Coefficient of viscosity Before we can define the

Coefficient of viscosity, η Before we can define the coefficient of viscosity, we need to look at viscosity in more detail. What causes viscosity in liquids? We need to consider what is happening on a molecular level. As we have discussed already, streamline flow behaves as if there are layers of liquid passing over each other. In a liquid with very low viscosity, there will be little interaction between the layers. In more viscous liquids, there will be interaction. As one layer passes over another, energy will be transferred. How does this interaction happen? 1 © TPS 2008

Coefficient of viscosity, η Before we can define the coefficient of viscosity, we need to look at viscosity in more detail. What causes viscosity in liquids? We need to consider what is happening on a molecular level. One model suggests that the molecules in in adjacent layers are attracted to each other. As the molecule in layer B passes the molecule in layer A, their mutual attraction speeds up A and slows down B. There is an exchange of energy between the layers. Layer B slows down and layer A speeds up. 2

Coefficient of viscosity, η Before we can define the coefficient of viscosity, we need to look at viscosity in more detail. What causes viscosity in liquids? We need to consider what is happening on a molecular level. A viscous liquid is “thick” because the layers seem to stick a little as they pass over each other due to intermolecular attractions. Increasing the temperature would make the molecules’ random thermal energy greater. They would be in each other’s near vicinity for a shorter time so the attractive forces would be less and the viscosity would decrease. 3

Coefficient of viscosity, η Before we can define the coefficient of viscosity, we need to look at viscosity in more detail. What causes viscosity in gasses? We need to consider what is happening on a molecular level. In a gas, molecules can drift under thermal motion as well as travel in the direction of bulk fluid flow. What happens to the layers of gas in streamline flow if a molecule passes from a slower layer to a faster one? • A faster molecule entering a slower layer will, on average, speed the slower layer up • A slower molecule entering a faster layer will, on average, slow the 4 faster layer down

Coefficient of viscosity, η Before we can define the coefficient of viscosity, we need to look at viscosity in more detail. What causes viscosity in gases? We need to consider what is happening on a molecular level. In a gas, molecules can drift under thermal motion, as well as travel in the direction of bulk fluid flow. What happens to the layers of gas in streamline flow, if a molecule passes from a slower layer to a faster one? • This manifests itself as an apparent frictional force between the adjacent layers, with energy being transferred between adjacent layers 5

Coefficient of viscosity, η Tangential stress: The force due to one plane within a streamline, on another adjacent plane, will be a viscous force. It will act tangentially against the streamline. Retarding force -F Relative velocity, Δv Area of contact, A Accelerating force F In this diagram, the upper streamline is travelling faster than the lower one and their relative velocity is Δv. The viscous drag forces between them form a Newton’s third law pair. The force on the upper streamline is in the opposite direction to the relative velocity so is a retarding force. The force on the lower 6 streamline accelerates the streamline.

Coefficient of viscosity, η Tangential stress: The force due to one plane within a streamline, on another adjacent plane, will be a viscous force. It will act tangentially against the streamline. Retarding force -F Relative velocity, Δv Area of contact, A Accelerating force F The tangential stress is sometimes called the shear or shearing stress and is defined by: Tangential stress = force / area τ=F/A The units of tangential stress are Pa (pascal). 7

Coefficient of viscosity, η Velocity gradient: Δv y Δy This tangential stress creates varying velocities, perpendicular to the direction of streamline flow, as represented by the Velocity vectors in this diagram. The rate of change of velocity across the streamlines is called the velocity gradient. Velocity gradient = Δv/Δy = Change in velocity / Change in y What are the units of velocity gradient? From the definition, they will be ms-1 / m. The units of velocity gradient are s-1. 8

Coefficient of viscosity, η Δv Velocity gradient: The greater the viscosity of the liquid, the smaller the velocity gradient. y Δy Velocity vectors Imagine a really viscous liquid, like treacle. If you have a cup full and spin it around, all the treacle moves too. This is due to the high viscosity. A large Δy would be required for even a small Δv. If you spin a cup of water, the water at the edge will turn immediately but as there is low viscosity and a high velocity gradient, the water further into the cup might not spin at all at first. 9

Coefficient of viscosity, η Coefficient of viscosity: So, viscosity is related to the tangential stress and the velocity gradient. We can define the coefficient of viscosity as follows: Coefficient of viscosity, η = tangential stress velocity gradient Where η = (F/A) / (Δv/Δy). What are its units? Pa s (pascal seconds) This is also Nsm-2 10

Coefficient of viscosity, η Coefficient of viscosity: If we plot tangential stress against velocity gradient, we should obtain a straight line. The gradient (slope) of the line will be the coefficient of viscosity. Note that the coefficient of velocity is extremely temperature dependent. Tangential stress Velocity gradient 11

Coefficient of viscosity, η Coefficient of viscosity: Fluids which obey this relationship are called Newtonian fluids. Fluids that do not are called non-Newtonian fluids. The graph will be non-linear for non. Newtonian fluids. Tangential stress Non-Newtonian fluid Velocity gradient Non-drip paint is a non-Newtonian fluid. With small tangential stresses, there is a large velocity gradient but this soon falls as the tangential stress increases. 12

Coefficient of viscosity, η Coefficient of viscosity: Non-drip paint is a non. Newtonian fluid. With small tangential stresses there is a large velocity gradient but this soon falls as the tangential stress increases. 13

Coefficient of viscosity, η Coefficient of viscosity: When on the brush and not being worked, it is very viscous – has a high viscosity. This means that it will not drip off the brush easily. 14

Coefficient of viscosity, η Coefficient of viscosity: As soon as it is worked, the viscosity falls and it becomes spreadable. 15

Coefficient of viscosity, η Coefficient of viscosity: Imagine the contrast between running through paint and running through water. Water is a Newtonian fluid. The faster you run, the greater the resistance you would feel from the water. It would be harder to overcome the resistive force of the water, the faster you ran. 16

Coefficient of viscosity, η Coefficient of viscosity: Imagine the contrast between running through paint and running through water. Running in non-drip paint, although a lot more messy, might actually be easier. Once you started moving, the viscous drag would decrease as nondrip paint is a non-Newtonian fluid. We would recommend that you do not try this experiment at home! 17

Coefficient of viscosity, η Coefficient of viscosity: The coefficient of viscosity of several fluids at 20°C is given below in Nsm-2. Air 1. 8 x 10 -5 Water 1. 0 x 10 -3 Mercury 1. 6 x 10 -3 Olive oil 8. 4 x 10 -2 Golden syrup 1. 0 x 102 18