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Notes by Category University Engineering

Mechanics & Stress Analysis*
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Forces in Fluids

There are many forces that act on solids, but there are only four main forces present in fluids:

  • Gravity – this is always present, but is sometimes negligible in comparison to other forces

  • Pressure – this is also always present, and is always modelled as compressive and normal to the surface

  • Viscous – this is always present as well, and similar to gravity, can sometimes be ignored

  • Surface Tension – this is only present where a liquid meets another medium, like in bubbles and sprays

Since fluids are constantly moving, the forces surrounding them are constantly changing. We split these forces into two types: body forces and surface forces.

Body Forces

Body forces are forces that act everywhere throughout the body. They occur when a body is subjected to an external field, and the magnitude of such forces depends on the volume of the body.

Gravity is a body force

The gravitational force per unit volume is given by:

Therefore, the magnitude of the force acting on an element of volume δV (from the continuum assumption) is given by:

Since acceleration due to gravity is constant, the gravitational force depends only on density and volume.

Surface Force

As the name suggests, a surface force is distributed across a surface, and (just like body forces) we use it in terms of intensity: the force per unit area. You may recognise this as pressure or stress – it is the same.

If the force is evenly distributed across the surface, the stress on the surface, τ, is given by:

However, when the force is not evenly distributed, we apply the continuum assumption to a small section of the surface, δA:

If the force is not normal to the surface, you need to find the perpendicular and parallel components: the normal and shear stresses respectively

Pressure Force

The pressure force is a surface force that is perpendicular to the surface. This means it is a normal stress:

From the molecular viewpoint, pressure is caused by the molecules colliding with the surface. From the continuum viewpoint, pressure is given as the normal force per unit area on an infinitely small surface.

Pressure is always compressive, and has the same magnitude in all directions

Viscous Force

The viscous force is the frictional force that opposes a fluid’s flow. Therefore, it is a shear force, as it is tangential to the velocity. From the molecular viewpoint, it is caused by the intermolecular forces and collisions, but we look at it as deformation instead.

The viscous force is only present when the fluid is in motion.

Once the external shear force causing the fluid to move is removed, the viscous force will bring the fluid to a halt. Therefore, the viscous force dissipates energy.

The No-Slip Condition

From velocity fields, we know that the deformation in fluids is different in different layers of the fluid:

  • the layer that is furthest from a wall has the highest velocity relative to the wall

  • the layer that is touching the wall has zero velocity relative to the wall

This is known as the no-slip condition.

Newton’s Law of Viscosity

Newton’s Law of Viscosity states that viscous stress is proportional to the local viscosity, the velocity gradient:

τ is still the shear stress, and μ is the viscosity.

Newton’s Law of Viscosity only applies to Newtonian Fluids.

Viscosity is different for all fluids, and depends on the conditions of the fluid, especially temperature. It is found from data tables.

Sometimes, Kinematic Viscosity, ν, is used instead of viscosity. This is defined in terms of density:

Fluids that do not obey this law are called non-Newtonian:

  • Pseudoplastics are an example, in which the viscosity decreases with strain rate. A common example is non-drip paint.

  • Dilatants are the precise opposite of Newtonian fluids: their viscosity increases with strain rate. The best example