Michael C HaslamElahian, Azadeh2025-07-232025-07-232025-05-092025-07-23https://hdl.handle.net/10315/43053This thesis presents an efficient Chebyshev-tau spectral solver solver to compute numerical solutions of the equations of motion of a viscous fluid in a circular pipe subject to the generalized Oldroyd-B stress model of Yeleswarapu [1] corresponding to the flow of a viscoelastic fluid with a shear rate dependent viscosity function. The code relies on an explicit fourth order Runge-Kutta time stepping procedure with O(N logN) operations per time step enabled by the use of fast cosine transforms. The resulting code is spectrally accurate in the number of spatial discetization points N. This solver is a significant improvement to what was previously available for this problem [1] and allows us to compare solutions corresponding to different shear rate dependent viscosity models by Powell-Eyring, Yeleswarapu and Carreau over long evolution times.Author owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.Mechanical engineeringApplied mathematicsBiomechanicsElectronic Thesis or Dissertation2025-07-23Shear stressStress profileChebyshev polynomialvelocity profileOldroyd-B stress modelChebyshev expansionstabilityLimit Cycle