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**General Introduction**

**Basics****States of matter**- Solid
- Liquid
- Gas

**Fluid**The basic definition of fluid is that it is a substance which is capable of flowing. Liquids and gasses come under the category of fluid.**Mechanics**Mechanics is the study of force and motion.**Fluid Mechanics**Fluid Mechanics is the science which deals with the behavior of fluids at rest and in motion.**Hydraulics**Hydraulics is the science which deals with the behavior of water at rest and in motion.**Branches of Fluid Mechanics**- Fluid Statics: Fluid statics is the study of fluids at rest.
- Fluid Dynamics: Fluid statistics is the study of fluids in motion. It is classified into two branches.
- Fluid Kinematics: Fluid Kinematics is the study of fluid motion without considering the causes of motion (forces).
- Fluid Kinetics: Fluid Kinetics is the study of fluid motion by considering the causes of motion (forces).

**Application of Fluid Mechanics**- Water distribution and sanitation
- Dams
- Irrigation
- Pumps and Turbine
- Water retaining structures
- Flood flow analysis
- Flow of air in and around buildings
- Bridge piers in rivers
- Ground-water flow

**Shear Stress in Moving Fluid**A stress is a force per unit area over which it acts. Stresses have both magnitude and direction, and the direction is relative to the surface on which the stress acts. There are two types of stresses:- Normal stress: The stress which acts perpendicular to the surface is normal stress.
- Tangential stress: The stress which acts along the surface is tangential stress. Shear stress is tangential stress.

**Differences Between Solid and Fluid**Their differences are here listed as:- Fluids lack the ability of solids to resist deformation.
- For a solid, strain is a function of applied stress, provided the elastic limit is not exceeded. For a fluid, the rate of strain is proportional to applied stress.
- In a fluid shear strain increases for as long as the shear stress is applied. This means the fluid flows as long as the forces act and will not recover its original position when the force is removed. In a solid shear strain is constant for a fixed shear stress, and if the elastic limit is not exceeded, the deformation disappears when the force is removed.

**System and Control Volume**

Control volume: A control volume is a fixed region in space through which fluid flows. The region is usually at a fixed location and fixed size. The boundary of the system is its control surface and its shape does not change with time. The element within the control volume obeys the physical laws. This approach makes mathematical analysis simpler.

As the fluid flows continuously, only a part of it is considered for analysis. The control volume is chosen arbitrarily for reasons of convenience of analysis. Control surface follows solid boundaries if present.**Continuum Concept**In Fluid Mechanics, a fluid is considered as a continuous substance. This concept is called continuum concept. In this concept, the molecular structure of the fluid is not considered and the separation between molecules is neglected. The fluid properties such as velocity and pressure are a continuous function of space and time. The fluid properties can be considered to be constant at any point in space, which is the average of a large number of molecules surrounding that point within a characteristic distance. Using continuum concept, the mathematical equations relating the physical laws can be derived easily as we don’t need to consider the motion of the individual molecule. This concept is not valid if the mean free path of molecules is greater than the characteristic dimension of fluid considered for analysis.**Mass Density**Density is about the compactness in the molecular arrangement in any substance which decides how heavier or lighter any substance is relative to other. The density of a fluid is defined as its mass per unit volume.

Where m = mass, V = Volume and D = density

Unit: kg/m3

Dimension: ML-3

Density decreases with the increase of temperature and increases with the increase of pressure. As the temperature increases, molecular activity increases and spacing between molecules increases, thus increasing volume and reducing density. If pressure is increased, a large number of molecules can be forced into a given volume, thus reducing volume and increasing density.**Specific Weight**The specific weight (also known as the unit weight) is the weight per unit volume of a material.The symbol of specific weight is \(\gamma\). where the specific weight of the material (weight per unit volume, typically N/m^{3}units) is the density of the material (mass per unit volume, typically kg/m^{3}),'g' is acceleration due to gravity (rate of change of velocity , given in m/s^{2}, and on Earth usually given as 9.81 m/s^{2}) Unit: N/m3

Dimension: ML-2T-2**specific Gravity**

Specific gravity (or relative density) is the ratio of specific weight (or density) of a fluid to that of standard fluid. In the case of liquid, the standard fluid is water at 40C.

Unit: As it is the ratio, it does not have the unit.**Specific Volume**The specific volume of a fluid is defined as its volume per unit mass.

the specific volume of a substance is the ratio of the substance's volume to its mass. It is the reciprocal of density and an intrinsic property of matter as well. Specific volume is defined as the number of cubic meters occupied by one kilogram of a particular substance. The standard unit is the cubic meter per kilogram .V= Volume, m = mass Unit: m3/kg Dimension: M-1L3**Viscosity**Viscosity is the property of a fluid due to which it offers resistance to shear. It is a measure of internal friction which causes resistance to flow.

The molecules of gas are not rigidly constrained and cohesive forces are small. So, the molecular mass interchange (momentum) is the cause of viscosity in a gas. As cohesive forces are significant in a liquid, both mass interchange and cohesion contribute to the viscosity of the liquid.

