W1_Lecture2_FM_SD_IITG.pdf Fluid Mechanics
- 1. Indian Institute of Technology Guwahati Department of Civil Engineering, Guwahati, Assam 781039 FLUID MECHANICS for Civil and Mechanical Engineering by Prof. Dr. Subashisa Dutta subashisa@iitg.ac.in Title: Fluid Properties
- 2. S Dutta/Professor/CE/IIT/Guwahati Course Content 1. Introduction, Basic Concepts and : Week 1 Properties of Fluids 2. Pressure and Fluid Statistics : Week 2 3. Fluid Kinematics : Week 3 4. Mass, Bernoulli, and Energy Equations : Week 4 & Week 5 5. Momentum Analysis of Flow Systems : Week 6 6. Dimensional Analysis And Modeling : Week 7 7. Flow in Pipes : Week 8
- 3. Contents of Lecture 1. Recap of previous lecture 2. The microscopic point of view and fluid as a continuum 3. Microscopic and macroscopic uncertainty 4. Definitions of density, specific volume, specific gravity and specific weight 5. Newton's law of viscosity: Microscopic and macroscopic point of view 6. Effect of temperature and pressure on viscosity 7. Surface tension 8. Summary
- 4. Recap of the Previous Lecture 1. FLUID MECHANICS is the science that deals with behavior of fluids at rest or in motion and the interaction of fluids with solids or other fluids at the boundaries 2. A fluid in direct contact with a solid surface sticks to the surface and there is no slip. This condition is called as NO-SLIP CONDITION Classification of Fluid Flows: 1. Flow of unbounded fluid over surface 2. Flow of fluid in pipe or duct External flow Internal flow 3. Density of fluid flow is constant 4. Density variability in fluid flow Incompressible flow Compressible flow 5. No change in flow with time 6. Change in flow with time Steady flow Unsteady flow 7. Depending on flow properties changing in directions 1D, 2D, 3D flows 8. Smooth layered flow 9. Rough flow with eddies and mixing between the layers Laminar flow Turbulent flows
- 5. The Microscopic point of view and the fluid as a continuum Fluid : Composed of molecules in constant motion and collision. Mean free path : The average distance travelled of a molecule before its collision. E.g. Number of O2 molecules in 1mm3 volume = 3 1016 Mean free path of oxygen molecules at 1 atm pressure and 20o C temperature = 6.3 10-8 m Mean free path 200 diameter of the oxygen molecule
- 6. Let us examine the fluid density, ratio between the mass m and its volume V = 𝛿𝑚 𝛿𝑉 As sampling volume V is very small, random molecular motion dominates. varies considerably due to microscopic uncertainty as shown in the Figure 10-9 mm3
- 7. Macroscopic uncertainty : If sampling volume V is too large, there is density variation within the volume. Then. the average density () must differ from the actual density at the centre of the sampling volume. It is caused by spatial variation of the density. This is called Macroscopic Uncertainty. That’s why, the sampling volume (V*) not too large nor too tiny and must contain around million molecules. the density at a point in space in defined as = lim 𝑉→𝑉∗ 𝑚 𝑉 can be defined as a continuous and continuously differentiate function = (x, y, z, t) Then, Pressure p = p(x, y, z, t), u = u(x, y, z, t), v = v(x, y, z, t), w = w(x, y, z, t) Velocity Vector 𝑉 = u Ƹ 𝑖 + v Ƹ 𝑗 + w 𝑘 Non Validation : Rarefied gases (where pressure close to zero), microfluidic applications, the flow devices at the micro and nanoscales, the continuum hypothesis does not valid.
- 8. Density, Specific Volume,Specific Gravity and Specific weight– Density is defined as mass per unit volume = 𝑚 𝑉 (kg/m3) Specific Volume is defined as volume per unit mass = 𝑉 𝑚 = 1 (m3/kg) Specific Gravity is the ratio of density of substance to density of well known substance ( water considered here) SG = 𝜌 𝜌𝑤𝑎𝑡𝑒𝑟 Specific Weight is the weight of unit volume of a substance = g (N/m3) Weight of a fluid (weight of air in a room) = gv (volume of the room) Weight of the total fluid column = σ𝑖=1 𝑛 𝜌i g Vi
- 9. Newton’s law of viscosity : Laminar flow (well ordered parallel flow) between plates. Microscopic Analysis : Layer A, tending to reduce the velocity due to the molecular mass and momentum flux exchange whereas Layer B due to the exchange, tending to increase the velocity for attending the equilibrium. Thus, Shear stress, 𝜏, developed at the interface of element (A) and element (B).
