Exact Navier-Stokes Solution for the Pulsatory Viscous Channel Flow with Arbitrary Pressure Gradient

Abstract

In this article, an exact solution of the Navier–Stokes equations is presented for the motion of an incompressible viscous fluidinachannelwitharbitrarypressuredistribution.Ageneralizationispursuedbyexpressingthepressure signal in terms of Fourier coefficients. The flow is characterized by two principal parameters: the pulsation parameter based on the periodic pressure gradient, and the kinetic Reynolds number based on the pulsation frequency. By way of verification, it is shown that for large kinetic Reynolds numbers, the Poiseuille flow may be recovered. Conversely, the purely oscillatory solution by Rott is regained for large pulsation parameters. For sinusoidalpulsations,the flowrateisdeterminedandcomparedwiththesteady flowanalogobtainedunderthesame pressure gradient. The amount of flow amplification or attenuation is determined as a function of frequency and pulsation parameters. For large frequencies, the maximum flow rate is evaluated and related to the pulsation parameter. By characterizing the velocity, vorticity, and shear stress distributions, cases of flow reversal are identified. Conditions leading to flow reversal are delineated for small and large kinetic Reynolds numbers in both planar and axisymmetric chambers. For an appreciable pulsation rate, we find that the flow reverses when the pulsation parameter is increased to the point of exceeding the Stokes number. By evaluating the skin friction coefficient and its limiting value, design criteria for minimizing viscous losses are realized. The optimal frequency that maximizes the flow rate during pulsing is also determined. Finally, to elucidate the effect of curvature, comparisons between planar and axisymmetric flow results are undertaken. The family of exact solutions presented here can thus be useful in verifying and validating computational models of complex unsteady motions in both propulsiveandnonpropulsive applications. Theycan alsobeused toguide the designoffuel injectors andcontrolled experiments aimed at investigating the transitional behavior of periodic flows.