The Fluid Mechanics course of Gate and Engineering services has the following chapters in its syllabus.
Introduction
Fluid statics
Kinematics of fluid
Conservation equation for fluid flow
Fluid flow applications
Viscous compressible flow
Application of viscous compressible flow through pipes
Principle of similarity
Flow of Ideal fluid
Flow with a free surface
Introduction to laminar boundary layer Introduction to turbulent flow
Textbook Recommended
These are some important text books recommended for the preparation of Engineering services and Gate exams.
Fluid Mechanics and fluid machinery by RK Bansal
Introduction to Fluid Mechanics and fluid machines by SK Som and Biswas
Fluid Mechanics by PN Modi
Let's start our very first topic in fluid mechanics in which we will cover the comparison of solid and fluid, the concept of Continuum and the various properties of fluid like viscosity, surface tension, capillarity etc.
Comparison of solid and fluid
Solids can resist shear force or stress under static equilibrium condition with a finite deformation. It can partially or fully regain its original shape and size after the removal of stress.
On another hand fluid goes on continuously deforming under the application of shear stress i.e. fluid resist shear stress only under dynamic condition and not under static condition hence continuously formation takes place in fluid.
Also after the removal of shear stress or load the fluid cannot come back to its original position except a very special case of fluid which is known as visco elastic fluid there is why Fluids are also called zero memory substance because it forget its memory that where it was before the application of shear stress.
Concept of Continuum or continuous medium
Under normal temperature and pressure conditions there are about 3x10^25 molecules per cubic metres which is very very high and under such conditions the concept of Continuum easily holds. Under entire Fluid Mechanics we will only consider the Continuum cases and all the formulae and concepts of Fluid Mechanics are made for such Continuum cases. There is a very important parameter called Knudsen number which defines when the Continuum hold and when it is not.
Mean free path - It is simply the average distance between two successive collision of a molecule.
If 'm' is mean free path and 'L' is characteristic dimension of the problem then 'm/L' is defined as the Knudsen number. Knudsen number (Kn) serves as a criteria for continuum.
Kn = m/L
If, kn < 0.01 continuum is valid
0.01 < Kn < 0.1 slip flow
0.1 < Kn < 10 transition flow
Kn > 10 free molecular flow.
Viscosity - it is a property of fluid which comes into play when fluid is in motion. It has nothing to do with fluid at rest. It is the property by which a fluid resist the motion of one layer of fluid over another.
Effect of temperature on the viscosity of liquid and gases
In any fluid the viscosity depends upon the Cohesive forces between the molecules and the molecular Momentum transfer between the molecules. On increasing the temperature the Cohesive forces decreases why the molecular Momentum transfer increases. In liquids the Cohesive forces dominate while in gases in the molecular Momentum transfer predominant. so on increasing the temperature the Cohesive forces in liquid decreases and so is the viscosity while in gases on increasing temperature the molecular Momentum transfer increases and hence the viscosity increases in the gases on increasing the temperature.
Newton's law of viscosity - According to this love this year is stress is directly proportional to the rate of shear strain in any fluid. All those flutes which obey Newton's law of viscosity are called newtonian fluids. Examples of newtonian fluids are water, mercury, oil etc. While the fridge which do not follow the Newton's law of viscosity are called non newtonian fluids. Examples of such fluids milk, blood, polymer solution, higher molecular oil, Ink etc.
Ostwald-de-waele model - All the fluids either newtonian and non newtonian can be expressed in a generalized form and this form is called Ostwald-de-waele model. This is expressed by the expression given below.
Where 'm' is called flow behaviour index while 'n' is called follow consistency index.
Note that for newtonian fluid viscosity is the property of fluid while viscosity is not a property of fluid for non newtonian fluid.
Ideal fluid - A fluid having zero viscosity is called ideal fluid. Such fluids are hypothetical and do not exist in reality.
