In my PhD, I am assigned mathematical modeling topics and my supervisor's work is related to mathematical biology. But I don't want to work in mathematical biology. Because actually there is not so much real application of that and I don't like biology. Instead, I want to work in fluid dynamics. But my supervisor's research area is totally different. Is it possible to do independent research work without supervision of supervisor?
In summary: is it possible? Yes. Should one do it? No.
You need an advisor to tell you which problem is important to solve, which direction may not be feasible to investigate, which conference to submit your work etc etc. If you have to do all these alone, you may waste years going nowhere.
That being said, I happen to know a couple of people who successfully did their PhD without working with their advisors. All are very talented, and extremely confident with their research. One of them is a very young PhD student in my former school, who has published 3 solo papers (and co-authored some others), and invented a new sub-field of cryptography.
You need someone to help you with the technical side of your work, and that person needs to be an expert in the field (remember, for a PhD you need to do new research, which google won't do for you).
However, the person who supervises you in practice does not ultimately have to be your supervisor in name. One option to consider would be co-supervision, so one supervisor brings experience of academia and research, while another brings technical knowledge. If it works out, that can also be a benefit to a young academic who is new to supervision. I've also known people whose supervisors were sufficiently absent that they weren't really much use. Much of the supervision was actually done by postdocs who were working in the same lab.
Theoretically speaking it is possible to do a PhD under a supervisor who is not an expert in your area but it can get very difficult for times when you need help and "supervision." PhD students are not completely independent researchers and generally follow the directions set out by their supervisors. So if you really want to do something different, the best way would be to change supervisors.
No, this is not possible.
Having a supervisor is not some formal paperwork you need to do to please the management. A supervisor exists for a reason. If you work alone, your work in fluid dynamics is unlikely to be successful. Most probably it will be about "reinventing the wheel", as proposing a doable topic with the potential of scientific novelty requires lots of competence. You cannot have also your PhD defence without the competent opponents; they will come and may find lots of issues.
A supervisor may only be partially competent if the project bridges multiple disciplines. For instance, if you are doing mathematical modelling in biology, your professor may be highly competent in mathematical modelling or in biology, but maybe not in the both areas at once. Also difficult but I have seen this working, and professor should normally try to arrange at least some co-operation with researchers competent in the complementing area (how successfully, another question). However simply ignoring the supervisor and going just on your own does not make any sense.
It depends on the field.
Just as a positive example from my field, I know people in geophysics who regularly do PhDs and research based masters with people who have little knowledge of their specific project at the start.
For example, one of my colleagues is doing a PhD with a very very well known geophysicist, who is known more for is observational and experimental work, but has developed a modeling project with him. Because my colleague has very strong computer skills, and access to advice from other people, the actual "modeling" part he doesn't need as much advising in as the actual general scientific skills. In my opinion, his dissertation work has turned out to be very good because of this, with many more productive collaborations than the normal PhD students tend to have out of necessity. Another strong point about his program is that because he is using unusual methods for his lab, he already shows strong independence from his PhD advisor, which might look good on job applications.
So I will say that it might be smart to do this depending on the quality of scientist you are at the beginning of your PhD work, and the quality of your advisor. These are two things that are very hard to evaluate, and I think that it turns out bad more than it turns out good, but there is clear upside and potential to create a truly unique educational experience regaurdless.
I think it's easier to do this in experimental topics than theoretical ones. Big physical lab groups really have little supervision, apprenticeship anyways.
Overall, it is going to depend on how independent you are, how feasible the problem is (pick something doable). Also that the advisor is at least not an active hurdle--it's one thing if you pull your own sled but if you have someone fouling the runners?