Boundary integral formulations are considered as robust and efficient methods used to solve the linearized Navier-Stokes equation. The partial differential equations are transformed into integral equations by Green’s identities, where the velocity field is represented as a combination of hydrodynamic potentials of single and double-layer. BIM formulation could also be extended in order to solve for non-Newtonian fluids. I would like to know whether similar methods can be derived as well for the full NS equation where the nonlinear inertial forces cannot be neglected compared next to the linear viscous forces? Thank you.
My answer is no as non-linearity creates problem in the formulation, else it would be an efficient method to solve NS equations.
Abdalahh. May you present the two versions of the NS equation you mention: the linear one and the non-linear one? Does the nonlinear one have an acceptable solution, or do you have some hypothesis about that solution?
Thank you both for your answers. Emilio, I mean by the linear equation the steady linearized NS equation i.e. Stokes equation. The NL equation stands for the complete NS equation by considering the convective term. There is indeed a serious problem in the formulation because of the presence of the nonlinearity but it works well for the non-Newtonian fluids?
I am not sure, however it is possible to solve problems strongly dominated by inertia using the Boundary Element Method, here is a beautiful example:
Ha-Ngoc, H. & Fabre, J. 2004 Test-case No. 29B: The velocity and shape of 2D long bubbles in inclined channels or in vertical tubes (PA, PN) Part II: In a flowing liquid.
Multiphase Sci.Technol.16, 191–206
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