Viscosity is practically independent upon pressure and depends on temperature only. If the temperature increases, the molecular interchange will increase. Therefore, the viscosity of a gas will increase with the increase in temperature. Cohesion is the predominant cause of viscosity in liquid and since cohesion decreases with temperature, the viscosity of liquid decreases with increase in temperature.**Newton's law of Viscosity**The shear stress on a fluid element layer is directly proportional to the rate of strain (or) velocity gradient, the constant of proportionality being called the coefficient of viscosity.

shear stress ( t ) = \(m(\frac{du}{dy})\)**No Slip Boundary Condition**A fluid flowing over a stationary surface comes to a complete stop at the surface because of the no-slip condition as shown in fig below.**Concept Of Newton's Law of Viscocity**

We know that, Distance = \(Velocity\times time\)

Hence, Distance = \(V\times t\) = \(du\times dt\)

For small angles, \(tan(d\theta)\) = \(d\theta)\)

Hence, (\(\frac{d\theta}{dt}\)) = rate of angular deformation

And, \((\frac{du}{dy})\) = velocity gradient

It is hence concluded that it is found that tangential stress is proportional to velocity gradient (in a direction perpendicular to the layers), Here stress between two layers.gradient (in a direction perpendicular to the layers), Here stress between two layers.

Hence,T= \(\mu(\frac{du}{dy})\) = \(\mu (\frac{d\theta}{dt})\) --------------> Law of Viscosity (i.e Newton's law of viscosity)

where, \(\mu\) = Dynamic viscosity or Coefficient of dynamic viscosity

T = shear stress and \((\frac{du}{dy})\) = Velocity Gradient

This formula doesn't hold good for every fluid. Those fluids who follow this law are called Newtonian fluids and those who don’t follow this law are called non-Newtonian fluids.**Causes Of Viscosity**Viscosity is caused by shear stress in a moving fluid. For a fluid at rest, there is no shear stress. When one layer of the fluid moves relative to an adjacent layer, transfer of molecular momentum sets up shear stress which resists the relative motion. The measure of the motion of one layer relative to an adjacent layer is velocity gradient, du/dy. According to Newton’s law of viscosity, shear stress varies linearly with the velocity gradient.

**compressibility**Measure of volume change a fluid as a response to pressure (or stress) change is called compressibility..**Capillarity and Surface Tension****Surface Tension**Surface tension is defined as the tensile force per unit length acting on a line lying in the interface of two fluids. The force is normal to the imaginary line in the surface, tangent to the free surface and is same at all points. Surface tension is constant at any given temperature for the surface of the separation of two particular substances but it decreases with increase in temperature because attractive force becomes apparent as the average kinetic energy of molecules increases. Intermolecular attraction is the cause of surface tension. A molecule within the body of a liquid is equally attracted in all directions by the other molecules surrounding it. At the interface between two fluids, the upward and downward attractions are unbalanced, and the surface molecules are pulled inward making the surface like an elastic membrane. The effect of surface tension is to reduce the surface of a free body of a liquid to a minimum (formation of spherical drop).

unit: N/m

Dimension: MT-2**Capillarity**Capillarity is the rise or fall of liquid in a column of very small diameter when the latter is dipped in it. It is caused by surface tension as well as adhesion (attraction between molecules of different substances) and cohesion (attraction between molecules of same liquid).**Derivation of Formula for Capillarity**

\(F_{up}\) = \(\gamma(2\pi a)cos\theta\)

Here \(\gamma\) = surface tension (of liquid air ) at NTP, \(2\pi a\) = circumference of the tube, and \(\theta\) = angle of contact.

Now the downward force due to gravity is given by:

\(F_{down}\) = \(\rho g(h\pi a^2)\)

Here, \(\rho\) = 1000 kg/m3=density of water, g = 9.8 m/s2 = acceleration due to gravity, and \((hπa^2)\) = volume of the water in the column.

Equating above two equation, we get: \(\gamma\) = \(\frac{\rho g a}{2} \frac{h}{cos \theta}\)

For water and glass(pure and clean), the contact angle is nearly zero. However, it varies so term cos0=1

\(\gamma \approx \frac{\rho ga}{2}h\)

**Cavitation and Vapour Pressure**Liquid evaporates because of molecules with sufficient kinetic energy escaping from the liquid surface. The vapor molecules exert a partial pressure in the space, which is called vapor pressure. Vapor pressure depends on temperature and increases with it. In equilibrium, the number of molecules striking the surface and condensing is equal to the number of escaping molecules. When the pressure above a liquid equals the vapor pressure of the liquid, boiling occurs. When the flow of liquid passes through a region having the pressure less than vapor pressure, there will be local boiling and a cloud of vapor bubbles will form. This phenomenon is known as cavitation. The bubbles of low-pressure zone move towards the high-pressure zone and collapse under that pressure. If this occurs in contact with a solid surface, serious damage can result. Cavitation can affect the performance of hydraulic machineries such as propellers, turbines and pumps and the impact of collapsing bubbles can cause local erosion of metal surface.**Classification Of Fluid****Newtonian and Non-Newtonian Fluids**Fluids which obey Newton’s law of viscosity are called Newtonian fluids. E.g. water, light oil, air, milk, glycerin, kerosene. For Newtonian fluids, the viscosity is constant i.e. viscosity depends on temperature only. Fluids which do not obey Newton’s law of viscosity are called Non-Newtonian fluids. E.g. paint, sewage sludge, crude oil. Viscosity is not constant for Non-Newtonian fluids i.e. viscosity depends on temperature, rate of strain and time.

**Bibliography**

Bansal, R. K. (2005). A Textbook of fluid mechanics. New Delhi: Laxmi Publications.

Douglas, J. F., Swaffield, J., Gasoriek, J. M., & Jack, L. (2008). Fluid Mechanics. London: Pearson.

Dulal, K. N. (2013). A complete note on fluid mechanics. Kathmandu: Printed.

Modi, P. N., & Seth, S. N. (2013). Hydraulics and fluid mechanics including hydraulic machine. Banglore: Standard Publishers Distributors.

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