ºÝºÝߣ

ºÝºÝߣShare a Scribd company logo

Properties of Matter - U3 - Viscosity-PPT 1

•
•
Dhivya Ramachandran
Dhivya Ramachandran

1. The document discusses viscosity and different methods to measure viscosity, including the coefficient of viscosity, Poiseuille's method using capillary tubes, and Searle's rotating cylinder method. 2. It provides definitions and explanations of concepts like laminar and turbulent flow, Reynolds number, critical velocity, terminal velocity, and Stoke's formula. 3. Different methods to calculate viscosity are presented, along with corrections to Poiseuille's formula accounting for kinetic energy and non-ideal flow conditions. Searle's rotating cylinder method relates viscosity to the torque measured on a rotating cylinder in a fluid.

1 of 28
Unit III - Viscosity Ms Dhivya R Assistant Professor Department of Physics Sri Ramakrishna College of Arts and Science Coimbatore - 641 006 Tamil Nadu, India 1
Coefficient of Viscosity Viscosity: Viscosity is the resistance of a fluid (liquid or gas) to a change in shape or movement of neighboring portions relative to one another. Viscosity denotes opposition to flow. 2 Sri Ramakrishna College of Arts and Science
Coefficient of Viscosity 3 Sri Ramakrishna College of Arts and Science
Coefficient of Viscosity The viscosity is calculated in terms of the coefficient of viscosity. It is constant for a liquid and depends on its liquid’s nature. The Poiseuille’s method is formally used to estimate the coefficient of viscosity, in which the liquid flows through a tube at the different level of pressures. 4 Sri Ramakrishna College of Arts and Science
Coefficient of Viscosity 5 Sri Ramakrishna College of Arts and Science
Streamline Flow Streamline flow in fluids is defined as the flow in which the fluids flow in parallel layers such that there is no disruption or intermixing of the layers and at a given point, the velocity of each fluid particle passing by remains constant with time. (Laminar Flow) In steady (Streamline/Laminar) flow, the density of the fluid remains constant at each point. In unsteady (Turbulent) flow, the velocity of the fluid varies between any given two points. 6 Sri Ramakrishna College of Arts and Science
Turbulent Flow Turbulent flow is a type of fluid (gas or liquid) flow in which the fluid undergoes irregular fluctuations, or mixing, in contrast to laminar flow, in which the fluid moves in smooth paths or layers. In turbulent flow the speed of the fluid at a point is continuously undergoing changes in both magnitude and direction. 7 Sri Ramakrishna College of Arts and Science
Streamline & Turbulent Flow 8 Sri Ramakrishna College of Arts and Science
