Dear scholars,
I am working on finding a numerical solution for an equation set with implicit boundary conditions on Matlab. The equations are expressing a nano liquid flow problem which are derived based on including continuity, momentum and heat transfer equations. The equation set and the related boundary conditions are attached. Except f and θ which are the functions of η, all the parameters are the properties of the fluid and known.
So, could you please tell me which numerical method is proper to treat this problem?
The boundary conditions aren't implicit-that doesn't make sense. They're inhomogeneous.
The only boundary condition that is relevant for the discussion, imposes a linear relation between f'(0) and f''(0), that takes the form f'(0) - K f''(0) = 1, where K is β*/(1-φ)2.5 .
This is a Robin-type boundary condition for f'(x). So by defining F'(x) = f'(x) + Ax +B, it's possible to find A and B, such that F'(0)-KF'''(0)=0.
To deal with the boundary condition at infinity, it's necessary to ``compactify'' the independent variable, e.g. by setting y = x/(1+x) and shifting the function, so that the Dirichlet boundary condition at x=infinity, i.e. y = 1, becomes homogeneous, also.
The next step is to expand f(x) (or F(x)) in a basis of functions that do satisfy the boundary conditions, using, for example, Chebyshev polynomials.
Looks like some sort of convective boundary layer problem from a cylinder.
These equations and boundary conditions form a typical voundary value problem. The equations only look complicated because of a large number of constants due to the nanofluid presence. If you are able to solve the Blasius boundary layer equations using a shooting method, or, better, the boundary layer equations for convection from a vertical surface - these date back to the 1950s, then this is just more of the same with no extra difficulties.
Re the boundary condition at infinity, a sufficiently large eta_max will do, but one will have to increase that gradually in order to check that your chosen eta_max is large enough. But I have a feeling that the solutions for theta and f' may well decay algebraically (a negative power of eta) rather than exponentially, so the "compactified" rescaling may eventually need to be done. But I would try without it first in order to make sure that your code runs and gives physically realistic looking solutions.
Dear D. Andrew S. Rees and Stam Nicolis ,
Thank you very much for detailed explanations.
The equations were taken from a published paper. In the following link you can find it.
Article On framing potential features of SWCNTs and MWCNTs in mixed ...
Best Regards.
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