Viscosity It is the property of a liquid due to which it flows in the form of layers and each layer opposes the motion of its adjacent layer. Cause of viscosity Consider two neighboring liquid layers A and B. Suppose A faster than B. B would tend to retard the motion of A. On the other hand, A would try to accelerate NB. Due to these two different tendencies, a backward tangential force is set up. This force tends to destroy the relative motion between the two layers. This accounts for the viscous behavior of both liquids and gases. Newton s law of viscous drag It states that force of viscous drag between two liquid layers is 1. Directly proportional to difference in velocity of two layers 2. Directly proportional to area of two layers 3. Inversely proportional to separation between two layers Combining these three equations we get Where η is called coefficient of viscosity Negative sign shows that direction of viscous drag is opposite to relative velocity between two layers. We have Definition of coefficient of viscosity Consider modulus Putting A = 1 m 2, dv/dx = 1 s -1 we get F = η 1 1 F = η Thus coefficient of viscosity of a liquid is defined as the force of viscous drag between any two layers of liquid having unit area in contact and having unit velocity gradient. Units of η 1
SI unit of η is Pas (i.e. Pascal second) and cgs unit is poise 1Pas = 10 poise One poise Putting F = 1 dyne, A = 1 cm 2, and dv/dx = 1 s -1 we get 1dyne = η 1cm 2 1s -1 i.e. Therefore coefficient of viscosity of a liquid is said to be one poise if there exists a force of viscous drag of 1 dyne between any two layers of liquid having 1 cm 2 area in contact and velocity gradient 1 s -1. 1. A metal plate 0.02 m 2 in area is lying on a liquid layer of thickness 10-3 m and coefficient of viscosity of 12 decapoise. Calculate the horizontal force required to move the plate with a speed of 0.025 ms -1. [Ans: 6 N] 2. A square plate of 0.1 m side moves parallel to another plate with a velocity of 0.1 ms -1, both the plates being immersed in water. If the viscous force is 2 10-3 N and viscosity of water is 10-3 decapose, wthat is their distance apart? [Ans: 5 10-4 m] 3. A metal plate of area 5 10-4 m 2 is placed on a 5 10-4 m thick castor oil layer. If a force of 2.25 newton is needed to move the plate with a velocity 3 10-2 ms -1, calculate the coefficient of viscosity of castor oil. [Ans: 75 decapoise] 4. A flat plate of area 10-2 m 2 is separated from a large plate by a layer 10-3 m thick. If the coefficient of visicoty of the liquid is 10-3 decapoise, what force is required to keep the plate moving with a velocity 0.05 ms -1? [Ans: 5 10-4 N] 5. A flat plate of area 1 m 2 is separated from a bigger flat plate of rest by a uniform layer of liquid 1 mm thick. If a tangential force of 5 N is required to move the smaller plate with a constant speed of 2.5 cm/s, what is the coefficient of viscosity of the liquid? [Ans: 0.2 Pas] Stokes Law It states that force of viscous drag experienced by a spherical body moving inside liquid at rest is 1. Directly proportional to radius of body i.e. 2. Directly proportional to velocity of body 3. Directly proportional to coefficient of viscosity of liquid Combining three factors we get This formula is called Stokes law Poiseuielle s Equation It states that rate of flow of liquid in a cylindrical tube of uniform crossection is 1. Directly proportional to the fourth power of radius of tube i.e. 2
2. Directly proportional to pressure difference across two ends to tube 3. Inversely proportional to length of tube 4. Inversely proportional to coefficient of viscosity of liquid Combining these four parameters we get This formula is called Poiseuielle s Formula; V has SI units m 3 / s 6. Calculate the mass of water flowing in 10 minute through a tube of radius 10-2 m, one metre in length and having a constant pressure head of 0.20 m of water. Coefficient of viscosity = 9 10-4 decapoise, g = 9.8 ms -2. [Ans: 5.13 10 3 kg] 7. A liquid flows through a pipe of 10-3 m radius and 0.1 m length under a pressure of 10 3 Nm -2. Calculate the rate of