Perturbation-numericalanalysis Of The Stokes Flow In Rough Micro-channels And Establishment Of A Roughness Detection Model And Method

With rapid development of manufacturing technology,applications of microfluidic devices are more and more extensive.The whole system can be integrated on a chip of centimeter or even the micron level.At this scale,the influence of surface roughness is an unavoidable problem.Surface roughness arises due to the limitation of machining accuracy and deposition,wear and corrosion in the working procedure.On one hand,surface roughness has an impact on the flow characteristics of the fluid in the microchannel,rendering it no longer fulfilment of the original performance requirements.On the other hand,surface roughness may have great impact on mechanical properties of components such as the forces and torques acting on them by the fluid.Therefore,surface roughness detection is important for judging whether the equipment needs to be replaced or maintained,so as to avoid economic losses caused by equipment failure and damage.The commonly used roughness detection techniques include atomic force microscope,scanning tunneling microscope,ultrasonic method and optical measurement method,etc.Although they have high accuracy,the equipments needed are in general expensive and may have difficulty to directly detect the surface roughness of internal components in microfluidic devices.Therefore,it is of practical importance to develop an economic/less expensive method,which could still capture the overall or mean characteristics of surface roughness.In this paper,from the perspective of fluid mechanics,the boundary perturbation method is carried out to solve the problems of Couette flow and Poiseuille flow in a rough microchannel.The roughness considered is a product of two co-sinusoidal functions,which represents a general component of the full two-dimendional Fourier waveform.This model leaves three parameters to be determined in detection of surface roughness:?:the mean oscillation,n:the mean azimuthal wave number,and?:the mean longitudinal wave number.The accuracy of the analytical solution is verified by numerical method,and the results are extended to expand the roughness amplitude range of proposed roughness detection method.Based on the research results for this model,a new surface roughness detection method of the bumpy shaft is constructed,which is easy to operate,economical and has certain universality and practical value.The specific work is as follows:1.The product cosine function is applied to simulate the surface roughness of a model microchannel which is composed of an outer smooth cylinder and an inner bumpy shaft.The ratio?of the surface roughness amplitude to the average radius of the inner shaft is taken as a small parameter to perform perturbation expansion pursued up to second order.Two flow problems:(i)the Couette flow generated by rotation of the smooth cylinder and(ii)the Poiseuille flow generated by the applying external pressure gradient are solved analytically,respectively.As the Reynolds number is small,the inertia term can be ignored,so the flow can be treated as the Stokes flow.The second-order modified expressions in?of the average torque M on the bumpy shaft of Couette flow and the total flow rate Q in the microchannel of Poiseuille flow are derivated.2.According to the analytical solutions obtained,the effects of dimensionless mean roughness amplitude?of the shaft,the azimuthal wave number n,and the longitudinal wave number?on M and Q under different mean radii b of the shaft are further studied,and the related phenomena are analyzed and explained.The results show that the net effect of roughness on M is always positive,which manifests as the increase of torque and the net effect increases monotonically with increasing b.The net effect on Q is more complicated.For a given b,under the appropriate combination of(n,?),M may have a minimum value while Q may have a maximal value.Moreover,given A_c,different b may have a minimal M or maximal Q at the same(n,?).In addition,due to the limitation of perturbation analysis,the wave numbers n and?are limited to a reasonable range,which essentially depends on b.3.In order to verify the accuracy of perturbation analysis method,the commercial software COMSOL Multiphysics is used for numerical simulation verification.The results showed that for the most part,when?=0.01,0.025,0.05 and 0.1,the error between the analytical solution and the numerical results is small,that is,the effective application range of perturbation analysis method is??0.1.The results were further extended to?=0.15,0.2,0.25,0.3 and 0.4 by numerical method,so as to expand the application scope of the proposed roughness measurement method.4.As the above research results achieved,a surface roughness detection method of the bumpy shaft based on fluid mechanics is constructed by the relationship of mean radius of rough shaft,roughness amplitude,azimuthal wave number and longitudinal wave number with average torque and total flow rate.Based on this method,the mean roughness amplitude,azimuthal wave number and axial wave number can be estimated by the combined measurements of average torque and total flow rate,which can overcome some difficulties of traditional roughness detection technology,laying the foundation for realizing the surface roughness rapid detection of microfluidic devices without disassembly.