I would like to derive the Navier Stokes Equation from three first order ordinary differential equations (shown in the attachment). I would be glad to have your expert opinions and suggestions.
It isn't possible. The Navier-Stokes equations are a set of coupled, nonlinear partial differential equations. Your quoted equations are uncoupled, linear ordinary differential equations. It may be that your system has been derived from the NS equations, but this would have been subject to all sorts of simplifying assumptions for a special type of flow. Therefore, at best, your equations form a very small subset of the NS equations, and the latter cannot be reconstructed from the former.
Deriving from 1D is challenging i guess....Searched the inventory didn't found anything...plz let me knw if u got answers
I am not sure what the equations are, maybe you are considering a material derivative for d/dt? You should give us more details and references for the system ..
It isn't possible. The Navier-Stokes equations are a set of coupled, nonlinear partial differential equations. Your quoted equations are uncoupled, linear ordinary differential equations. It may be that your system has been derived from the NS equations, but this would have been subject to all sorts of simplifying assumptions for a special type of flow. Therefore, at best, your equations form a very small subset of the NS equations, and the latter cannot be reconstructed from the former.
Thanks everyone. Really appreciate your great responses!
@ Manigandan Sekar, Prof. Filippo Denaro and Prof. Andrew S. Rees, I am actually trying to study a fluid flow system in which the acceleration (in several directions) can be quantified . The initial equations (NSE.docx) describe the acceleration as functions of the directional velocity and some control parameters. Our hope is that if the NSE could be derived from these differential equations, we may be able to obtain new properties of this system. However, in accordance with the observation of Prof. Rees, we have remodeled these ODEs by including some feedback parameters in the constant parameters a, b and c (shown in the attached NSE Modified.docx). I hope this detail helps in clarifying the challenge. Thanks!
@ Hayat Rezgui, thank you for all the wonderful books you have made available to me. Thanks.
Michael, I do not believe you are on the right way if you want to consider your system as a prototypal of NS equations... Even assuming that you work with the lagrangian acceleration (local acceleration plus the convective acceleration), ideal fluid, you can see that in the NS equation you have the acceleration provided by the pressure gradients. How do you take into account for the isotropic stress?
The best way to work for getting a simplified model is starting from the original NS equations, introduce your hypotheses to simplify the system and express the resulting acceleration.
it is impossible. Why you worry if the solution is available.
Thanks Prof. Najah. Actually, the research work of one of my graduate students seems to point to deriving NSE from the ODEs but from the expert opinions everyone has afforded us, this is not feasible. We have reworked our mathematical methods accordingly. However, we are presently trying to explore the suggestions of Prof. Andrew Rees by making our model equations a subset of the NSE; that is, breaking the NSE into our ODEs (we could find a way of making the three ODEs coupled actually). Your suggestions on this latter task would be highly appreciated.
The only procedure I could think similar about your trying is the semi-discrete formulation where the spatial terms in the NS are discretized and the system reduces to a set of ODE. But I don't think that is what you are actually trying to do...
Thank you for your kind contribution, Prof. Denaro. We are actually trying to simulate flow features of a magnetic nanoparticle within some specific fluid. The ODEs are descriptive of the magnetic field of these particles which we need to monitor (in 3D) over time. The problem is controlling these particles from outside the system because their magnetic fields are significantly reduced with interaction with fluid particles. Therefore, we wish to model the flow with NSE while monitoring the influences+changes in the magnetic field (for effective tracking within the fluid). Thank you also for your suggestion on semi-descrete formulation. I have already got some text on this formulation but I still would not mind text recommendations from you.
- Badges
- Science topic
- Similar topics
- Mathematics
- Applied Mathematics
- Differential Equations