Fluid mechanics

Fluid Mechanics: An applied science

Part A

Abstract

This paper identifies fluid mechanics as an essential component of engineering and its applications in everyday life. The fundamental formula and equations have been presented in a lucid style so that the student develops keen interest in the basics and can use his imagination to the maximum for best understanding of concepts.

Fluid mechanics 

It is that part of applied science that deals with the ways fluids behave during rest, motion, subjected to pressure or temperature. It is a sub-discipline of continum mechanics which refers to the study of stressed solids.

Fluid is any substance which does not permanently resist deformation.

Compressible and incompressible fluids

If the density changes slightly with moderate changes in temperature and pressure, the fluid is said to be incompressible. Generally liquids are incompressible. The density of a gas generally changes significantly with thermal and pressure changes and so gases fall under the category of compressible fluids. This property of gases facilitates their storage in limited space and accounts for CNG fuelled vehicles.

Fluids may be categorized as Newtonian and Non-Newtonian.

Newton’s Law:
According to this law, the shear stress relative to two adjacent layers is proportional to the velocity and inversely proportional to the distance between layers. 

τ = μ du/dy

The constant of proportionality is called the coefficient of viscosity.

Higher the viscosity, higher will be the value of shear stress. 

The fluids which obey Newton’s law are referred to as Newtonian fluids.

Analogy between Hooke’s Law and Newton’s Law

Hooke’s Law states that stress is directly proportional to strain, the constant of proportionality being referred to as the coefficient of elasticity. While Hooke’s Law applies to solid materials, Newton’s Law applies to fluids.

Elasticity is an index of strength of solid as it indicates the ability of a body to get restored to its initial form once the deforming force is removed. Viscosity is an index of liquid strength. It reflects the ability of fluid particles to move closely and in continuum. Viscosity is an intensive property and depends upon the fluid, generally not affected by external force. However an external change that affects viscosity is temperature. When the temperature of liquid is increased, it becomes less viscous.

Pressure:

It refers to force per unit surface area. It exists at every point within the entire fluid volume. Pressure energy also known as hydraulic energy plays a significant role in designing pumps and compressors. In those devices, the kinetic energy of water is transformed into pressure energy helping in the lifting of water over a given height.

From the first law of energy-

Pressure head + Kinetic head+ height = constant

Barometric pressure:

It is atmospheric pressure, first stated by Torricelli. Its value is 1 atm. Pressure applied by vacuum is called vacuum pressure. Whenever a vacuum is formed, air is drawn inwards to make up for pressure difference. This principle is utilized in the making of vacuum cleaners. The power of vacuum machines can be evaluated by knowing the vacuum pressure.

Hydrostatic equilibrium

A thin section of a vertical column of liquid of height h and cross sectional area S is acted upon by the resultant of three forces:

1. Force from pressure p acting upwards i.e., pS.
2. Force from pressure p acting downwards i.e., (p + dp)S.

iii. Gravitational force acting downwards.

Hence, pS – (p+dp)S – gρSdZ = 0

or, dp + gρdZ  = 0
Note that forces are written as per sign convention.
In case of incompressible fluids, i.e. for constant density the second equation is integrated, which yields

p/ ρ + gZ = constant.

Manometer: It is a device used for measuring pressure differences. The working principle here is that the meniscus in one branch is higher than the other. 

Potential flow (Irrotational flow):
Fluid kinetics is highly affected by the presence of solid boundary. If the impact produced by solid wall is less, the shear stress would become negligible and approach the behaviour of ideal fluid which is incompressible. This type of flow is known as potential flow and obeys the principles of Newtonian mechanics and energy conservation.

Characteristics of potential flow:

1. There is no eddy formation within the stream.
2. There is no friction and hence no heat dissipation.

Prandtl’s boundary layer theory:

The layer of fluid in contact with the solid wall is called boundary layer. It is the boundary layer which is subjected to shearing forces. Except the boundary layer, the remaining part of the fluid is under potential flow conditions.

Due to the presence of solid boundary, the results can be:

1. Coupling of velocity gradients and shear stress fields.
2. Onset of non laminarity/ turbulence.
3. Formation of boundary layers.
4. Separation of boundary layers.

Velocity field: The fluid flowing past a solid wall adheres to it leading to interfacial tension at the surface.

If the solid boundary is at rest, velocity is zero. But it increases gradually as we move away from the solid wall. Hence velocity at a given point is a function of the spatial coordinates and time. 

Steady flow: It refers to the case of fluid flow where velocity at each location is constant and field is invariant with time.

One dimensional flow: If the flow has only single velocity component, such flow is called one dimensional flow eg. Flow through straight pipe.

When fluids are flowing with very low velocities, lateral mixing is minimal. Adjacent layers slide past one another and no cross currents are formed.

Onset of turbulence/ Reynold’s experiment:
A glass tube was immersed in a tank of water with a solid glass boundary. A fine filament of colored water is made to enter in the tank. When flow rate is less, the filament was observed to flow along the mainstream and there was no mixing. Osborne Reynolds observed that at a certain velocity known as critical velocity, the thread became wavy and the color diffused throughout the water. Hence the motion of water has changed from laminar to turbulent.

