Boundary layer fluid flow containing microorganism is my research topic.So i need some suggestions regarding numerical methods by which i can do my research so easily.
The answer to your question depends on how you would like to model your system. If your model is a system of odes or if it is a system of pdes will dictate what class of solver you will use. Unfortunately, there is no one method that works for all situations.
Once again, depending on the structure of the odes, you will be using either an explicit (a method that only uses pass solutions) or an implicit (a method that uses the new solution to calculate the new solution) solver.
If you have access to Matlab, you will first want to try their ode45 solver. It is an embedded (uses two different order methods, 4th and 5th, to automatically adjust its step size) solver using the Dormand-Prince Runge Kutta method.
If this method is slow or you know your equations are "stiff", you will want to use an implicit solver. Matlab's go to solver is ode15s which uses numerical difference method, (not to be confused with backward difference methods although you can flag Matlab to use backward difference).
If your system has to conserve energy, you will then be looking into Verlet integration. Matlab's only version, that I know of, is ode23tb which uses the trapezoidal rule and is implicit. There are other types but I am unfamiliar with these.
I derived equations to compute the flow around very small objects settling in fluids. I also derived the relations defining the dimensions of the boundary layers in creeping flow. The Reynolds number would be less than 1.
I do not know the scale of your project and do not know about the geometries of the systems, but, if small enough, you may do some use of these relationships. I know that viscous boundary flows in water can be up to 260 micrometers. The paper Part III in my profile has the information. Good luck.
Fractional step methods. Put the time evolution at the same importance of space dicretization. But you may use it for parabolic PDE, that is the case of boundary layers.
You can use Crank Nicolson method for solving boundary layers pde equations which is implicit scheme and unconditionaly stable.
Hi Nayema,
If you have access to Comsol Multiphysics, it is very easy to develop models for Boundary layer problems. I have done some if you are interested:
Article Numerical Modeling of Solid-Phase Microextraction: Binding M...
Good results have been found with the method of lines. Lines along the solid surface, parallelly. Then using Runge-Kutta for lines as a system of ordinary differential equations. Transversal direction may be discretized using Finite Difference, Finite Volumes or Finite elements. As you want.
If your boundary layer equations are a set of ODEs, then a shooting method based on a good Runge-Kutta scheme is a good first try. This will allow you to compute the unknown initial conditions. Other methods exist, of course, but a fourth order Runge-Kutta with a constant step-length will provide extremely good solutions in most cases. Do make sure that your computational domain is sufficiently large for the exponential decay to be seen.
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