# non-newtonian fluid

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 January 8, 2009, 16:29 non-newtonian fluid #1 meda Guest   Posts: n/a Dear all, could you recommend me a book about non-newtonian fluids? I wonder how I can write Navier-Stokes Equations for a non-newtonian fluid. a paper which gives governing equation is also welcome. thanks in advance

 January 9, 2009, 02:58 Re: non-newtonian fluid #2 Jed Guest   Posts: n/a What sort of non-newtonian fluid? In the most general case, the viscosity becomes a rank 4 tensor which is a function of the velocity and perhaps strain. The weak form looks like is find (u,p) \in V \times P such that \int_\Omega \Big[ \rho v \cdot (u \cdot \nabla u) + \eta \DD v : \DD u - p \nabla\cdot v - q \nabla\cdot u - v \cdot \bm f \Big] + \int_{\Gamma} v \cdot (p I - \eta \DD u) \cdot n = 0 for all (v,q) \in V_0\times P where \DD u = \frac 1 2 (\nabla u + (\nabla u)^T) is symmetric gradient, \rho is density, \eta is viscosity (a rank 4 tensor in the most general case, often a scalar). The boundary integral is over the non-Dirichlet portion of the boundary and is usually replaced by a boundary condition, although it's admissible as an outflow condition (this has nothing to do with non-Newtonian flows). A good book on visco-elastics is @book{owens2002cr, title={{Computational Rheology}}, author={Owens, R.G. and Phillips, T.N.}, year={2002}, publisher={Imperial College Press} } I'm not aware of any good books on anisotropy, examples of power-law fluids (arguably the most common non-Newtonian fluids) include the earth's mantle, ice, blood, and paint. Polymers are the usual visco-elastic fluid.

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