, with the help of this post. We will also see here the basic of flow net with the help of this post.

Equipotential line

Equipotential line is basically defined as the line along which the velocity potential (ϕ) is constant.
For equipotential line, ϕ = Constant

Streamline

Streamline is basically defined as the line along which the Streamline function (Ψ) is constant.
For streamline, Ψ = Constant
Let us consider the equation of equipotential line and stream line. We will conclude here that the product of slope of equipotential line and slope of stream line at the point of interseaction will be equivalent to -1.

 
Therefore we can say that equipotential line and stream line will be perpendicular to each other at the point of intersection.

Flow net

If we will draw equipotential lines and stream lines for a fluid flow, we will see that both lines will intersect each other at right angle or orthogonally and will develop one grid or net and that grid will be termed as flow net.
Flow net is a very important tool to analyse the two dimensional irrotational flow problems.

We will discuss another term in fluid mechanics in our next post.

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

Reference:

Fluid mechanics, By R. K. Bansal
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Wednesday, 28 March 2018

VELOCITY POTENTIAL FUNCTION AND STREAM FUNCTION

We were discussing the basic concept of  Types of fluid flowDischarge or flow rateContinuity equation in three dimensionscontinuity equation in cylindrical polar coordinates and total acceleration, in the subject of fluid mechanics, in our recent posts.

Now we will go ahead to understand the basic concept of velocity potential function and stream function, in the field of fluid mechanics, with the help of this post.

 
Let us consider that V is the resultant velocity of a fluid particle at a point in a flow filed. Let us assume that u, v and w are the components of the resultant velocity V in x, y and z direction respectively.

We can define the components of resultant velocity V as a function of space and time as mentioned here.

Velocity potential function

Velocity potential function is basically defined as a scalar function of space and time such that it’s negative derivative with respect to any direction will provide us the velocity of the fluid particle in that direction.

Velocity potential function will be represented by the symbol ϕ i.e. phi.

Velocity components in cylindrical polar-coordinates in terms of velocity potential function will be given as mentioned here.
Where,
ur is the velocity component in radial direction and uθ is the velocity component in tangential direction.

We can write here the continuity equation for incompressible steady flow in terms of velocity potential function as mentioned here.

Stream function

 
Stream function is basically defined as a scalar function of space and time such that it’s partial derivative with respect to any direction will provide us the velocity component at perpendicular to that direction.

Stream function will be represented by Ψ i.e. psi. It is defined only for two dimensional flow.

Velocity components in cylindrical polar-coordinates in terms of stream function will be given as mentioned here.
Where,
ur is the velocity component in radial direction and uθ is the velocity component in tangential direction.

Let us use the value of u and v in continuity equation; we will have following equation as mentioned here.
Therefore existence of stream function (Ψ) indicates a possible case of fluid flow. Flow might be rotational or irrotational.

If stream function (Ψ) satisfies the Laplace equation, it will be a possible case of an irrotational flow.

We will discuss another term i.e. “Equipotential line and streamline” in 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

Newton’s law of viscosity
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Tuesday, 27 March 2018

LOCAL ACCELERATION AND CONVECTIVE ACCELERATION

We were discussing the basic concept of Lagrangian and Eulerian methodTypes of fluid flowDischarge or flow rateContinuity equation in three dimensions and continuity equation in cylindrical polar coordinates, in the subject of fluid mechanics, in our recent posts.

Now we will go ahead to understand the basic concept of local acceleration and convective acceleration, in the field of fluid mechanics, with the help of this post.

 
Before going ahead, we need to find out the term total acceleration of a fluid particle in a flow field.
Let us consider that V is the resultant velocity of a fluid particle at a point in a flow filed. Let us assume that u, v and w are the components of the resultant velocity V in x, y and z direction respectively.

We can define the components of resultant velocity V as a function of space and time as mentioned here.

Total acceleration is basically divided in two components i.e. local acceleration and convective acceleration.

We will first see here the basic concept of local acceleration and later we will find out the convective acceleration.

Local acceleration

Local acceleration is basically defined as the rate of increase of velocity with respect to time at a given point in the flow filed.

 
Local acceleration is basically due to the change in local velocity of the fluid particle as function of time.

Convective acceleration

Convective acceleration is basically defined as the rate of change of velocity due to change of position of fluid particle in fluid flow field.

Convective acceleration is basically due to the fluid particle being convected from one given location to another location in fluid flow field and second location being of higher or lower velocity.

Convective acceleration is also termed as advective acceleration.

We will discuss another term i.e. "Velocity potential function and streamline function", in our next post, in the field of fluid mechanics. 

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

 

Reference:

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

Also read

Continue Reading

Monday, 26 March 2018

TOTAL ACCELERATION IN FLUID MECHANICS

We were discussing the basic concept of Lagrangian and Eulerian methodTypes of fluid flow, Discharge or flow rate, Continuity equation in three dimensions and continuity equation in cylindrical polar coordinates, in the subject of fluid mechanics, in our recent posts.

Now we will go ahead to understand the basic concept of total acceleration in fluid mechanics. Further we will find out the term total acceleration and convective acceleration, in the field of fluid mechanics, with the help of this post.

 
So let us first discuss here the term total acceleration of a fluid particle in a flow field.

Total acceleration

Let us consider that V is the resultant velocity of a fluid particle at a point in a flow filed. Let us assume that u, v and w are the components of the resultant velocity V in x, y and z direction respectively.

We can define the components of resultant velocity V as a function of space and time as mentioned here.

Total acceleration of a fluid particle in a direction will be equal to the rate of change of velocity of that fluid particle in that direction in a flow field.

 
Let us consider that ax, ay and az are the total acceleration in x, y and z direction respectively. Considering the chain rule of differentiation, we will have the following equation of total acceleration as mentioned here.

Total acceleration is basically divided in two components i.e. local acceleration and convective acceleration.

We will now see the basic concept of local acceleration and convective acceleration, in the field 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

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