Critical Velocity & Reynolds number 9 Sri Ramakrishna College of Arts and Science Critical velocity is the speed and direction at which the flow of a liquid through a tube changes from smooth to turbulent. Determining the critical velocity depends on multiple variables, but it is the Reynolds number that characterizes the flow of the liquid through a tube as either turbulent or laminar. The Reynolds number is a dimensionless variable, which means that it has no units attached to it.
Critical Velocity & Reynolds number 10 Sri Ramakrishna College of Arts and Science Critical Velocity: ð‘£ð‘ = 𑘠⋅ 𜂠ðœŒð‘Ÿ Reynolds number 𑘠= ð‘£ð‘ðœŒð‘Ÿ â‹… ðœ‚
Poiseuille’s formula for the flow of liquid through a capillary tube 11 Sri Ramakrishna College of Arts and Science
Poiseuille’s formula for the flow of liquid through a capillary tube Now the velocity of the liquid at a distance ð‘Ÿ from the axis is ð‘£ and at a distance ð‘Ÿ + ð‘‘ð‘Ÿ is 𑣠− ð‘‘ð‘£. So, the velocity gradient=− ð‘‘ð‘£/ð‘‘ð‘Ÿ Surface area of the cylinder, A=2ðœ‹ð‘Ÿð‘™ According to Newton’s law viscosity, the viscous Force between two layer is given by ð¹1 = −ðœ‚ð´ð‘‘ð‘£/ð‘‘ð‘Ÿ = −𜂠× 2ðœ‹ð‘Ÿð‘™ ×( ð‘‘ð‘£/ð‘‘ð‘Ÿ) ----------(1) Where ðœ‚= coefficient of viscosity, Now, the forward push due to the difference of pressure P on two sides of the tube of radius r is ð¹2 = 𑃠× ð´ð‘Ÿð‘’ð‘Ž = 𑃠× ðœ‹ð‘Ÿ2 ---------- (2) 12 Sri Ramakrishna College of Arts and Science
Poiseuille’s formula for the flow of liquid through a capillary tube For steady flow, F1= F2 𑃠× ðœ‹ð‘Ÿ2 = 𜂠× 2ðœ‹ð‘Ÿð‘™ ×( ð‘‘ð‘£/ð‘‘ð‘Ÿ) Or ⅆ𒗠= −𒑠ðŸðœ¼ð’ ð’“ð’…ð’“------------(3) Integrating ⅆ𒗠= 🎠𒓠−𒑠ðŸðœ¼ð’ ð’“ð’…ð’“ 13 Sri Ramakrishna College of Arts and Science
Poiseuille’s formula for the flow of liquid through a capillary tube ⅆ𒗠= 🎠𒓠−𒑠ðŸðœ¼ð’ ð’“ð’…ð’“ ð’— = −ð’‘ð’“ðŸ ðŸ’ðœ¼ð’ + 𑪠----- (4) Where r=a & v=0 🎠= −ð’‘ð’‚ðŸ ðŸ’ðœ¼ð’ + 𑪠ð‚ = ð’‘ð’‚ðŸ ðŸ’ðœ¼ð’ ð‚ = ð’‘ð’‚ðŸ ðŸ’ðœ¼ð’ ----- (5) ∴ ð¯ = ð’‘ ðŸ’ðœ¼ð’ (ð’‚ðŸ − ð’“ðŸ) – (6) 14 Sri Ramakrishna College of Arts and Science
Poiseuille’s formula for the flow of liquid through a capillary tube Volume of the liquid flowing per second ð‘‘𑉠= ð´ð‘Ÿð‘’ð‘Ž ð‘œð‘“ ð‘ð‘Ÿð‘œð‘ ð‘  ð‘†ð‘’ð‘ð‘¡ð‘–ð‘œð‘› ð‘œð‘“ ð‘¡â„Žð‘’ ð‘ â„Žð‘’ð‘™ð‘™ ð‘œð‘“ ð‘Ÿð‘Žð‘‘ð‘–ð‘¢ð‘  ð‘Ÿ ð‘Žð‘›ð‘‘ ð‘¡â„Žð‘–ð‘ð‘˜ð‘’ð‘›ð‘’ð‘ ð‘  ð‘‘𑟠× ð‘‰ð‘’ð‘™ð‘œð‘ð‘–ð‘¡ð‘¦ ð‘œð‘“ ð‘™ð‘œð‘¤ ð‘‘𑉠= 2ðœ‹ð‘Ÿð‘‘ð‘Ÿ ð‘ 4ðœ‚ð‘™ (ð‘Ž2 − ð‘Ÿ2) = ðœ‹ð‘ 2ðœ‚ð‘™ (ð‘Ž2𑟠− ð‘Ÿ3) ð‘‘𑟠𑉠= 0 ð‘Ž ðœ‹ð‘ 2ðœ‚ð‘™ (ð‘Ž2𑟠− ð‘Ÿ3) ð‘‘ð‘Ÿ = ðœ‹ð‘ 2ðœ‚ð‘™ ð‘Ž2 ð‘Ÿ2 2 − ð‘Ÿ4 4 0 ð‘Ž 15 Sri Ramakrishna College of Arts and Science
Poiseuille’s formula for the flow of liquid through a capillary tube 𑉠= ðœ‹ð‘ 2ðœ‚ð‘™ ð‘Ž2 ð‘Ÿ2 2 − ð‘Ÿ4 4 0 ð‘Ž = ðœ‹ð‘ 2ðœ‚ð‘™ ð‘Ž4 4 𑉠= ðœ‹ð‘ð‘Ž4 8ðœ‚ð‘™ 16 Sri Ramakrishna College of Arts and Science
Corrections to Poiseuille’s Formula Correction for pressure head: Less Effective pressure due to KE acquired ð‘1 = ð‘ − ð‘‰2 𜌠ðœ‹2ð‘Ž4 The KE ð¸â€² = 0 ð‘Ž 1 2 2ðœ‹ð‘‘ð‘Ÿð‘£ðœŒ ð‘£2 = ðœ‹ðœŒ 0 ð‘Ž ð‘Ÿð‘£3 ð‘‘ð‘Ÿ But ð‘£ = ð’‘ ðŸ’ðœ¼ð’ (ð’‚ðŸ − ð’“ðŸ) ∴ ð¸â€²= ðœ‹ðœŒ 0 ð‘Ž ð‘Ÿ ð’‘ ðŸ’ðœ¼ð’ 3 (ð’‚ðŸ − ð’“ðŸ)3ð‘‘ð‘Ÿ 17 Sri Ramakrishna College of Arts and Science
