Fluids
Unit 1 – Topic 2 (materials)
Learning Objectives
Specification points:
(23) be able to use the equation density
𝑚
ρ = m𝑉
(24) understand how to use the relationship upthrust = weight of fluid displaced
(25) (a) be able to use the equation for viscous drag (Stokes’ Law), F = 6πηrv.
(b) understand that this equation applies only to small spherical objects moving at low speeds with laminar flow (or in the
absence of turbulent flow) and that viscosity is temperature dependent.
To be able to:
• Understand and use the density formula.
• State that an upthrust is provided by the fluid displaced by a submerged or floating object.
• Calculate the upthrust in terms of the weight of the displaced fluid.
• Recall and apply the principle that, for an object floating in equilibrium, the upthrust is equal to the
weight of the new object to new situations or to solve related problems.
• Define what viscosity is
• Understand and use stokes law.
Success criteria
Successfully…………
 A – Apply Stokes’ law in terminal velocity conditions to calculate missing variables and/or
to explain different relevant scenarios.
 B/C – Using viscosity terminology , such as rate of flow, to explain certain scenarios and
solve problems.
 C/D – Use Stokes’ law to calculate viscous drag.
 E – Use the density formula to calculate density.
Density
Fluids
What is a fluid?
A fluid is any substance that can flow. Usually that’s liquids and gases
For example; water, air , oil
Density
The density of a substance is the relationship between the mass of the
substance and how much space it takes up (volume).
Density is measure of the mass per unit volume of a substance
Unit of density(ρ):
kilograms per cubic meter (in SI units) (kg/m3)
Upthrust
When an object is submerged in a fluid, upthrust is the upward force that a liquid or gas exerts on a body floating
in it.
According to Archimedes' Principle, the upthrust on an object in a fluid is equal to the weight of the fluid
displaced. So the volume displaced multiplied by the density of the fluid.
Upthrust = Weight of Fluid Displaced
The volume of this amount of liquid is equal to the volume of the object itself. The weight of fluid displaced
and therefore the upthrust will be bigger if the density of the liquid is large.
When do things float?
• Any object which is less dense than water will float
• When floating, an object will sink until it has displaced an equal weight of water to its own total weight.
• If weight is then added to the object, it will sink until it has displaced enough water to again equal its own
weight.
• If weight is removed, the upthrust will now be greater than its weight, and it will accelerate upwards until
the upthrust is again equal to its weight.
Ensure that you use the terms weight and mass correctly! Weight is
the force, mass is the amount of matter.
Therefore, the factors affecting upthrust are:
- density of the fluid
- volume of the object immersed
Unit of Upthrust:
newton (N) or kgm/s2
Worked example
A ball of mass 10 kg is held under the surface of a pool. The instant its released, it has an instantaneous
acceleration of 4 m/s2 towards the bottom of the pool. What is the volume of the ball?
• The net force on the ball is expressed as:
Fnet= ma
• Since its accelerating downward, the force of gravity is larger
then the buoyant force(upthrust). Therefore;
Fnet = Weight- Upthrust
• Therefore;
Fnet = mg - (Vball * ρwater * g)
• (10*4 ) = (10g) – (Vball * 1000 * g)
• Rearrange;
(10g- 40)/1000g = Vball
Answer: 0.0059m3
Fluid movement
Fluid flows can be divided into two different types:
laminar flow and turbulent flow
Laminar flow ( streamline flow)
The term streamline flow is descriptive of the flow because, in laminar flow, there is virtually no mixing
between layers, fluid particles move in definite and observable paths or streamlines.
In summary,
Laminar flow is characterized by :
•
•
•
•
the velocity of the fluid at a point is constant
No abrupt change in the direction/speed of flow
The lines of flow do not cross
layers/flowlines/streamlines are parallel
Labelling and drawing fluid flow diagrams
is not needed for the new specification.
Fluid movement
Turbulent flow
Turbulent flow, type of fluid 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.
