In this paper a mathematical model is developed for the description of thepulsatile blood flow through an axisymmetric artery of mild varying cross-section,taking into account the tethering effect of the sorrounding tissues.Based on thelinear Navier-Stokes equation and boundary conditions,a set of differential equa-tions for the modulus of phase velocity,blood velocities,pressure and deformationof the vessel wall are deduced and the numerical solutions are found by utilizingthe Runge-Kutta method.The axial velocity profiles and the peak wall shear stressesare given for both mild tapering tube and mild stenosed tube.It is shown thatthe peak wall shear stresses in the tethered tube are higher than those in thefree tube.It is also shown that in a series of previously research work on peakwall shear stresses of varying cross-section vessels the free vessel assumptionhas the tendency of under assessing the peak wall shear stresses and vice versafor the rigid vessel assumption. In this paper a mathematical model is developed for the description of the pulsatile blood flow through an axisymmetric artery of mild varying cross-section, taking into account the tethering effect of the sorrounding tissues. Based on the linear Navier-Stokes equation and boundary conditions, a set of differential equa-tions for the modulus of phase velocity, blood velocities, pressure and deformation of the vessel wall are deduced and the numerical solutions are found by utilizing the Runge-Kutta method. axial velocity profiles and the peak wall It is shown that the peak wall shear stresses in the tethered tube are higher than those in the free tube. It is also shown in that a series of previously research work on peakwall shear stresses of varying cross-section vessels the free vessel assumptionally the tendency of under assessing the peak wall shear stresses and vice versa for the rigid vessel assumption.