From biology to engineering, while numerous applications are based on capillary phenomena in tubes having roughened surfaces, such as blood transport, paper-based rapid diagnostics, microfluidic fuel cells, and shale gas transport, the dynamics of such capillary flow remains poorly understood. We present a theoretical model for a circular undulated tube that has an idealized cosine-type inner wall characterized by two key morphological parameters: undulation amplitude and axial wave number. With the tube oriented at an arbitrary angle, we first characterize the apparent contact angle of the fluid as a function of local distortion angle and then establish a theoretical model involving inertia, viscosity, and gravity to describe the dynamics of capillary flow. A dimensionless number combining the three forces is introduced to quantify their influence. The model predictions reveal that, in an undulated tube with large wave numbers, the capillary height in equilibrium state is generally lower than that in a smooth tube of similar dimensions, whereas the reverse holds if the wave number becomes relatively small. When the viscosity of fluid is sufficiently small, capillary oscillation in an undulated tube is alleviated relative to that in a smooth tube, and hence stable capillary flow forms more easily in the former.

To machine precision, a micro-duct cannot be fabricated without producing surface roughness. It is of essential importance to examine the effects and predict the level of roughness on electro-osmotic (EO) pumping for ducts of fundamental shapes. In this study, we consider a bumpy microtube with its wall shape modeled by the product of two sinusoidal functions. Boundary perturbation is carried out with respect to the amplitude roughness ε (relative to the Debye length) up to the second-order by considering the Debye-Hückel approximation and viscous Stokes equation for the electrolyte transport. Besides the amplitude roughness ε, the key parameters include the azimuthal wave number n and the axial wave number α of the bumpiness, as well as the non-dimensional electrokinetic width K. It is shown that the EO pumping rate Q is modified by a second-order term −ε²πχ, namely, Q = Q₀ − ε²πχ, where Q₀ denotes the pumping rate through the smooth tube. The net effect χ = χ₁ + χ₂ comprises two components: χ₁ = χ₁(K) < 0 increases with increasing K, representing a pure gain, while χ₂ has no definite sign and is a complex function of K, n, and α. In particular, χ is negative at small α whilst being positive at large α, and the dividing line of signs also depends on K. For small α (1), χ decreases with increasing n at large K (>20). For a given number of oscillations A_c = nα (>1), there exists an intermediate n at which the EO pumping rate is maximized at small K (<20). Moreover, we identify a long-wavelength limit singularity in the EO pumping rate as α → 0 for all n, i.e., in the longitudinal sense. In addition, the velocity component along the tube axis is modified by a second-order term of the roughness, though the same velocity component near the wavy wall exhibits periodic behaviors in phase with the wall roughness. Physical reasoning is given to all the derived mathematical results, and their implication in practical applications as a model for predicting tube roughness is explained. As the tube shape represents a conduit of practical use, a particular emphasis is placed upon potential applications of the derived result.

A numerical investigation was conducted in this study utilizing Force Element Analysis to explore the vortex-induced vibration (VIV) mechanism of side-by-side dual cylinders under the conditions of Reynolds number Re = 100, mass ratio m* = 10, and spacing ratios L/D ranging from 3 to 6. The hydrodynamic forces by force element formulas were incorporated into the vibration response calculations of elastically supported rigid cylinders using a User-Defined Function (UDF) and the fourth-order Runge–Kutta method. A comprehensive analysis was performed to elucidate the combined effects of the spacing ratio L/D and reduced velocity U_r on the vibration responses, quantifying the hydrodynamic forces involved in the mutual interaction during VIV for side-by-side dual cylinders. The influence mechanisms of inter-cylinder interaction and their effects on the resultant hydrodynamic phenomena were discussed. It was revealed that for side-by-side arranged dual cylinders outside the “lock-in region”, the lift and drag forces are predominantly supplied by the volume vorticity forces in conjunction with surface vortices (including frictional) forces. However, within the “lock-in region”, the surface acceleration lift forces provide greater force contributions, and the volume vorticity lift force contributes significantly to negative values. Notably, alterations to the spacing ratio do not change the proportion of force element components. The amplitudes of the cylinders’ mutual interaction forces are identical in magnitude but opposite in phase. Additionally, the “slapping” phenomenon near the “lock-in region” leads to “bounded” trajectories of cylinders.

