Howdy Milne Ando, "by any other name a rose would smell as sweet," and I do try to be accurate with names, Yi-Shian Ciou.

Your questions are a delight. As a self-study artist, I affirm that if one is willing to be bewildered for a few thousand pages it will become clear what "they" mean by those words. Good show!

Yes, buoyancy is, and it must be included in science when one treats nature! Approximations, however, may lead to results that are very valuable. Were I to wonder, I would suspect that you are in a hurry. So, I affirm your course with the very strong reservation: Make it yours! You take a few moments longer to appreciate the efforts of others at each stage and "catch up" by reading and thought. I expect great things from you, when you have found the time. The equation: pffffttt, engineers will do most anything, but you get to understand and do it right (?).

Be special: any canyon is capable of echoes. According to theory, bumblebees cannot fly, but how can one survey their opinions if they're all up in the air?

Happy Trails, Len

Hello Milne Ando, The use of Archimedes' principle in the book can be explained as follows:

Even though you are correct that the pressure gradient around a falling sphere is not exactly equal to the hydrostatic pressure gradient $(ρ_f)g$, Archimedes' principle can still be applied as a reasonable approximation when the fluid flow around the sphere is not too complicated (i.e., in cases where the flow is slow, steady, and incompressible).

The buoyancy force term $(-m_f)g$ is not redundant, and it represents the net effect of the pressure distribution acting on the sphere's surface. The gravitational force acting on the sphere $(m_f)g$ is counteracted by the net effect of the pressure distribution acting on the sphere's surface, and the remaining pressure force contributes to the drag force acting on the sphere. Therefore, Archimedes' principle helps to simplify the equation of motion.

Your method, which considers the pressure force and shear stress force separately, is also valid. However, it might be less convenient in some cases, since it requires dealing with the detailed pressure distribution on the sphere's surface. The author's approach simplifies the problem by decomposing the forces into simpler terms (gravity, buoyancy, and drag) that can be handled more easily.

So, the author's use of Archimedes' principle is a valid approximation under the assumptions of slow, steady, and incompressible fluid flow. The buoyancy term is not redundant, and it represents the net effect of the pressure distribution acting on the sphere's surface.

Your approach is also valid but might be less convenient in some cases, as it requires dealing with detailed pressure distributions on the sphere's surface.

Regards,

Alessandro Rizzo hi,

I got confused because in Aerodynamics textbook the way to calculate Lift/drag is just integrating components of (p, τ) over the surface of airfoil but no buoyancy is mentioned in that section. Hmm, I guess that's because the p in the integral equation is total pressure which includes static pressure, dynamic pressure and hydrostatic pressure, namely, the effect of gradient of hydrostatic pressure (buoyancy) has been included; thus, calculating a buoyancy term is not necessary(?)

If my understanding is wrong, please correct me. Thank you very much!

Hi Milne Ando ,

Your understanding is indeed correct. In aerodynamics, the lift and drag forces acting on an object (such as an airfoil) are often calculated by integrating the pressure and shear stress components over the object's surface. The total pressure, which includes static, dynamic, and hydrostatic pressure, accounts for the buoyancy effect implicitly.

When studying lift and drag forces in aerodynamics, the buoyancy force is typically not mentioned separately, as it is inherently included in the pressure distribution acting on the surface of the airfoil. This is in contrast to the example of a free-falling sphere in a fluid, where buoyancy is considered explicitly.

In both cases, the same principles apply, but the focus is different. For the free-falling sphere, buoyancy is a primary force that needs to be considered, while for an airfoil, lift and drag are the primary forces of interest. However, the calculations involving pressure and shear stress components on the surface of the object account for buoyancy in both cases.

So, your understanding is correct, and the treatment of buoyancy in aerodynamics is implicitly included within the pressure distribution acting on the surface of the airfoil.

Howdy Folks,

I have recommended Alessandro Rizzo's answers and affirm Milne Ando's comment. This is so much better than "being bewildered for a few thousand pages!" I have found similar helpful clarity here with others as well and rejoice that online communication has made it possible.

My dad remembered breaking the ice in a wash basin to clean up before walking to a small country school. When his family moved into a town (706 population) he was set back a grade because the town folk were so far ahead. I learned Fortran IV from a minicomputer manual while writing an harmonic analysis program for energy fluxes at the earth's surface. Now, one wise enough to ask receives excellent, swift answers about existing programs! I think it's just great.

Happy Trails, Len

Dear Leonard,

Thank you for your kind words. Your appreciation means a lot to me and I'm glad that I could be of help.

Best regards, Alessandro