i heard that fluid dynamics and thermo dynamics are the hardest subjects in mechanical engineering and physics, can somebody elaborate on this issue ?
Thermodynamics does have this tendency to really baffle some people. One great quote I've read is: "Thermodynamics is a funny subject. The first time you go through it, you don't understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don't understand it, but by that time you are so used to it, it doesn't bother you any more. "
I don't know if they are the "hardest subjects" or if they are simply not commonly taught as part of the undergraduate Physics curriculum. The mathematics of fluid dynamics (classical field theory) is mostly identical to general relativity- differential geometry and tensor analysis- but I'm starting to see more and more functional analysis: http://www.powells.com/biblio?isbn=9783211826874 Thermodynamics is also a classical field theory, but typical physics curricula spend as little time as possible on this, preferring instead to focus instead on statistical formulations. I was fortunate to have taken a continuum mechanics class through the Mech E department, and I know Chem E departments have excellent courses on thermodynamics/transport theory. If you want to learn those subjects, you may be better off taking engineering classes for the material. I honestly think the 'canonical' US undergraduate physics curriculum is outdated and needs a new structure. For example, I would cut out about half of the typical introductory textbook and replace that with (introductory) quantum mechanics/solid state, computational/modeling, and multidisciplinary topics like colloids and polymers.
Why don't engineers learn either of these two techniques then? Fluid dynamics being as ubiquitous as it is in aerospace/mechanical degrees, I'm very curious what's going on here with engineering education in light of your statement.
Honestly, I think it depends a lot on how well the subject is taught and on the individual student. I personally found fluid mechanics and thermodynamics to be two of the most enjoyable subjects that I've had to this point. However, they do require some thinking and analysis that is very different from a lot of what most students have done prior to those classes, and as a result, a lot of people do struggle. The mathematics of fluid dynamics is largely boundary value problems and tensor analysis, with a decent amount of integrals in vector and scalar fields. Some of it depends on the exact course (compressible vs incompressible, as well as differential vs integral formulations of the equations), but that covers the majority of what is needed. Differential geometry might be somewhat applicable, but I would say that it would be far less helpful for fluid mechanics than a good course on tensor analysis and more advanced techniques for differential equations (preferably including numerical methods). Oh, and at least at my school, tensor analysis is taught to the aerospace engineering graduate students. The undergraduate curriculum only includes one semester of fluid mechanics, which doesn't go into enough detail to require tensors. The graduate students are basically required to learn it though, and my graduate fluids courses have been heavily tensor based.
Two reasons: 1.) Anything useful is numerical simulations (most of which have been written for you) 2.) Non math majors are not in the best position to learn a lot of the mathematics involved.
Fluid mechanics is definitely one of the most interesting subjects to me in Mechanical Engineering, and it involves dealing with lot of equations. I believe that's why some people think it is hard. The math behind those equations are not very deep (at least in the undergraduate intro fluid course), but you have to be very careful to understand the assumptions made when those equations are derived, or you will suffer the consequence of applying it wrongly. In a UG level ME intro fluid course, you are unlikely to see tensor analysis. I don't think you will even see how the Navier Stoke equation is derived. In general, in any UG ME continuum mechanics class (solid/fluid etc), tensor technique might be used for some of the analysis, but I don't believe anyone will tell you that's call tensor. On the other hand, I felt very differently when I took UG ME thermodynamics class. A big part of it involve analyzing engine cycles and related processes, which got boring pretty quick. It has a very different taste than the thermo-physics/statistical mechanics course offered by physics dept. The ME course is much more narrow in scope.
I spoke with a graduate student(in physics) that did his undergraduate studies in aerospace engineering. He said that they didn't ever do tensors, just a lot of calc 3 and some differential equations as far as math goes. He seemed to indicate that the use of tensors came in, in more advanced studies of aerospace(graduate school).
He was right. Differential equations are definitely useful, and you use a ton of calc 3, but tensors don't really show up until grad school, at least in my experience. (I'm an aerospace masters student focusing on fluid mechanics and propulsion)
I don't have an engineering degree (and I don't have an appointment in an engineering department), so I can't comment about "engineering education". The engineers I have worked with all used finite element analysis for fluid and heat transfer problems. Good engineers, like good scientists, don't limit their knowledge to what was taught in class. "lifelong learning" is a silly buzzword, but it's a requirement to stay employable.
Very interesting, thanks. I'd like to go into fluid mechanics and propulsion myself when I hit the grad school stage, so it's nice to know what I'll end up seeing.
Yep. That's because finite element, finite volume, and similar methods are the only way to solve many of the problems people would like to solve. Analytical solutions are nice, but they are only available for a very small, specialized subset of problems. It's still worthwhile to study analytical solutions to fluids problems though, since it helps build intuition. Also, many practical problems can be approximated in ways that allow you to use the analytic solutions, which is a great method for double checking results of something like a CFD solver.
Because they turn out to be useless for real world problems. Deep mathematical principles are useful if you can find a symmetry that simplifies the problem. If you are in the real world and you want to know how fluid flows around a house, then you can't use symmetry to simplify the problem.
Don't know about hardest, but one thing that makes fluid dynamics and thermodynamics interesting is that we don't understand all of it. If you do Newtonian mechanics, the basic principles are known, and so it's a matter of applying what is known. In fluids and thermo, there are some basic problems for which we don't have a fundamental understanding. For example, if you pump water down a pipe and the Reynold's number goes past a critical value, the flow will become turbulent. Now you can simulate this with various degrees of accuracy, and you can put fluid down a pipe and just see what happens, but no one knows how to calculate the critical Reynolds number from first principles.
They do use tensors. No one points out to them that they are using tensors, but they are doing it anyway.
Same here. My astrophysics Ph.D. had a very heavy component of CFD. The difference between the astrophysics CFD and aerospace CFD is that in astrophysics, the boundary conditions are simple, but the interaction of the fluid is complicated. In most aerospace CFD, the fluids are simple, but the boundary conditions are complex. Once you have complex boundary conditions, then it's usually difficult/impossible to do the problem without numerical techniques, usually finite element. Also, there is stuff in the physics curriculum that isn't in the engineering curriculum, but the reverse is also true. Most physicists don't get much exposure to finite element methods, which are essential for most engineering. Any time you do multi-dimensional PDE's, you are using tensors. Now it may be that no one mentions that you are using tensors, but you are using tensors.
True enough. It would probably be more accurate to state that we didn't explicitly use tensors until my graduate classes (where they were used heavily).
Yup. In engineering maths classes we are taught methods to solve problems that we are likely to come across either in other undergraduate classes, post graduate degrees/research and industry. But it isn't pointed out to us that this method is called X or Y or whatever.
To quote Werner Heisenberg: "When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first." Proving the existence (or non-existence) of a general solution to the Navier-Stokes equation is one of the great problems of mathematics. We don't even know if a general solution does exist. The math involved depends heavily on what you are doing, but includes basically all calculus 1, 2 and 3, ODE's and PDE's, nonlinear dynamics and chaos, perturbation methods, complex variables, linear algebra, tensor calculus, Fourier analysis, and more. Honestly, I didn't find it hard (at least the basic classes) but the earlier comment was right on that it is a different sort of class than most first-timers have had before and they often struggle. It seems to be a "love-it-or-hate-it" subject.