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# Note for FLUID MECHANICS AND MACHINERY - FMM by BIBEKANANDA SAHOO

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#### Note for FLUID MECHANICS AND MACHINERY - FMM by BIBEKANANDA SAHOO

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www.Vidyarthiplus.com S K Mondal’s 1. Properties of Fluids Chapter 1 Pr oper ties of Fluids Contents of this chapter 1. Definition of Fluid 2. Characteristics of Fluid 3. Ideal and Real Fluids 4. Viscosity 5. Units of Viscosity 6. Kinematic Viscosity 7. Units of Kinematic Viscosity 8. Classification of Fluids 9. Effect of Temperature on Viscosity 10. Effect of Pressure on Viscosity 11. Surface Tension 12. Pressure Inside a Curved Surface 13. Capillarity 14. Derive the Expression for Capillary Rise Page 3 of 307 www.Vidyarthiplus.com

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www.Vidyarthiplus.com S K Mondal’s Properties of Fluids Chapter 1 Theory at a Glance (for IES, GATE, PSU) Definition of Fluid A fluid is a substance which deforms continuously when subjected to external shearing forces. Characteristics of Fluid 1. It has no definite shape of its own, but conforms to the shape of the containing vessel. 2. Even a small amount of shear force exerted on a fluid will cause it to undergo a deformation which continues as long as the force continues to be applied. 3. It is interesting to note that a solid suffers strain when subjected to shear forces whereas a fluid suffers Rate of Strain i.e. it flows under similar circumstances. Concept of Continuum The concept of continuum is a kind of idealization of the continuous description of matter where the properties of the matter are considered as continuous functions of space variables. Although any matter is composed of several molecules, the concept of continuum assumes a continuous distribution of mass within the matter or system with no empty space, instead of the actual conglomeration of separate molecules. Describing a fluid flow quantitatively makes it necessary to assume that flow variables (pressure, velocity etc.) and fluid properties vary continuously from one point to another. Mathematical descriptions of flow on this basis have proved to be reliable and treatment of fluid medium as a continuum has firmly become established. For example density at a point is normally defined as Here Δ is the volume of the fluid element and m is the mass If Δ is very large ρ is affected by the inhomogeneities in the fluid medium. Considering another extreme if Δ is very small, random movement of atoms (or molecules) would change their number at different times. In the continuum approximation point density is defined at the smallest magnitude of Δ , before statistical fluctuations become significant. This is called continuum limit and is denoted by Δ c. One of the factors considered important in determining the validity of continuum model is molecular density. It is the distance between the molecules which is Page 4 of 307 www.Vidyarthiplus.com

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www.Vidyarthiplus.com S K Mondal’s Properties of Fluids Chapter 1 characterised by mean free path ( λ ). It is calculated by finding statistical average distance the molecules travel between two successive collisions. If the mean free path is very small as compared with some characteristic length in the flow domain (i.e., the molecular density is very high) then the gas can be treated as a continuous medium. If the mean free path is large in comparison to some characteristic length, the gas cannot be considered continuous and it should be analysed by the molecular theory. A dimensionless parameter known as Knudsen number, Kn = λ / L, where λ is the mean free path and L is the characteristic length. It describes the degree of departure from continuum. Usually when Kn> 0.01, the concept of continuum does not hold good. Beyond this critical range of Knudsen number, the flows are known as slip flow (0.01 < Kn < 0.1), transition flow (0.1 < Kn < 10) and free-molecule flow (Kn > 10). However, for the flow regimes considered in this course, K n is always less than 0.01 and it is usual to say that the fluid is a continuum. Other factor which checks the validity of continuum is the elapsed time between collisions. The time should be small enough so that the random statistical description of molecular activity holds good. In continuum approach, fluid properties such as density, viscosity, thermal conductivity, temperature, etc. can be expressed as continuous functions of space and time. Ideal and Real Fluids 1. Ideal Fluid An ideal fluid is one which has no viscosity no surface tension and incompressible 2. Real Fluid An Real fluid is one which has viscosity surface tension and compressible Naturally available all fluids are real fluid. Viscosity Definition: Viscosity is the property of a fluid which determines its resistance to shearing stresses. Cause of Viscosity: It is due to cohesion and molecular momentum exchange between fluid layers. Newton’s Law of Viscosity: It states that the shear stress (τ) on a fluid element layer is directly proportional to the rate of shear strain. The constant of proportionality is called the co-efficient of viscosity. Page 5 of 307 www.Vidyarthiplus.com

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www.Vidyarthiplus.com Properties of Fluids S K Mondal’s Chapter 1 When two layers of fluid, at a distance ‘dy’ apart, move one over the other at different velocities, say u and u+du. Velocity gradient = du dy According to Newton’s law τ∞ du dy Velocity Variation near a solid boundary or du τ=µ dy Where µ = constant of proportionality and is known as co-efficient of Dynamic viscosity or only Viscosity Thus viscosity may also be defined as the shear stress required As µ= τ  du   dy    producing unit rate of shear strain. Units of Viscosity S.I. Units: Pa.s or N.s/m2 C.G.S Unit of viscosity is Poise= dyne-sec/cm2 One Poise= 0.1 Pa.s 1/100 Poise is called centipoises. Dynamic viscosity of water at 20oC is approx= 1 cP Kinematic Viscosity It is the ratio between the dynamic viscosity and density of fluid and denoted by Mathematically ν= dynamic viscosity µ = density ρ Units of Kinematic Viscosity S.I units: m2/s C.G.S units: stoke = cm2/sec One stoke = 10-4 m2/s Page 6 of 307 www.Vidyarthiplus.com