No slip condition
Relative velocity of the fluid with respect to a solid surface is zero at the solid surface. This is precisely because of the frictional interaction between the fluid and the solid surface. And this is called no slip condition. The following figures shows various flow conditions and all shows the no slip condition. The various profiles of fluid flow are generated only because of no slip condition. Note that no slip condition is the result of viscosity of fluid.
Different flows showing no slip condition
Compressibility - it is the reciprocal of bulk modulus of elasticity. And bulk modulus of elasticity is given by,
Note that hire is the bulk modulus of elasticity means greater pressure would be required to cause a little change in the volume or density. For example for water bulk modulus of elasticity is 2x10^6 kilo Newton per square metre and the bulk modulus of elasticity of air at STP is 101 kilo Newton per square metre. Which air is approximately 2x10^4 times more compressible than water.
Note that the flow is ideal or not can be decided only on the basis of fluid that is when the fluid will be ideal the flow of fluid will be ideal and vice versa. But the flow is compressible or incompressible it cannot be decided on the basis of without the fluid is compressible or incompressible. So then how will we decide that whether the flow is compressible or incompressible and let us see it in our next section.
Compressible and incompressible fluid flows
A flow is said to be incompressible when the change in density volume of the fluid is less than 5%. If the change in density of the volume of the fluid flow is greater than 5% then the flow is said to be compressible. There is another parameter called Mach number which decides whether the flow is compressible or incompressible. Mach number is defined as the ratio of velocity of fluid flow to the acoustic velocity in that medium or fluid.
For air, the Mach number corresponding to 5% change in volume or density is equals 0.33. So for the flow of the air under STP condition when the Mach number is less than 0.33 then the flow is called incompressible flow and when the Mach number is greater than 0.33 then the flow is called compressible flow. the velocity of a corresponding to 0.33 Mach number is 110 metre per second has the acoustic velocity is 335 metre per second in air at STP condition. So, the flow of air at a speed less than 110 metre per second then the flow would be incompressible and if the velocity of air his more than 110 metre per second then the flow would be compressible.
Introduction
Fluid statics
Kinematics of fluid
Conservation equation for fluid flow
Fluid flow applications
Viscous compressible flow
Application of viscous compressible flow through pipes
Principle of similarity
Flow of Ideal fluid
Flow with a free surface
Introduction to laminar boundary layer Introduction to turbulent flow
Textbook Recommended
These are some important text books recommended for the preparation of Engineering services and Gate exams.
Fluid Mechanics and fluid machinery by RK Bansal
Introduction to Fluid Mechanics and fluid machines by SK Som and Biswas
Fluid Mechanics by PN Modi
Let's start our very first topic in fluid mechanics in which we will cover the comparison of solid and fluid, the concept of Continuum and the various properties of fluid like viscosity, surface tension, capillarity etc.
Comparison of solid and fluid
Solids can resist shear force or stress under static equilibrium condition with a finite deformation. It can partially or fully regain its original shape and size after the removal of stress.
On another hand fluid goes on continuously deforming under the application of shear stress i.e. fluid resist shear stress only under dynamic condition and not under static condition hence continuously formation takes place in fluid.
Also after the removal of shear stress or load the fluid cannot come back to its original position except a very special case of fluid which is known as visco elastic fluid there is why Fluids are also called zero memory substance because it forget its memory that where it was before the application of shear stress.
Concept of Continuum or continuous medium
Under normal temperature and pressure conditions there are about 3x10^25 molecules per cubic metres which is very very high and under such conditions the concept of Continuum easily holds. Under entire Fluid Mechanics we will only consider the Continuum cases and all the formulae and concepts of Fluid Mechanics are made for such Continuum cases. There is a very important parameter called Knudsen number which defines when the Continuum hold and when it is not.
Mean free path - It is simply the average distance between two successive collision of a molecule.
If 'm' is mean free path and 'L' is characteristic dimension of the problem then 'm/L' is defined as the Knudsen number. Knudsen number (Kn) serves as a criteria for continuum.