flow and the speed of the liquid coming out of the pipe. The coefficient of viscosity of the liquid is 1.25 10-3 decapoise. [Ans: 1 ms -1 ] Terminal velocity It is the maximum velocity with which a body falls freely under the effect of gravity inside a liquid at rest. Consider a spherical body of radius r falling freely inside a liquid of coefficient of viscosity η. Suppose ρ is the density of body and ρ 0 is the density of liquid. When body acquires terminal velocity we observe that total upward force acting on the body is balanced by total downward force i.e. Total upward force = Total downward force i.e. F B + F V = mg Substituting values for Buoyancy, Viscous drag we get Vρ 0 g + 6πηrv = Vρg Putting for volume of spherical body we get After cancelling we get 3
8. The terminal velocity of a copper ball of radius 2.0 mm falling through a tank of oil at 20 C 0 is 6.5 cms -1. Compute the viscosity of the oil at 20 C 0. Density of oil is 1.5 10 3 kg m -3, density of copper is 8.9 10 3 kgm -3. [Ans: 0.99 kgm -1 s -1 ] 9. A solid ball of volume V is dropped in a viscous liquid. It experiences a viscous force F. If a solid ball of volume 2V and of the same material is dropped in the same liquid, then what would be the viscous force? [Ans: 2F] 10. A rain drop of radius 0.5 mm has a terminal velocity of 2 ms -1 in air. The viscosity of air is 18 10-5 poise. Calculate the viscous force on the rain drop. [Ans: 0.034 dyne] 11. A gas bubble of diameter 2 10-2 m rises steadily at the rate of 2.5 10-3 ms -1 through a solution of density 2250 kgm -3. Calculate the coefficient of viscosity of the solution. Neglect the density of the gas. Given g = 9.8 ms -2 [Ans: 196 decapoise] 12. A drop of water of diameter 0.02 10-3 m is falling through a medium of density 1.2 kgm -3 and coefficient of viscosity 1.8 10-5 decapoise. Calculate the terminal velocity of the drop. [Ans: 1.21 10-2 ms -1 ] 13. A solid sphere of radius 2 mm and density 10.5 gcm -3 is dropped in glycerin of coefficient of viscosity 9.8 poise and density 1.5gcm -3. Calculate the terminal velocity of the sphere. [Ans: 8 cms -1 ] 14. Two drops of equal size are falling through air with a steady velocity of 0.1 ms -1. If the drops coalesce what would be the new terminal velocity? [Ans: 0.159 ms -1 ] 15. Find the radius of a small oil drop falling in air with a terminal velocity of 1 mm/s. Given specific gravity of oil = 0.900 and η of air = 1.8 10-5 decapoise. Neglect the density of air. [Ans: 3 10-6 m] 16. An air bubble of 0.01 m radius is rising at a steady rate of 5 10-3 ms -1 through a liquid of density of 800 kg m -3. Calculate the coefficient of viscosity of the liquid. Neglect the density of air. [Ans: 34.88 decapoise] 17. Fine particles of sand are shaken up in water contained in a tall cylinder. If the depth of water in the cylinder is 0.3 m, calculate the size of the largest particle of sand that can remain suspended after the expiry of 40 minute. Given: density of sand = 2600 km -3 and viscosity of water = 10-3 decapoise. [Ans: 5.986 10-6 m] 18. Eight rain drops of radius 10-3 m each falling down with a terminal velocity of 0.05 ms -1 coalesce to form a bigger drop. Calculate the terminal velocity of the bigger drop. [Ans: 0.2 ms -1 ] 19. A spherical ball of radius 1 10-4 m and density 10 4 kgm -3 falls freely under gravity through a distance h before entering a tank of water. If, after entering the water, the velocity of the ball does not change, find h. The coefficient of viscosity of water is 9.8 10-6 Pas. Take g = 9.8 ms -2. [Ans: 20.4 m] Streamline flow The flow of a fluid is said to be streamline if every particle of the fluid follows exactly the path of its preceding particle and has the same velocity (both in magnitude and direction) as that of its preceding particle when crossing that point. Note: 1. The fixed path followed by an orderly procession of particles in the steady flow of a liquid is called streamline. 2. A streamline may be defined as a path, straight or curved, such that tangent to it at any point indicates the direction of flow of the liquid at that point. 3. A group of streamlines is called a tube of flow. 4
