There are several concepts that need to be introduced before we embark on a course in fluid mechanics. For example, what exactly is a fluid?
Herein we will provide definitions of:
Concepts such as pressure will be covered in detail in a separate notebook.
Of the three states of matter — solid, liquid and gas — the latter two are classified as fluids. What defines a solid? How are fluids different?
Unlike solids, fluids flow!
A solid is comprised of densely packed molecules, often in precise crystal structures such as found in metals or when water is frozen into ice. In Solid Mechancis you will have learned about stress, strain and elastic and plastic deformation. The video below illustrates how the densely packed atoms of a solid respond elastically to longidutional, torsion amd shear loads.
Solids can also undergo plastic as well as elastic defromation. Here is a simulation of a three point bend. As you can see the beam only partially returns to its original shape due to plastic deformation.
Brittle solids can fracture as cracks from and propagate resulting in component failures.
Fluids on the other hand can be easily deformed from one shape into another. The intermolecular forces do not hold the molecules in a rigid structure but cohesion due to these forces means that molecules of the same species are attracted to each other and can slide over each other. A liquid fills the container it is poured into and a gas quickly spreads out in all directions to fill a container it is released into.
The liquid state is a form of matter where molecules are attracted to each other and are packed densely (although less so than in a solid) and the molecules remain disordered and can be readily deformed. We are familiar with liquids such as water from $0$ to $100^\circ C$, but there are many different liquids such as liquid mercury at room temperature, oils that float on water and 'thick' liquids like honey and paint. The video below shows a simulation of pouring tea. The liquid exits the spout of the white teapot in a single curved stream and fills the volume of the glass teapot. The surface of the liquid in the glass teapot sloshes as the stream impacts it.
As we've seen, liquids fill their container and easily flow through conduits such as teapot spouts and pipes. Here we see a liquid flow through a complex pipe geometry and into a secondary container. Notice how the leading front of the liquid is thrown outwards in the pipe due to centrifugal forces. As it fills the lower container it slowly spreads out to cover the bottom.
This ability of fluids to pass through pipes and tubes of various shapes makes them very important in engineering. We can route liquids through machines and buildings to transfer heat, pressure and the fluid itself.
When a fluid changes phase from a liquid to a gas due to the application of heat (or a reduction in pressure) the thermal agitation of the molecules allows them to overcome the cohesion that keeps a liquid densely packed. The ensemble of these microscopic thermal fluctuations are what you know as temperature. The molecules in a substance are constantly jostling about. In a solid this movement is very small and the intermolecular forces keeps the material tightly packed. The addition of energy allows the material to change state to liquid as the balance of competition between the forces changes until finally with enough energy the molecules can breakaway as a gas.
The gas molecules will spread out in all directions and so the density of the fluid will be reduced limited only by the volume of the container. The video below shows a group of particles representing molecules where the cohesion forces are reduced so that the particles are able to spread out. The motion you observe is called Brownian motion. It is a random motion of the molecules as they collide with each other and is due to the thermal agitation discussed above.
As with liquids, gases are extremely important in engineering. The air around us is a gas and we build aircraft that can easily fly through it and use that air to generate a lift force keeping them aloft. We compress useful gases such as propane into containers so we can transport large quantities in a small volume which can be released as chemical energy when combusted.
A fundamental assumption of fluid mechanics relates to the way in which a fluid interacts with a surface. While two rigid solids moving relative to each other can be considered to slip past each other with a certain degree of friction a Newtonian fluid obeys what is called the zero-slip condition. Here we assume that the infinitesimal layer of fluid in contact with a surface is perfectly adhered to that surface. It does not slip over that surface nor does it move through the surface, the velocity at the wall is zero. As one moves away from the wall the velocity can change. How it changes depends on a fluid property called viscosity. Understanding the ideas presented below is important for understanding concepts such as aerodynamic drag. In the case of the above video notice how particles bounce off the walls of the cube, never penetrate it and never travel along the surface parallel to it.
Fluids, like all forms of matter exert and experience forces and are subject to Newton's laws. An oar is forced through water to propel a boat; pressurised gas stretches the material of a balloon; a swimmer experiences buoyancy in a body of water. Unlike solids fluids can respond differently to the rate at which a force is applied. Consider again the video of the three point bend above. How would this change if the bending force was applied over a period of one minute or one year? Rapid deformation of the beam may heat the beam and repeated deformations can result in fatigue failure but from a fundamental perspective the rate of the deformation is unimportant.
The video below shows a fluid container being rocked back and forth much like you might do with a bottle of water. The fluid and the degree of rocking is identical for each container all that is changing is the rate of rocking which is increased by 25\% for each case. Notice how the fluid response changes.
For the slowest cases the surface of the orange fluid remains mostly flat as the fluid volume is distributed due to gravitational forces. As the rocking motion increases the forces from the sidewalls pushing the fluid results in the generation of surface waves until the fluid surface becomes highly disturbed.
It is clear that the rate which we deform a fluid can have dramatic effects!
Now we will consider how a fluid exerts a force on a surface with a bit more rigour. Consider a force, $F$, acting on a surface as shown in the 3D model below. We can easily decompose that force into its tangential and normal components relative to that surface.
#! pip3 install PyGEL3D import os from PyGEL3D import gel from PyGEL3D import js js.set_export_mode() m = gel.obj_load("media/1.1/forces.obj") js.display(m, smooth=False, data=None, wireframe=True)