Corrections to Poiseuille’s Formula ð¸â€² = ðœ‹ðœŒ ð’‘ ðŸ’ðœ¼ð’ 3 ð‘Ž8 8 = ðœ‹ð‘ð‘Ž4 8ðœ‚𑙠𜌠ðœ‹2ð‘Ž4 ð¸â€² = ð‘‰3 𜌠ðœ‹2ð‘Ž4 Correction for pressure head: ð©ð• = ð©ðŸð• + ð‘‰3𜌠ðœ‹2ð‘Ž4 Less Effective pressure due to KE acquired ð’‘ðŸ = 𒑠− ð‘½ðŸ ð† ð…ðŸð’‚💠= ð’ˆð† 𒉠− ð‘½ðŸ ð…ðŸð’‚ðŸ’ð’ˆ 18 Sri Ramakrishna College of Arts and Science
Corrections to Poiseuille’s Formula Correction for length of the tube: Not streamlined flow for some distance Effective length is increased from ð‘™ ð‘¡ð‘œ ð‘™ + 1.64ð‘Ž. 𜂠= ðœ‹ð‘ð‘Ž4 8ð‘‰ð‘™ 𜼠= ð…ð’‚💠ðŸ–ð‘½(ð’ + ðŸ. ðŸ”ðŸ’ð’‚) 𒉠− ð‘½ðŸ ð…ðŸð’‚ðŸ’ð’ˆ ð’ˆð† 19 Sri Ramakrishna College of Arts and Science
Terminal Velocity The uniform velocity acquired by a body while moving through a highly viscous liquid is called terminal velocity. F = k ð‘£ð‘Žð‘Ÿð‘ðœ‚ð‘r F = MLT-2; ð‘£=LT-1; ð‘Ÿ=L; ðœ‚=ML-1T-1; 20 Sri Ramakrishna College of Arts and Science
Stoke’s Formula MLT-2=(LT-1)a (L)b (ML-1T-1)c MLT-2=Mc La+b-c T-a-c C = 1 ; a+b+c = 1; -a-c=-2 F=kð‘£ð‘Ÿðœ‚; 𑘠= 6𜋠F= 6ðœ‹ð‘£rðœ‚; 21 Sri Ramakrishna College of Arts and Science
Expression for Terminal Velocity Weight of the ball = 4 3 ðœ‹ð‘Ÿ3ðœŒð‘” Weight of the displaced liquid = 4 3 ðœ‹ð‘Ÿ3ðœŒâ€²ð‘” Apparent weight of the ball = 4 3 ðœ‹ð‘Ÿ3ðœŒð‘”- 4 3 ðœ‹ð‘Ÿ3ðœŒâ€²ð‘” = 4 3 ðœ‹ð‘Ÿ3(𜌠− ðœŒâ€²)ð‘” When the ball attains its terminal velocity, 6ðœ‹ð‘£rðœ‚= 4 3 ðœ‹ð‘Ÿ3(𜌠− ðœŒâ€²)ð‘” ð‘£ = 2 9 ð‘Ÿ2 𜂠(𜌠− ðœŒâ€²)ð‘” 22 Sri Ramakrishna College of Arts and Science
Assumptions made by Stoke  The medium through which the body falls is infinite in extent  The moving body is perfectly rigid and smooth  There is no slip between the moving body and the medium  There are no eddy current or waves set up. The object moves very slowly in the medium 23 Sri Ramakrishna College of Arts and Science
Stoke’s Method Terminal velocity ð‘£ = ð‘¥ ð‘¡ Coefficient of Viscosity 𜂠= 2 9 ð‘Ÿ2 ð‘£ (𜌠− ðœŒâ€²)ð‘” 24 Sri Ramakrishna College of Arts and Science
Searle’s Viscometer: Rotating Cylinder Method Angular velocity = ω Linear velocity = v = r ω Velocity Gradient = ð‘‘ð‘£ ð‘‘ð‘Ÿ = ð‘‘ ð‘‘ð‘Ÿ ð‘Ÿðœ” = 𜔠ð‘‘ð‘Ÿ ð‘‘ð‘Ÿ + ð‘Ÿ ð‘‘𜔠ð‘‘ð‘Ÿ =𜔠+ ð‘Ÿ ð‘‘𜔠ð‘‘ð‘Ÿ 25 Sri Ramakrishna College of Arts and Science
Searle’s Viscometer: Rotating Cylinder Method Since 𜔠is necessary for preventing slip of a layer, this is not involved in viscous drag. Hence the effective Velocity Gradient =ð‘Ÿ ð‘‘𜔠ð‘‘ð‘Ÿ According to newtons formula, F = ηð´ ð‘‘ð‘£ ð‘‘ð‘Ÿ F = η2ðœ‹ð‘Ÿð‘™ð‘Ÿ ð‘‘𜔠ð‘‘ð‘Ÿ Moment of this force = C = η2ðœ‹ð‘Ÿð‘™ð‘Ÿ ð‘‘𜔠ð‘‘𑟠× ð‘Ÿ 26 Sri Ramakrishna College of Arts and Science