Turbulence is also characterized by eddies, and apparent randomness.
In summary,
Turbulent flow is characterized by:
•
•
•
•
the velocity at a point keeps on changing
Abrupt changes in speed/direction
Eddies/whirlpools form
Layers/flowlines/streamlines mix/cross
Labelling and drawing fluid diagrams is
not needed for the new specification.
Viscosity
A fluids resistance to flow
• In a fluid, each 'layer' exerts a force of friction on another 'layer’.
• This frictional force is also present when solid object moves through a liquid.
This force is termed Viscous Drag.
• Viscous Drag is greater in Turbulent Flow than Laminar Flow.
We say a thick fluid has high viscosity or is very
viscous, while a thin fluid would have low viscosity
Viscosity
The size of the Viscous Drag in a fluid depends on the (coefficient of) Viscosity of that fluid.
𝑘𝑔
(coefficient of) Viscosity is given the letter η(Eta) and is measured in
/Pa s.
𝑚𝑠
The greater the Viscosity, the greater the Viscous Drag, therefore the lower the rate of flow (rate of flow
decreases as viscosity increases).
Coefficient of viscosity; a numerical value given to a fluid to indicate
how much it resists flow
• In most liquids, Viscosity decreases as
temperature increases, (temperature is
inversely proportional to viscosity)
• whereas in most gases, Viscosity increases as
temperature increases.
Stokes’ law
It is possible to calculate the drag force exerted on a spherical object in a fluid using Stoke's Law:
F = 6πηrv
• r is the radius of sphere (m)
• η is the viscosity (Pa s)
• v is velocity (m/s)
Conditions:
- Laminar flow,
- slow moving(reach its terminal velocity)
- Small smooth spherical object
Terminal velocity
Consider a sphere falling through a viscous fluid. As the sphere falls , its velocity increases until it reaches a velocity known
as the terminal velocity.
Terminal velocity is achieved, therefore, when the speed of a moving object is no longer increasing or
decreasing; the object’s acceleration is zero
At this velocity the frictional drag (due to viscous forces ) and upthrust is balanced by the gravitational force and the
velocity is constant
• This means that the forces acting on the object are balanced. An equation
can be formed by equating Weight with Upthrust and Viscous Drag
• weight – upthrust – viscous drag = 0
or
• upthrust + viscous drag = weight
Terminal velocity
Terminal velocity graphs
Velocity-time graph
Acceleration-time graph
Terminal velocity
weight = upthrust + viscous drag
• It can be expressed as the following;
• This formula can be rearranged to solve for v ( terminal velocity);
Terminal velocity
When carrying out an experiment to determine the viscosity, by dropping a spherical object in a fluid for example, a
graphical method can be used to determine a value
Some measurements need to be taken/ obtained first, such as;
• radius of object
• density of object ( mass can be rather obtained)
• density of fluid
• Calculate the velocity of object ( terminal velocity)
Then,
• Plot a graph of v against 𝑟 2 in (y –axis) and (x- axis) respectively
• measure the gradient
Hence, the coefficient of viscosity can be obtained using the following formula;
Success criteria
Which grade are you on?
Successfully…………
 A – Apply Stokes’ law in terminal velocity conditions to calculate missing variables and/or
to explain different relevant scenarios.
 B/C – Using viscosity terminology , such as rate of flow, to explain certain scenarios and
solve problems.
 C/D – Use Stokes’ law to calculate viscous drag.
 E – Use the density formula to calculate density.
Past paper
Q&A
Question 1
Answer 1
Question 2
Answer 2
Question 3
Answer 3
Question 4
Answer 4
Question 5
Answer 5
Question 6
Answer 6
Question 7
Answer 7
Question 8
Answer 8
Question 9
Answer 9
Question 10
Answer 10
Question 11
Answer 11
Question 12
Answer 12
Question 13
Answer 13
Question 14
Answer 14
Question 15
Answer 15
Question 16
Answer 16
Question 17
Answer 17