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Note for Fluid Mechanics - FM By Ravichandran Rao

  • Fluid Mechanics - FM
  • Note
  • West Bengal University of technology - WBUT
  • Mechanical Engineering
  • 5 Topics
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The relative density (or specific gravity) is the ratio of a fluid density to the density of a standard reference fluid maintained at the same temperature and pressure: ρ gas For gases: RDgas = For liquids: RDliquid = ρ air = ρ liquid ρ water ρ gas 1205 . kg / m 3 = ρ liquid 1000 kg / m 3 Viscosity Viscosity is a measure of a fluid’s resistance to flow. The viscosity of a liquid is related to the ease with which the molecules can move with respect to one another. Thus the viscosity of a liquid depends on the: • Strength of attractive forces between molecules, which depend on their composition, size, and shape. • The kinetic energy of the molecules, which depend on the temperature. Viscosity is not a strong function of pressure; hence the effects of pressure on viscosity can be neglected. However, viscosity depends greatly on temperature. For liquids, the viscosity decreases with temperature, whereas for gases, the viscosity increases with temperature. For example, crude oil is often heated to a higher temperature to reduce the viscosity for transport. Consider the situation below, where the top plate is moved by a force F moving at a constant rate of V (m/s). G\ 0RYLQJ SODWH 9 PV ) 9HORFLW\ GY )L[HG SODWH The shear stress τ is given by: τ = F/A The rate of deformation dv (or the magnitude of the velocity component) will increase with distance above the fixed plate. Hence: τ = constant x (dv / dy) 2

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where the constant of proportionality is known as the Dynamic viscosity (µ) of the particular fluid separating the two plates. τ = µ x ( V / y) Where V is the velocity of the moving plate, and y is the distance separating the two plates. The units of dynamic viscosity are kg/ms or Pa s. A non-SI unit in common usage is the poise where 1 poise = 10-1 kg/ms Kinematic viscosity (ν) is defined as the ratio of dynamic viscosity to density. i.e. ν = µ/ρ (1.1) The units of kinematic viscosity are m2/s. Another non-SI unit commonly encountered is the “stoke” where 1 stoke = 10-4 m2/s. 3

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Dynamic Viscosity Kinematic Viscosity Centipoise* (cp) Centistokes (cSt) Water 1 1 Vegetable oil 34.6 43.2 SAE 10 oil 88 110 SAE 30 oil 352 440 Glycerine 880 1100 SAE 50 oil 1561 1735 SAE 70 oil 17,640 19,600 Typical liquid Table 1.1 Viscosity of selected fluids at standard temperature and pressure Note: 1 cp = 10-3kg/ms and 1cSt = 10-6 m2/s Figure 1.1 Variation of the Viscosity of some common fluids with temperature Worked Example 1.1 The temperature dependence of liquid viscosity is the phenomenon by which liquid viscosity tends to decrease as its temperature increases. Viscosity of water can be predicted with accuracy to within 2.5% from 0 °C to 370 °C by the following expression: 4

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μ (kg/ms)= 2.414*10^-5 * 10^(247.8 K/(Temp - 140 K)) Calculate the dynamic viscosity and kinematic viscosity of water at 20 oC respectively. You may assume that water is incompressible, and its density is 1000 kg/m3. Compare the result with that you find from the viscosity chart and comment on the difference. Solution a) Using the expression given: μ (kg/ms) = 2.414*10 -5 * 10(247.8 K/(Temp - 140 K)) = 2.414x10-5x10(247.8/(20+273-140) = 1.005x10-3 kg/ms Kinematic viscosity = dynamic viscosity / density = 1.005x10-3/1000 = 1.005x10-6 m2/s b) From the kinematic viscosity chart, for water at 20 is 1.0x10-6 m2/s. The difference is small, and observation errors may be part of it. Worked Example 1.2 A shaft 100 mm diameter (D) runs in a bearing 200 mm long (L). The two surfaces are separated by an oil film 2.5 mm thick (c). Take the oil viscosity (µ) as 0.25 kg/ms. if the shaft rotates at a speed of (N) revolutions per minute. a) Show that the torque exerted on the bearing is given as: 7RUTXH P[S  [1[/  [F ['  b) Calculate the torque necessary to rotate the shaft at 600 rpm. Solution: a) The viscous shear stress is the ratio of viscous force divided by area of contact 5

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