Searle’s Viscometer: Rotating Cylinder Method Moment of this force = C = 2ðœ‹ð‘™Î·ð‘Ÿ3 ð‘‘𜔠ð‘‘ð‘Ÿ ð‘œð‘Ÿ C ð‘‘ð‘Ÿ ð‘Ÿ3 = 2ðœ‹ð‘™Î·ð‘‘𜔠C ð‘ ð‘Ž ð‘‘ð‘Ÿ ð‘Ÿ3 = 2ðœ‹ð‘™Î· 0 ðœ”1 ð‘‘𜔠C − 1 2ð‘Ÿ2 ð‘ ð‘Ž = 2ðœ‹ð‘™Î·ðœ”1 C 2 1 ð‘2 − 1 ð‘Ž2 ð‘ ð‘Ž = 2ðœ‹ð‘™Î·ðœ”1 ð¶ = 4ðœ‹ðœ‚ðœ”1ð‘Ž2ð‘2 ð‘Ž2 − ð‘2 ð‘™ 27 Sri Ramakrishna College of Arts and Science
Searle’s Viscometer: Rotating Cylinder Method 28 Sri Ramakrishna College of Arts and Science

More Related Content

Properties of Matter - U3 - Viscosity-PPT 1

  • 1. Unit III - Viscosity Ms Dhivya R Assistant Professor Department of Physics Sri Ramakrishna College of Arts and Science Coimbatore - 641 006 Tamil Nadu, India 1
  • 2. Coefficient of Viscosity Viscosity: Viscosity is the resistance of a fluid (liquid or gas) to a change in shape or movement of neighboring portions relative to one another. Viscosity denotes opposition to flow. 2 Sri Ramakrishna College of Arts and Science
  • 3. Coefficient of Viscosity 3 Sri Ramakrishna College of Arts and Science
  • 4. Coefficient of Viscosity The viscosity is calculated in terms of the coefficient of viscosity. It is constant for a liquid and depends on its liquid’s nature. The Poiseuille’s method is formally used to estimate the coefficient of viscosity, in which the liquid flows through a tube at the different level of pressures. 4 Sri Ramakrishna College of Arts and Science
  • 5. Coefficient of Viscosity 5 Sri Ramakrishna College of Arts and Science
  • 6. Streamline Flow Streamline flow in fluids is defined as the flow in which the fluids flow in parallel layers such that there is no disruption or intermixing of the layers and at a given point, the velocity of each fluid particle passing by remains constant with time. (Laminar Flow) In steady (Streamline/Laminar) flow, the density of the fluid remains constant at each point. In unsteady (Turbulent) flow, the velocity of the fluid varies between any given two points. 6 Sri Ramakrishna College of Arts and Science
  • 7. Turbulent Flow Turbulent flow is a type of fluid (gas or liquid) flow in which the fluid undergoes irregular fluctuations, or mixing, in contrast to laminar flow, in which the fluid moves in smooth paths or layers. In turbulent flow the speed of the fluid at a point is continuously undergoing changes in both magnitude and direction. 7 Sri Ramakrishna College of Arts and Science
  • 8. Streamline & Turbulent Flow 8 Sri Ramakrishna College of Arts and Science
  • 9. Critical Velocity & Reynolds number 9 Sri Ramakrishna College of Arts and Science Critical velocity is the speed and direction at which the flow of a liquid through a tube changes from smooth to turbulent. Determining the critical velocity depends on multiple variables, but it is the Reynolds number that characterizes the flow of the liquid through a tube as either turbulent or laminar. The Reynolds number is a dimensionless variable, which means that it has no units attached to it.