Kn = m/L
If, kn < 0.01 continuum is valid
0.01 < Kn < 0.1 slip flow
0.1 < Kn < 10 transition flow
Kn > 10 free molecular flow.
Viscosity - it is a property of fluid which comes into play when fluid is in motion. It has nothing to do with fluid at rest. It is the property by which a fluid resist the motion of one layer of fluid over another.
Effect of temperature on the viscosity of liquid and gases
In any fluid the viscosity depends upon the Cohesive forces between the molecules and the molecular Momentum transfer between the molecules. On increasing the temperature the Cohesive forces decreases why the molecular Momentum transfer increases. In liquids the Cohesive forces dominate while in gases in the molecular Momentum transfer predominant. so on increasing the temperature the Cohesive forces in liquid decreases and so is the viscosity while in gases on increasing temperature the molecular Momentum transfer increases and hence the viscosity increases in the gases on increasing the temperature.
Newton's law of viscosity - According to this love this year is stress is directly proportional to the rate of shear strain in any fluid. All those flutes which obey Newton's law of viscosity are called newtonian fluids. Examples of newtonian fluids are water, mercury, oil etc. While the fridge which do not follow the Newton's law of viscosity are called non newtonian fluids. Examples of such fluids milk, blood, polymer solution, higher molecular oil, Ink etc.
Ostwald-de-waele model - All the fluids either newtonian and non newtonian can be expressed in a generalized form and this form is called Ostwald-de-waele model. This is expressed by the expression given below.
Where 'm' is called flow behaviour index while 'n' is called follow consistency index.
Note that for newtonian fluid viscosity is the property of fluid while viscosity is not a property of fluid for non newtonian fluid.
Ideal fluid - A fluid having zero viscosity is called ideal fluid. Such fluids are hypothetical and do not exist in reality.
No slip condition
Relative velocity of the fluid with respect to a solid surface is zero at the solid surface. This is precisely because of the frictional interaction between the fluid and the solid surface. And this is called no slip condition. The following figures shows various flow conditions and all shows the no slip condition. The various profiles of fluid flow are generated only because of no slip condition. Note that no slip condition is the result of viscosity of fluid.
Different flows showing no slip condition
Compressibility - it is the reciprocal of bulk modulus of elasticity. And bulk modulus of elasticity is given by,
Note that hire is the bulk modulus of elasticity means greater pressure would be required to cause a little change in the volume or density. For example for water bulk modulus of elasticity is 2x10^6 kilo Newton per square metre and the bulk modulus of elasticity of air at STP is 101 kilo Newton per square metre. Which air is approximately 2x10^4 times more compressible than water.
Note that the flow is ideal or not can be decided only on the basis of fluid that is when the fluid will be ideal the flow of fluid will be ideal and vice versa. But the flow is compressible or incompressible it cannot be decided on the basis of without the fluid is compressible or incompressible. So then how will we decide that whether the flow is compressible or incompressible and let us see it in our next section.
Compressible and incompressible fluid flows
A flow is said to be incompressible when the change in density volume of the fluid is less than 5%. If the change in density of the volume of the fluid flow is greater than 5% then the flow is said to be compressible. There is another parameter called Mach number which decides whether the flow is compressible or incompressible. Mach number is defined as the ratio of velocity of fluid flow to the acoustic velocity in that medium or fluid.
For air, the Mach number corresponding to 5% change in volume or density is equals 0.33. So for the flow of the air under STP condition when the Mach number is less than 0.33 then the flow is called incompressible flow and when the Mach number is greater than 0.33 then the flow is called compressible flow. the velocity of a corresponding to 0.33 Mach number is 110 metre per second has the acoustic velocity is 335 metre per second in air at STP condition. So, the flow of air at a speed less than 110 metre per second then the flow would be incompressible and if the velocity of air his more than 110 metre per second then the flow would be compressible.
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