- 10. Experimentally established, the shear stress,𝜏, is direct proportional to the velocity gradient ( 𝜕v 𝜕𝑥 ) 𝜏 α 𝑑𝑣 𝑑𝑥 𝜏 = μ 𝑑𝑣 𝑑𝑥 Where μ is coefficient of viscosity. The concept is similar to frictional resistance in motion of solid objects. 1 poise = 1g/cm.s in CGS system
- 11. Microscopic point of view Let us consider a fluid element (ABCD) subjected to contact shear force. After time t, the element (ABCD) has gone under deformation to the new shape (EBFC). The shear deformation () can be approximated as = 𝑉∆𝑡 𝑙 or 𝜕 𝜕𝑡 = 𝑉 𝑙 ……………………....1 As approximation of linear velocity distribution 𝑢 𝑦 = V 𝑙 𝑑𝑢 𝑑𝑦 = 𝑉 𝑙 ………………………. 2 Then the shear rate is 𝜕 𝜕𝑡 = 𝑑𝑢 𝑑𝑦 Newton’s law of viscosity, Shear stress shear strain rate 𝜏 α 𝜕 𝜕𝑡 or 𝜏 α 𝑑𝑢 𝑑𝑦 𝜏 = μ 𝑑𝑢 𝑑𝑦
- 12. Effect of temperature and pressure variation on viscosity For liquids, Change of pressure -> very nominal change in μ. Increase of temperature ≈ Reducing inter molecular forces Decrease trend of μ. For gases, Change of pressure ≈ very nominal change in μ. Increase of temperature Increase of random motion of molecule Increase trend of μ. The Sutherland correlation (from the US Standard Atmosphere) μ = 𝑎𝑇 ൗ 1 2 1+𝑏𝑇 for gases μ = 𝑎10b∕ T−c for liquids where, T is absolute temperature. a, b and c are experimentally estimated constants.
- 13. Variation of 𝑪𝒐𝒆𝒇𝒇𝒊𝒄𝒆𝒏𝒕 𝒐𝒇 𝒗𝒊𝒔𝒄𝒐𝒔𝒊𝒕𝒚 𝛍 𝜏 = μ 𝑑𝑢 𝑑𝑦 If 𝑑𝑢 𝑑𝑦 = 1 𝜏 = μ For Non- Newtonian fluids, apparent viscosity (μ), the slope at the particular deformation rate is used. Increase of μ with increase of “deformation rate” Shear thickening fluid (Dilatant). E.g. solutions with suspended starch or sand. Decrease of μ, with increase of deformation rate -> shear thinning fluids, Pseudoplastic. E.g. paints, polymer solution etc. Bingham plastic, resist a finite shear stress and then deform continuously where the shear stress exceeds the yield stress E.g. tooth paste.
- 14. Surface Tension At the interface of liquid and gas, net force due to imbalance of cohesive ( like molecules) and adhesive forces ( unlike molecules) as shown in the Figure • The interface surface of the liquid acts like a stretched elastic membrane under tension • The pulling force that causes this tension acts parallel to the surface and is due to the attractive forces between the molecules of the liquid. • The magnitude of this force per unit length is called surface tension Unit: N/m Dimension: MT -2 Typical values: Water 0.074 N/m at 20oC with air surface tension decreases with the liquid temperature because intermolecular cohesive forces decreases
- 15. Surface Tension Pressure difference at the interface consider a small spherical droplet of a fluid at rest. 𝑃𝑙𝑖𝑞 − 𝑃𝑔𝑎𝑠 𝜋𝑟2 = 𝜎(2𝜋𝑟) Angle of Contact The angle between the solid surface and the tangent to the surface of the liquid at the contact point Non-wetting front Wetting front
- 16. Summary of the Lecture 1. MICROSCOPIC and MACROSCOPIC point of view in Fluid Mechanics 2. Newton's Law of Viscosity 𝜏 = μ 𝑑𝑢 𝑑𝑦 Definitions: 1. Density Mass per unit volume 2. Specific Volume Volume per unit mass 3. Specific Gravity Ratio of density of substance to density of well known substance 4. Specific Weight Weight of unit volume of a substance 5. Surface Tension Force acting per unit length at the interface Fluid Effect of Temperature Effect of Pressure Liquids Viscosity Decreases Very Nominal Gases Viscosity Increases Very Nominal