4. Tangent at any point of the streamline gives the direction of velocity of the liquid at that point. 5. Two streamlines cannot intersect. If two streamlines intersect, then it would mean two different directions of velocity at given point. This is physically impossible. Laminar flow It is a type of flow of liquid in which it flows in the form of layers. Where each layer slides on the other. Laminar flow occurs at a greater velocity than streamline flow. Turbulent flow It is that flow of liquid in which molecules of liquid moves irregularly. Flow of liquid becomes turbulent when its velocity is greater than its critical velocity. Note: 1. When velocity of a liquid exceeds the critical velocity, the paths and velocities of the liquid particles begin to change continuously flow loses all its orderliness and is called turbulent flow. 2. For example the wakes left in water by moving ships is turbulent flow. 3. Sounds produced by whistling and by wind instruments result from the turbulent flow of air. Critical velocity The critical velocity of a liquid is that velocity of liquid up to which its flow is streamlined and above which its flow becomes turbulent. It is found that critical velocity is 1. Directly proportional to coefficient of viscosity of liquid 2. Inversely proportional to radius of tube 3. Inversely proportional to density of liquid Combining 1, 2 and 3 we get Where k is constant Note: This equation can be proved using dimensional formulae as: v c (r) a (ρ) b (η) c After putting values of dimensions of various quantities we can find expression for v c Reynolds s number It is a pure number which is used to predict the type of flow of liquid in a tube It is given by: It is found that if 5
1. Reynolds s number is less than 2000 then flow of liquid will be streamline. 2. Reynolds s number is between 2000 and 3000 then flow of liquid will be laminar. 3. Reynolds s number is greater than 3000 then flow will be turbulent. 20. Calculate the critical velocity for air flowing through a tube of 1 10-2 m radius. For air ρ = 1.3 kgm -3 and η = 181 10-7 decapo9ise. [Ans: 1.39ms -1 ] 21. What should be the maximum average velocity of water in a tube of diameter 2 cm so that the flow is laminar? Given viscosity of water = 10-3 Nm -2 [Ans: 0.05 ms -1 ] 22. Water at 20 C 0 flows through a tube of diameter 2 10-2 m with a velocity of 0.12 ms -1. Given η = 10-3 decapoise, ρ of water = 0.99 10 3 kgm -3. Comment on the nature of flow. [Ans: Laminar] 23. The flow rate from a tap of diameter 1.25 cm is 3l / min. The coefficient of viscosity of water is 10-3 Pas. Characterize the flow. [Ans: turbulent] Take a Test 24. An oil drop of radius 4 10-4 mm falls freely in air whose coefficient of viscosity is 1.8 10-4 pose. Calculate its terminal velocity if the density of oil is 0.9 gcm -3 and that of air is 1.293 g lt -1, g = 980 cms -2 [Ans: 1.74 10-3 ] 25. A spherical ball of radius 1 10-4 m and of density 10-4 kg m -3 fall freely under gravity through a distance before entering a tank of water. If after entering the water, the velocity of the ball does not change, find h. The coefficient of viscosity of water is 9.8 10-6 Nsm -2. [Ans: 9.8 10-6 Ns -1 m -1 ] 26. In Millikan s oil drop experiment, what is the terminal speed of a drop of radius 2.0 10-5 m and density 1.2 10-5 Nms -1. How much is the viscous force on the drop at that speed? Neglect buoyancy of the drop due to air. [Ans: 3.93 10-10 N] 27. Eight spherical rain drops of equal size are falling vertically through air with a terminal velocity of 0.10 ms - 1. What should be the velocity if these drops were to combine to form one large spherical drop? [Ans: 0.4 ms -1 ] 28. A plate of metal 10 cm 2 rests on a layer of castor oil 2mm thick whose coefficient of viscosity is 15.5 poise. Calculate the horizontal force required to move the plate with a speed of 3 cms -1. [Ans: 23250 dyne.] 29. What should be the average velocity of water in a tube of diameter 2.0 cm so that the flow is laminar? The viscosity of water is 0.001 Nm -2 s. [Ans: 0.1ms -1 ] 6