Sailor Al said:
I came unstuck at: "that is, the total enthalpy is constant along a streamline."
Reading Anderson's definition of a streamline (p. 18, 6th ed):
"a moving fluid element traces out a fixed path in space. As long as the flow is steady (i.e., as long as it does not fluctuate with time), this path is called a streamline of the flow. "
"a fixed path" sounds to me like a geometrical line - i.e. one with no thickness. How can a line of no thickness have enthalpy?
Let's describe a scalar field to describe the temperature of a sheet of metal. Each of the points at the surface will have a temperature so that ##T(x,y)## if it is only dependent on the position or ##T(x,y,t)## if it also changes in time.
For the moment, let's consider the stationary case that doesn't change with time so ##T(x,y)##. Such a field may be represented as a 3D surface where the height represents the temperature or a 2D surface with colors representing the temperature.
These graphs have no relation with one another. They are the first I could find to help transmit what I'm trying to say. By the way, I am aware the 3D surface has color too which is technically not necessary unless you're representing an additional variable but it's what I found on the internet.
The main idea is that each point has a temperature value. However,
according to your reasoning, how could that be possible? Temperature is a property of mass and a point has no volume hence no mass by definition. Therefore, points should not have any temperature associated
(I'm not sure about black holes but let's leave that for another day).
That clearly makes not much sense so the initial reasoning is faulty.
If we study the case from a numerical/practical perspective by dividing the metal sheet and associated temperature field into chunks and then start increasing the resolution of the mesh by making the chunks smaller and smaller, at the very end we reach continuity which is a useful mathematical concept to describe many situations.
This can cause quite some headaches. For example, with integration, you're fundamentally adding up an infinite number of vertical rectangles with infinitely small withs and this turns up to a number. It is weird but it works and mathematicians put some effort into making it consistent and solid so roll with it.