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

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 <code>(u,p) \in V \times P</code> 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 <code>(v,q) \in V_0\times P</code> where <code>\DD u = \frac 1 2 (\nabla u + (\nabla u)^T)</code> is symmetric gradient, <code>\rho</code> is density, <code>\eta</code> 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.