  • 10. Critical Velocity & Reynolds number 10 Sri Ramakrishna College of Arts and Science Critical Velocity: ð‘£ð‘ = 𑘠⋅ 𜂠ðœŒð‘Ÿ Reynolds number 𑘠= ð‘£ð‘ðœŒð‘Ÿ â‹… ðœ‚
  • 11. Poiseuille’s formula for the flow of liquid through a capillary tube 11 Sri Ramakrishna College of Arts and Science
  • 12. Poiseuille’s formula for the flow of liquid through a capillary tube Now the velocity of the liquid at a distance ð‘Ÿ from the axis is ð‘£ and at a distance ð‘Ÿ + ð‘‘ð‘Ÿ is 𑣠− ð‘‘ð‘£. So, the velocity gradient=− ð‘‘ð‘£/ð‘‘ð‘Ÿ Surface area of the cylinder, A=2ðœ‹ð‘Ÿð‘™ According to Newton’s law viscosity, the viscous Force between two layer is given by ð¹1 = −ðœ‚ð´ð‘‘ð‘£/ð‘‘ð‘Ÿ = −𜂠× 2ðœ‹ð‘Ÿð‘™ ×( ð‘‘ð‘£/ð‘‘ð‘Ÿ) ----------(1) Where ðœ‚= coefficient of viscosity, Now, the forward push due to the difference of pressure P on two sides of the tube of radius r is ð¹2 = 𑃠× ð´ð‘Ÿð‘’ð‘Ž = 𑃠× ðœ‹ð‘Ÿ2 ---------- (2) 12 Sri Ramakrishna College of Arts and Science
  • 13. Poiseuille’s formula for the flow of liquid through a capillary tube For steady flow, F1= F2 𑃠× ðœ‹ð‘Ÿ2 = 𜂠× 2ðœ‹ð‘Ÿð‘™ ×( ð‘‘ð‘£/ð‘‘ð‘Ÿ) Or ⅆ𒗠= −𒑠ðŸðœ¼ð’ ð’“ð’…ð’“------------(3) Integrating ⅆ𒗠= 🎠𒓠−𒑠ðŸðœ¼ð’ ð’“ð’…ð’“ 13 Sri Ramakrishna College of Arts and Science
  • 14. Poiseuille’s formula for the flow of liquid through a capillary tube ⅆ𒗠= 🎠𒓠−𒑠ðŸðœ¼ð’ ð’“ð’…ð’“ ð’— = −ð’‘ð’“ðŸ ðŸ’ðœ¼ð’ + 𑪠----- (4) Where r=a & v=0 🎠= −ð’‘ð’‚ðŸ ðŸ’ðœ¼ð’ + 𑪠ð‚ = ð’‘ð’‚ðŸ ðŸ’ðœ¼ð’ ð‚ = ð’‘ð’‚ðŸ ðŸ’ðœ¼ð’ ----- (5) ∴ ð¯ = ð’‘ ðŸ’ðœ¼ð’ (ð’‚ðŸ − ð’“ðŸ) – (6) 14 Sri Ramakrishna College of Arts and Science
  • 15. Poiseuille’s formula for the flow of liquid through a capillary tube Volume of the liquid flowing per second ð‘‘𑉠= ð´ð‘Ÿð‘’ð‘Ž ð‘œð‘“ ð‘ð‘Ÿð‘œð‘ ð‘  ð‘†ð‘’ð‘ð‘¡ð‘–ð‘œð‘› ð‘œð‘“ ð‘¡â„Žð‘’ ð‘ â„Žð‘’ð‘™ð‘™ ð‘œð‘“ ð‘Ÿð‘Žð‘‘ð‘–ð‘¢ð‘  ð‘Ÿ ð‘Žð‘›ð‘‘ ð‘¡â„Žð‘–ð‘ð‘˜ð‘’ð‘›ð‘’ð‘ ð‘  ð‘‘𑟠× ð‘‰ð‘’ð‘™ð‘œð‘ð‘–ð‘¡ð‘¦ ð‘œð‘“ ð‘™ð‘œð‘¤ ð‘‘𑉠= 2ðœ‹ð‘Ÿð‘‘ð‘Ÿ ð‘ 4ðœ‚ð‘™ (ð‘Ž2 − ð‘Ÿ2) = ðœ‹ð‘ 2ðœ‚ð‘™ (ð‘Ž2𑟠− ð‘Ÿ3) ð‘‘𑟠𑉠= 0 ð‘Ž ðœ‹ð‘ 2ðœ‚ð‘™ (ð‘Ž2𑟠− ð‘Ÿ3) ð‘‘ð‘Ÿ = ðœ‹ð‘ 2ðœ‚ð‘™ ð‘Ž2 ð‘Ÿ2 2 − ð‘Ÿ4 4 0 ð‘Ž 15 Sri Ramakrishna College of Arts and Science
