A fast numerical scheme is proposed to determine the velocity field of an incompressible fluid in a concentric annulus under a constant pressure gradient. The idea behind the scheme is to find the radius R in the annulus where the shear stress becomes zero. In the region from the inner wall at $R=\kappa$ to R, the shear rate is positive, while it is negative from this radius to the outer wall at $r=1$. Integrating the velocity field from the inner wall, where it is zero, one determines its value at R. This acts as the initial value for the integration of the shear rate over the second region, where the velocity must decrease to zero at the outer wall. Choosing a value for R, iterations continue to find its optimal value till the velocity on the outer wall vanishes to within an acceptable error term, which is ${10}^{-10}$ here. The numerical method chosen here delivers this result within 5 to 10 iterations for generalised Newtonian and PTT fluids. For viscoplastic fluids, instead of finding the optimal value for R which lies within the plug, one has to find its counterpart r₁. Noting that the shear stress equals the yield stress at r₁ and that the shear rate is positive from κ to r₂ one finds the velocity at this radius. Since the width of the plug $r_2-r_1$ within the fluid is known and the velocity is constant across the plug, one integrates the velocity field from r₂ to the outer wall at $r=1$ till the velocity approaches zero to within the chosen error term. Once again, the number of iterations to find the velocity field in Bingham and Herschel-Bulkley fluids is small and lies between 5 and 8. Finally, application of the numerical scheme to determine the velocity field in helical flows is also suggested.

提出了一种快速数值方案来确定在恒定压力梯度下同心环空中不可压缩流体的速度场。该方案背后的想法是在剪切应力变为零的环带中找到半径R。在内壁的区域$\mathrm{[R}=\kappa$到R时，剪切速率为正，而从该半径到外壁处的剪切速率为负$\mathrm{[R}=\mathrm{1个}$。积分来自内壁的速度场（该速度场为零），可以确定其在R处的值。这是第二区域上剪切速率积分的初始值，在第二区域上外壁的速度必须降至零。选择R的值，迭代将继续找到其最佳值，直到外壁上的速度消失在可接受的误差范围内，即${10}^{-10}$这里。对于一般的牛顿流体和PTT流体，此处选择的数值方法可在5到10次迭代中提供此结果。对于粘塑性流体，必须找到其对应的r ₁，而不是找到位于塞子内的R的最佳值。注意到剪切应力等于在r ₁处的屈服应力，并且从κ到r ₂的剪切速率是正的，可以找到该半径处的速度。由于插头的宽度${\mathrm{[R}}_2-{\mathrm{[R}}_{\mathrm{1个}}$流体内部的速度是已知的，并且穿过塞子的速度是恒定的，一个对从r ₂到外壁的速度场进行积分$\mathrm{[R}=\mathrm{1个}$直到速度接近零，直到在选定的误差范围内。再一次，在宾厄姆和赫歇尔-比尔克利流体中找到速度场的迭代次数很小，介于5和8之间。最后，还建议应用数值方案确定螺旋流中的速度场。