We analyse the oscillatory flow with a zero-mean of a viscoelastic incompressible fluid over a wavy wall. Using the Maxwell model of viscoelasticity, the problem is transformed into a boundary layer flow by assuming that the amplitude of fluid oscillation as well as the Stokes layer thickness are very small compared with the wavelength of the wall. Analytical solutions are obtained with a perturbation method taking the ratio of the amplitude of fluid oscillation as a small parameter. The first order solution corresponds to the Stokes’ second problem for a Maxwell fluid, with the penetration depth and the wavelength of the oscillation motion modified by the Deborah number that characterizes the viscoelastic behavior. Further, the phase lag introduced by viscoelastic effects between the streamwise velocity component and driving pressure gradient is described by a parametric equation of an ellipse. At second order, non-linear Reynolds stresses in the unsteady boundary layer originate a steady streaming flow that leads to upper and lower steady regions of recirculation. As the Deborah number grows, the lower steady recirculations disappear while the magnitude of the steady tangential axial velocity increases which can positively affect mixing processes. A generalization of the Rayleigh’s law of streaming to the viscoelastic case is obtained and the thickness of the outer layer and the steady tangential velocity are estimated.

我们用零均值的粘弹性不可压缩流体在波状壁上分析振荡流。使用麦克斯韦粘弹性模型，通过假设流体振荡的幅度以及斯托克斯层厚度与壁的波长相比非常小，将问题转化为边界层流。以流体振荡幅值比为小参数，采用摄动法求得解析解。一阶解对应于麦克斯韦流体的斯托克斯第二问题，其中振荡运动的穿透深度和波长由表征粘弹性行为的德博拉数修改。更远，流向速度分量和驱动压力梯度之间的粘弹性效应引入的相位滞后由椭圆参数方程描述。在二阶上，不稳定边界层中的非线性雷诺应力产生稳定的流动，导致再循环的上部和下部稳定区域。随着德博拉数的增加，较低的稳定再循环消失，而稳定切向轴向速度的幅度增加，这可以对混合过程产生积极影响。得到了瑞利流动定律在粘弹性情况下的推广，并估计了外层的厚度和稳定切向速度。非稳定边界层中的非线性雷诺应力产生稳定的流动，导致上稳定再循环区域和下稳定区域。随着德博拉数的增加，较低的稳定再循环消失，而稳定切向轴向速度的幅度增加，这可以对混合过程产生积极影响。得到了瑞利流动定律在粘弹性情况下的推广，并估计了外层的厚度和稳定切向速度。非稳定边界层中的非线性雷诺应力产生稳定的流动，导致上稳定再循环区域和下稳定区域。随着德博拉数的增加，较低的稳定再循环消失，而稳定切向轴向速度的幅度增加，这可以对混合过程产生积极影响。得到了瑞利流动定律在粘弹性情况下的推广，并估计了外层的厚度和稳定切向速度。