  • 16. Poiseuille’s formula for the flow of liquid through a capillary tube 𑉠= ðœ‹ð‘ 2ðœ‚ð‘™ ð‘Ž2 ð‘Ÿ2 2 − ð‘Ÿ4 4 0 ð‘Ž = ðœ‹ð‘ 2ðœ‚ð‘™ ð‘Ž4 4 𑉠= ðœ‹ð‘ð‘Ž4 8ðœ‚ð‘™ 16 Sri Ramakrishna College of Arts and Science
  • 17. Corrections to Poiseuille’s Formula Correction for pressure head: Less Effective pressure due to KE acquired ð‘1 = ð‘ − ð‘‰2 𜌠ðœ‹2ð‘Ž4 The KE ð¸â€² = 0 ð‘Ž 1 2 2ðœ‹ð‘‘ð‘Ÿð‘£ðœŒ ð‘£2 = ðœ‹ðœŒ 0 ð‘Ž ð‘Ÿð‘£3 ð‘‘ð‘Ÿ But ð‘£ = ð’‘ ðŸ’ðœ¼ð’ (ð’‚ðŸ − ð’“ðŸ) ∴ ð¸â€²= ðœ‹ðœŒ 0 ð‘Ž ð‘Ÿ ð’‘ ðŸ’ðœ¼ð’ 3 (ð’‚ðŸ − ð’“ðŸ)3ð‘‘ð‘Ÿ 17 Sri Ramakrishna College of Arts and Science
  • 18. Corrections to Poiseuille’s Formula ð¸â€² = ðœ‹ðœŒ ð’‘ ðŸ’ðœ¼ð’ 3 ð‘Ž8 8 = ðœ‹ð‘ð‘Ž4 8ðœ‚𑙠𜌠ðœ‹2ð‘Ž4 ð¸â€² = ð‘‰3 𜌠ðœ‹2ð‘Ž4 Correction for pressure head: ð©ð• = ð©ðŸð• + ð‘‰3𜌠ðœ‹2ð‘Ž4 Less Effective pressure due to KE acquired ð’‘ðŸ = 𒑠− ð‘½ðŸ ð† ð…ðŸð’‚💠= ð’ˆð† 𒉠− ð‘½ðŸ ð…ðŸð’‚ðŸ’ð’ˆ 18 Sri Ramakrishna College of Arts and Science
  • 19. Corrections to Poiseuille’s Formula Correction for length of the tube: Not streamlined flow for some distance Effective length is increased from ð‘™ ð‘¡ð‘œ ð‘™ + 1.64ð‘Ž. 𜂠= ðœ‹ð‘ð‘Ž4 8ð‘‰ð‘™ 𜼠= ð…ð’‚💠ðŸ–ð‘½(ð’ + ðŸ. ðŸ”ðŸ’ð’‚) 𒉠− ð‘½ðŸ ð…ðŸð’‚ðŸ’ð’ˆ ð’ˆð† 19 Sri Ramakrishna College of Arts and Science
  • 20. Terminal Velocity The uniform velocity acquired by a body while moving through a highly viscous liquid is called terminal velocity. F = k ð‘£ð‘Žð‘Ÿð‘ðœ‚ð‘r F = MLT-2; ð‘£=LT-1; ð‘Ÿ=L; ðœ‚=ML-1T-1; 20 Sri Ramakrishna College of Arts and Science
  • 21. Stoke’s Formula MLT-2=(LT-1)a (L)b (ML-1T-1)c MLT-2=Mc La+b-c T-a-c C = 1 ; a+b+c = 1; -a-c=-2 F=kð‘£ð‘Ÿðœ‚; 𑘠= 6𜋠F= 6ðœ‹ð‘£rðœ‚; 21 Sri Ramakrishna College of Arts and Science
