Fluid Mechanics

Introduction to Fluids

Background

• There are three states of matter: solids, liquids and gases.
• Both liquids and gases are classified as fluids.
• Fluids do not resist a change in shape. Therefore fluids assume the shape of the container they  occupy.
• Liquids may be considered to have a fixed volume and therefore can have a free surface. Liquids are almost incompressible.
• Conversely, gases are easily compressed and will expand to fill a container they occupy.
• We will usually be interested in liquids, either at rest or in motion.

Definition

The strict definition of a fluid is:

A fluid is a substance which conforms continuously under the action of shearing forces.

To understand this, remind ourselves of what a shear force is:

Definition Applied to Static Fluids

According to this definition, if we apply a shear force to a fluid it will deform and take up a state in which no shear force exists. Therefore, we can say:

If a fluid is at rest there can be no shearing forces acting and therefore all forces in the fluid must be perpendicular to the planes in which they act.

Note here that we specify that the fluid must be at rest. This is because, it is found experimentally that fluids in motion can have slight resistance to shear force. This is the source of viscosity.

Definition Applied to Fluids in Motion

For example, consider the fluid shown flowing along a fixed surface. At the surface there will be little movement of the fluid (it will ‘stick’ to the surface), whilst further away from the surface the fluid flows faster (has greater velocity):

If one layer of is moving faster than another layer of fluid, there must be shear forces acting between them. For example, if we have fluid in contact with a conveyor belt that is moving we will get the behaviour shown:

When fluid is in motion, any difference in velocity between adjacent layers has the same effect as the conveyor belt does.
Therefore, to represent real fluids in motion we must consider the action of shear forces.

Consider the small element of fluid shown, which is subject to shear force and has a dimension s into the page. The force F acts over an area A = BC×s. Hence we have a shear stress applied:

Any stress causes a deformation, or strain, and a shear stress causes a shear strain.
This shear strain is measured by the angle φ .

Remember that a fluid continuously deforms when under the action of shear. This is different to a solid: a solid has a single value of φ for each value of τ . So the longer a shear stress is applied to a fluid, the more shear strain occurs. However, what is known from experiments is that the rate of shear strain (shear strain per unit time) is related to the shear stress:

In this graph the Newtonian fluid is represent by a straight line, the slope of which is μ . Some of the other fluids are:

• Plastic: Shear stress must reach a certain minimum before flow commences.
• Pseudo-plastic: No minimum shear stress necessary and the viscosity decreases with rate of shear, e.g. substances like clay, milk and cement.
• Dilatant substances; Viscosity increases with rate of shear, e.g. quicksand.
• Viscoelastic materials: Similar to Newtonian but if there is a sudden large change in shear they behave like plastic.
• Solids: Real solids do have a slight change of shear strain with time, whereas ideal solids (those we idealise for our theories) do not.

Lastly, we also consider the ideal fluid. This is a fluid which is assumed to have no viscosity and is very useful for developing theoretical solutions. It helps achieve some practically useful solutions.

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