Reynold’s no.:-
It is a dimensionless number which is given as the ratio of inertial to viscous forces. Reynold’s no. is a function of density, velocity, diameter and viscosity. Between Reynold’s no. of 2100 and 4000, the transition region is found where the nature of fluid flow shifts from laminar to turbulent. 

Turbulence is concerned with the flow of eddies of various sizes coexisting in the flowing stream. The causes of turbulence are-

1. Contact of flowing stream with solid boundary
2. Contact between two layers of fluid moving at different velocities. Eg. Flow of jet of water into a mass of stagnant fluid

The first case leads to wall turbulence while the second leads to free turbulence.

Turbulence is not disordered motion, but the transfer of energy between eddies. The energy of largest eddies is derived from potential energy of bulk fluid. Smaller eddies get lost due to impact of viscous shear. Inside a single eddy flow is laminar. Energy conversion by viscous action is called viscous dissipation. The velocity of turbulent fluid can be described with the help of statistical distribution.
The pressure at a selected point fluctuates rapidly and simultaneously with velocity fluctuations. Oscillographs are used for graphical observation and study of such pressure fluctuations. Studies show that the unpredictability associated with fluctuations is constrained between definite limits, displaying mathematical linear functions. Quantitative characterization of turbulence is done by the statistical analyses of frequency distributions.

Total instantaneous velocity can be split into two components-

1. Time average of component in direction of flow
2. Instantaneous fluctuation of component around the mean i.e. deviating velocity.

Consider one dimensional flow of fluid i.e. in x direction.
u_i  = u + u’ where u_i  is the instantaneous velocity , u is the constant velocity of stream in x direction and u’ the deviation velocity in x direction.

In case of fluctuating pressure, p_i  = p + p’.

Isotropic turbulence: This arises in case of absence of velocity gradient or in presence of constant velocity distribution. Isotropic turbulence can be observed at the centre of a pipe or outer edge of boundary layer.

The impact of shear stress is quite high in presence of velocity gradient across a shear plane as inside large eddies. The mechanism of turbulent shear depends upon deviating velocities in anisotropic turbulence. Turbulent shear stresses are also referred to as Reynolds stress.

Consider a fluid in turbulent flow moving in positive x direction. The flow is parallel to plane S. The mean velocity u increases with y and hence the velocity gradient is positive.

Shear stress in case of turbulent flow(τ_t) is related to the velocity gradient by the following equation.

τ_t =  E_{v *}  du/dy where E_v = eddy viscosity

It can be seen that eddy viscosity is analogous to viscosity(μ) in the Newtonian shear stress equation.

The total shear stress acting upon a fluid element shall be the resultant of viscosity and eddy viscosity.

Total shear stress = (μ + E_v ) * du/dy

Eddy viscosity is not only a fluid property but affected by factors influencing detailed patterns of turbulence, deviating velocities, location, scale and intensity. 

Boundary Layer theory

A boundary layer is defined as that part of a moving fluid in which the fluid motion is influenced by the presence of a solid boundary.
Consider the flow of fluid parallel with a thin plate. The velocity of the fluid upstream from the leading edge of the plate is uniform across the entire fluid stream. The velocity of the fluid at the interface between solid and fluid is zero. The velocity increases with distance from the plate.

Equations of fluid flow

The most important equations are based on the principles of mass balance or continuity. Consider an element of fluid i.e. part of a macroscopic flowing system. Then according to law of conservation of mass.

Rate of (mass inflow – mass outflow)= Rate of mass accumulation.

Flux is defined as rate of flow of any quantity per unit area.

One dimensional flow-

Streamline: It is an imaginary path in the mass of flowing fluid such that at every point the vector of net velocity along it is acting tangentially. So there is zero net flow across it. 

Streamtube: An imaginary pipe in the mass of fluid such that there is no net flow through the walls. Local velocity will vary across the cross section. 

Differential momentum balance

Rate of momentum accumulation = rate of momentum(entering – leaving) + sum of forces acting on the system

1. Navier Stokes equation: It applies to constant density and viscosity
2. Euler’s equation: It applies to fluids with constant density and zero viscosity.

 Couette flow

When the plates are horizontal, the fluid velocity varies linearly with distance from the stationary plate and the velocity gradient is constant. Viscosity in such a case becomes a function of shear stress.

Eg. Flow between a rotating cylinder and concentric stationary cylinder is referred to as Couette flow.

Mechanical energy equation-

Bernoulli’s equation: It states that the sum total of energies at a point are constant. The sum of pressure head, kinetic head and height at a point are constant. If one of the parameters is varied, the others are varied to make up for the difference.

Applications of Bernoulli’s equations-

1. Flight of an Airplane/Aerodynamics: Due to the streamlined shape, air has to cover greater distance over the curvilinear surface. This creates a velocity difference and thus a pressure difference, providing a lift to the airplane.       
2. Movement of spinning ball: Spin is created by using differential air pressure and the curvature of the ball.

Part B contents:-

1. Dimensionless no.s
2. Models and prototypes
3. more on floating bodies
4. pumps and compressors
5. surface tension

 and more.