## Investigation into the effect of temperature on viscosity

Introduction:

The task is to investigate into the effect of temperature on viscosity based on two main variables, the temperature of the honey (the fluid used) and the time it takes for a ball bearing to travel through a distance at its terminal velocity. Using Stoke’s law and knowledge of the flow of fluids investigate how the fluid temperature and the terminal velocities of a ball bearing are related to viscosity.

Basic setup:

First thoughts:

The above rough experimental design has much room for improvement. A greater level of accuracy can be achieved by using a combination of more sensitive equipment and a better overall setup. A formula linking viscosity and time will be derived and a graph of viscosity against temperature will be produced.

Derivation:

Diagram of variables and constants:

F= viscous drag force, N/force, N

r= radius, m

η= co-efficient of viscosity, Nsm-2

v= velocity, ms-1

W= weight, kg

g= gravitational field strength, 9.81ms-2

a= acceleration, ms-2

ρ= density, kg m-3

s= displacement, m

t = time, s

A formula linking η (viscosity) and t (time) must be produced.

The following consists of rules and principals that could be applied to this situation.

Archimedes’ Principle: When a body is partially or totally immersed in a fluid the upthrust is equal to the weight of fluid displaced.

Viscosity can be defined as a measure of resistance to flow that a fluid offers when subjected to shear stress. The viscosity is due to the friction between layers of the fluid.

If an object is dropped into a fluid it experiences two primary forces, weight, which acts in the downward direction and also upthrust (equal to the weight of fluid displaced). If these values are equal then there will be no resultant force. If the upthrust is greater than the objects weight the object it will float and if its weight is greater than the upthrust it will sink. Based only on the above, this would mean that a skydiver jumping from 3000m would be able to reach huge speeds, but such huge speeds are not possible and this is where viscous drag comes in.

The faster the object moves through the fluid the more the fluid resists flow (viscous drag, F= 6 Pi rηv). The magnitude of the frictional force increases from zero with the speed of the falling object. This is due to the way the fluid moves and means that the object will not constantly accelerate. It will reach a point where all forces balance out in equilibrium and the object can no longer speed up, it travels at a constant speed. This is called the terminal velocity.

Therefore care must be taken to allow the ball to reach terminal velocity before recording the time it takes to cover a distance, otherwise the application of stokes law could be considered invalid.

Terminal velocity occurs when the forces are balanced as below:

Upthrust + viscous drag = weight

Stoke’s law: F=6 Pi rηv (upthrust)

ρ= m

v

F=ma

W=mg

v=st

It would be sensible to derive the weights of honey and ball bearing in the same format so that the large formula (with all the variables) can then be cancelled down to one involving the main variables only.

F=ma

W=mg (downward gravitational acceleration of 9.81ms-2 )

Mass= density x volume

Although the mass of the ball bearing could be measured and the weight/force calculated it makes more sense to keep both the forces (upthrust and ball weight) in the same format to allow them to be easily combined.

Ball weight:

Mass= 4/3 Pi r3 ρsteel

Weight of the ball= (4/3 Pi r3 ρsteel) g

Upthrust (equal to weight of fluid displaced):

4/3 Pi r3 gives the volume displaced by the ball

Mass= 4/3 Pi r3 ρfluid

Weight of honey: (4/3 Pi r3 ρfluid) g

The viscous drag is stokes law the value that stokes law finds.

F=6 Pi rηv

We can now arrange this in the balanced forces (terminal velocity) formula:

Upthrust + viscous drag = weight

(4/3 Pi r3 ρfluid) g + 6 Ρi rηv = (4/3 Pi r3 ρsteel) g

And then rearrange to find the viscosity:

4 Pi r3 ρfluid g + 6 Ρi rηv = 4 Pi r3 ρsteel g cancel Pi and r

- 3

4 r² ρfluid g + 6 ηv = 4 r² ρsteel g x 3

3 3

4 r² ρfluid g + 18 ηv = 4 r² ρsteel g / 2

2 r² ρfluid g + 9 ηv = 2 r² ρsteel g -2 r² ρfluid g and cancel common factors

9 ηv= 2 r² (ρ steel - ρfluid)g divide by 9v

η= 2 r² (ρ steel - ρfluid) g substitute v for s/t (v=s/t)

9v

η = 2 r2 (ρ steel - ρfluid) g

9 (s/t)

Fair test:

There are many variables that may affect this investigation, all have been considered thoroughly and precautions will be taken to ensure that the experiment runs as smoothly as possible.

The viscosity is the one of the most important variables of the investigation. To keep a fair test it is important that the same type of honey is used each time (as it is likely that each individual sample of honey will have a slightly different density). Using the same honey would mean that the density remains constant throughout the experiment.

Thermal energy loss may occur as when heated the honey immediately begins losing heat to its surroundings (and gaining heat, when cold, to return to room temperature) Effects like this must be taken into account, so that as the temperature at which the recording is made, is as near as possible to the temperature stated.

Gravitational field strength, 9.81ms-2 affects the ball bearing in the tube causing it to accelerate.

The distance over which the ball bearing falls must be kept constant thus allowing a direct comparison of velocities.