  • 22. Expression for Terminal Velocity Weight of the ball = 4 3 ðœ‹ð‘Ÿ3ðœŒð‘” Weight of the displaced liquid = 4 3 ðœ‹ð‘Ÿ3ðœŒâ€²ð‘” Apparent weight of the ball = 4 3 ðœ‹ð‘Ÿ3ðœŒð‘”- 4 3 ðœ‹ð‘Ÿ3ðœŒâ€²ð‘” = 4 3 ðœ‹ð‘Ÿ3(𜌠− ðœŒâ€²)ð‘” When the ball attains its terminal velocity, 6ðœ‹ð‘£rðœ‚= 4 3 ðœ‹ð‘Ÿ3(𜌠− ðœŒâ€²)ð‘” ð‘£ = 2 9 ð‘Ÿ2 𜂠(𜌠− ðœŒâ€²)ð‘” 22 Sri Ramakrishna College of Arts and Science
  • 23. Assumptions made by Stoke  The medium through which the body falls is infinite in extent  The moving body is perfectly rigid and smooth  There is no slip between the moving body and the medium  There are no eddy current or waves set up. The object moves very slowly in the medium 23 Sri Ramakrishna College of Arts and Science
  • 24. Stoke’s Method Terminal velocity ð‘£ = ð‘¥ ð‘¡ Coefficient of Viscosity 𜂠= 2 9 ð‘Ÿ2 ð‘£ (𜌠− ðœŒâ€²)ð‘” 24 Sri Ramakrishna College of Arts and Science
  • 25. Searle’s Viscometer: Rotating Cylinder Method Angular velocity = ω Linear velocity = v = r ω Velocity Gradient = ð‘‘ð‘£ ð‘‘ð‘Ÿ = ð‘‘ ð‘‘ð‘Ÿ ð‘Ÿðœ” = 𜔠ð‘‘ð‘Ÿ ð‘‘ð‘Ÿ + ð‘Ÿ ð‘‘𜔠ð‘‘ð‘Ÿ =𜔠+ ð‘Ÿ ð‘‘𜔠ð‘‘ð‘Ÿ 25 Sri Ramakrishna College of Arts and Science
  • 26. Searle’s Viscometer: Rotating Cylinder Method Since 𜔠is necessary for preventing slip of a layer, this is not involved in viscous drag. Hence the effective Velocity Gradient =ð‘Ÿ ð‘‘𜔠ð‘‘ð‘Ÿ According to newtons formula, F = ηð´ ð‘‘ð‘£ ð‘‘ð‘Ÿ F = η2ðœ‹ð‘Ÿð‘™ð‘Ÿ ð‘‘𜔠ð‘‘ð‘Ÿ Moment of this force = C = η2ðœ‹ð‘Ÿð‘™ð‘Ÿ ð‘‘𜔠ð‘‘𑟠× ð‘Ÿ 26 Sri Ramakrishna College of Arts and Science
  • 27. Searle’s Viscometer: Rotating Cylinder Method Moment of this force = C = 2ðœ‹ð‘™Î·ð‘Ÿ3 ð‘‘𜔠ð‘‘ð‘Ÿ ð‘œð‘Ÿ C ð‘‘ð‘Ÿ ð‘Ÿ3 = 2ðœ‹ð‘™Î·ð‘‘𜔠C ð‘ ð‘Ž ð‘‘ð‘Ÿ ð‘Ÿ3 = 2ðœ‹ð‘™Î· 0 ðœ”1 ð‘‘𜔠C − 1 2ð‘Ÿ2 ð‘ ð‘Ž = 2ðœ‹ð‘™Î·ðœ”1 C 2 1 ð‘2 − 1 ð‘Ž2 ð‘ ð‘Ž = 2ðœ‹ð‘™Î·ðœ”1 ð¶ = 4ðœ‹ðœ‚ðœ”1ð‘Ž2ð‘2 ð‘Ž2 − ð‘2 ð‘™ 27 Sri Ramakrishna College of Arts and Science
  • 28. Searle’s Viscometer: Rotating Cylinder Method 28 Sri Ramakrishna College of Arts and Science