Steady flow of a Reiner-Rivlin fluid between rotating plates

This paper considers a von Kármán type axisymmetric flow between parallel plates, in which the top plate rotates and the bottom one is stationary. Between the plates is a weakly non-Newtonian fluid of Reiner-Rivlin type. A highly accurate spectral method is presented for solving the steady problem, and Newton’s method is used to find the Fourier coefficients and an eigenvalue. Multiple solutions are found, of which one is clearly of Batchelor type and another is clearly of Stewartson type, and these persist in the non-Newtonian regime. Such flows may be of practical use in viscometry, in which the coefficient of the non-Newtonian viscous term might be measured.