A microshaft may become rough due to corrosion, abrasion, and deposition when it has been operating in a viscous fluid. It is of importance to investigate the effects and to estimate the level of the shaft’s surface roughness. In this study, we consider a bumpy shaft with its shape modeled by the product of two cosinoidal functions; the roughness ε is defined to be the ratio of the amplitude of the product to the mean radius b of the shaft. First, we consider the Couette flow of the shaft in a viscous fluid enclosed by a rotating smooth cylinder. A perturbation analysis is carried out for the Stokes equation with respect to ε up to the second-order with the key parameters including the azimuthal wave number n and the axial wave number α of the roughness, as well as the mean radius b. In addition, a perturbation analysis is performed for the Poiseuille flow in the gap between the shaft and the shrouded cylinder so that we have complete information for estimating the mean roughness of the shaft. Moreover, numerical simulations are carried out for the torque acting on the shaft at selected b, ε, and wave numbers n, α for verifying the accuracy of the perturbation results. It is shown that the mean torque M acting on the unit area of the bumpy shaft and the total flow rate Q of the Poiseuille flow are both modified by a second-order term of roughness in ε, namely, M = M₀ + ε²η and Q = Q₀ − ε²2πχ, where M₀ and Q₀ denote the torque and the flow rate, respectively, for the smooth shaft. The net effects are conveniently written as η = η₁ + η₂ and χ = χ₁ + χ₂, both comprising two components: η₁ = η₁ (b) < 0 (pure deficit) increases with increasing b and χ₁ = χ₁ (b) first increases and then decreases again with increasing b, while η₂ and χ₂ are complex functions of b, n, and α. For a given density of roughness A_c = nα, there exists an intermediate n at which the mean torque M is minimized, while the total flow rate Q is maximized. The main results are thoroughly derived with all the steps of derivation explained physically, and their relationships to the various geometrical parameters are used to establish a simplified model for predicting the shaft roughness within the range of reasonable accuracy.

The present study is aimed to investigate the effects of wavy roughness on electro-osmotic (EO) flow over a wavy plate. The waviness of the plate is modeled by the product of two cosinoidal functions, and the roughness (ε) is defined to be the ratio of the wavy amplitude to the Debye length. The effects are examined with respect to the roughness ε and different wave numbers (α and β) of the plate waviness. The analysis of the EO flow over the wavy plate is carried out for the applied electric potential, the potential for the electric double layer, as well as the EO flow velocity and pressure field under the Debye-Hückel approximation by using a boundary perturbation method. It is found that the velocity component along the direction of the applied electric field is modified by a second-order term of the roughness, though the same velocity component near the wavy wall exhibits periodic behaviors in phase with the plate waviness. The mean flow rate deficit (ε²μ₂) due to the surface roughness presents a sophisticated dependence on the longitudinal wave number (α) and transverse wave number (β) of the plate waviness, yet the flow deficit is linear in α for small α at β = 0, and shows a long wavelength limit singularity at β = 0 for α ≠ 0.

In this study, we present an analysis of a Stokes micropump comprising a periodic array of parallel finite belts moved by rotating shafts. The geometry of the mechanical micropump is uniquely determined by the ratio of the length of the belts to the width between two neighboring belts (i.e., the aspect ratio a). The method of eigenfunction expansions with collocation is applied to solve the Stokes equation for the pumping rate, the stream function, and the velocity field as well as for the pressure gradient, which are all normalized by proper scales. It is found that with increasing a, the normalized pumping rate per unit micropump (or, simply abbreviated as a unit channel) first increases drastically and then decreases exponentially until it becomes a constant for large a, indicating that there exists a critical aspect ratio (a_c = 0.035) at which the maximum pumping rate (q_max = 0.861) occurs, while the limiting value of q at large a is 0.5. The steady flow is driven by the moving belts against the established pressure gradient, and the pressure gradient at the centerline reaches its maximum value at the channel center and vanishes at distances from the micropump. Moreover, it is shown that the average flow velocity component perpendicular to the moving direction of the belts is relatively small, so that the flow field in the channel is approximately a unidirectional laminar flow, and therefore, the results are not necessarily limited to very low Reynolds numbers.