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Basic Fluid Mechanics

This post is presented to fill the void in basic fluid mechanics to the die casting community. It was observed that knowledge in this area cannot be avoided. The design
of the process as well as the properties of casting (especially magnesium alloys) are determined by the fluid mechanics/heat transfer processes. It is hoped that others
will join to spread this knowledge. There are numerous books for introductory fluid mechanics but the Potto series book “Basic of Fluid Mechanics” is a good place to
start. This post is a summary of that book plus some pieces from the “Fundamentals of Compressible Flow Mechanics.” It is hoped that the reader will find this post
interesting and will further continue expanding his knowledge by reading the full Potto books on fluid mechanics and compressible flow.

First we will introduce the nature of fluids and basic concepts from thermodynamics.
Later the integral analysis will be discussed in which it will be divided into introduction of the control volume concept and Continuity equations. The energy equation will be explained in the
next section. Later, the momentum equation will be discussed. Lastly, the post will be dealing with the compressible flow gases. Here it will be refrained from dealing with topics such boundary layers, non–
viscous flow, machinery flow etc which are not essential to understand the rest of this book. Nevertheless, they are important and it is advisable that the reader will read on these topics as well.

What is fluid? Shear stress
Fluid, in this book, is considered as a substance that “moves” continuously and permanently when exposed to a shear stress. The liquid metals are an example of such substance. However, the liquid metals do not have to be in the liquidus phase to be
considered liquid. Aluminum at approximately 4000C is continuously deformed when shear stress are applied. The whole semi–solid die casting area deals with materials that “looks” solid but behaves as liquid.

What is Fluid?
The fluid is mainly divided into two categories: liquids and gases. The main difference between the liquids and gases state is that gas will occupy the whole volume while liquids has an almost fixed volume. This difference can be, for most practical purposes
considered, sharp even though in reality this difference isn’t sharp. The difference between a gas phase to a liquid phase above the critical point are practically minor. But below the critical point, the change of water pressure by 1000% only change the
volume by less than 1 percent. For example, a change in the volume by more than 5% will require tens of thousands percent change of the pressure. So, if the change of pressure is significantly less than that, then the change of volume is at best 5%. Hence,
the pressure will not affect the volume. In gaseous phase, any change in pressure directly affects the volume. The gas fills the volume and liquid cannot. Gas has no free interface/surface (since it does fill the entire volume).

What is Shear Stress?
The shear stress is part of the pressure tensor. However, here it will be treated as a separate issue. In solid mechanics, the shear stress is considered as the ratio of the force acting on area in the direction of the forces perpendicular to area. Different from solid,
fluid cannot pull directly but through a solid surface. Consider liquid that undergoes a shear stress between a short distance of two plates as shown in Figure.

U = f(A, F, h) (2.1)

Where A is the area, the F denotes the force, h is the distance between the plates. From solid mechanics study, it was shown that when the force per area increases, the velocity of the plate increases also. Experiments show that the increase of height will
increase the velocity up to a certain range. Consider moving the plate with a zero lubricant (h » 0) (results in large force) or a large amount of lubricant (smaller force). In this discussion, the aim is to develop differential equation, thus the small distance
analysis is applicable. For cases where the dependency is linear, the following can be written

U is directly proportional to hF and inversely proportional to A

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