Computing stationary free-surface shapes in microfluidics

The problem of determination of friction factor (or average fluid velocity) for a steady gravitational flow of fluid in an inclined open triangular groove is addressed. It is assumed that the angle of slope of the groove is known and the gravity is the only driving force of flow. The solution comprises of two parts. First, for a given groove’s basal angle, liquid-solid contact angle, and the Bond number, the shape of free surface is determined starting from the Young-Laplace equation. The optimization technique as developed formerly is applied. Then, having determined the shape of the free surface and slope of the groove, the distribution of fluid flow is determined. The boundary value problem is solved using boundary collocation method in least square sense. Given the distribution of fluid velocity the friction factor as a function of the other parameters of the model is analyzed.

Free surface deformation due to a source and a sink of equal strength in Stokes flow

A note on the relationship between the solutions of the stokes flow equations in a half-space with a no-slip boundary condition and with a slip boundary condition

When a fluid liquid interface joins a solid boundary a common line is formed, this line will be referre to as the contact line. The main objective of this paper is to obtain explicit solutions of the governing equations of fluid mechanics for viscous flow of a liquid, in the vicinity of a stationary contact line. In each case we have indicated the procedure for developing locally consistent asymptotic expansions for the flow field and the shape of the free surface, with explicit solutions mostly for the first two terms only. Apart from arbitrary constants of the type f'(0; Aj+1) y which are to be determined by the distant driving mechanisms, the flow field and the shape of free surface are completely determined by the contact angle 0(1 and the local arbitrary parameter pQ. However, mainly due to difficulties in labelling the eigen-values and partly because of the size of the paper, in this part we present results for wetting liquids only.

An analytical solution of the impulsive impact of a cylindrical body with an arbitrary cross-section submerged in an undisturbed water surface is obtained by solving a free boundary problem. The studied case of a rigid body moving in a fluid is kinematically equivalent to the case of a fluid moving around a fixed rigid body with acceleration. The problem is formulated in a non-inertial coordinate system attached to the body. After solving the problem, one can find all the flow characteristics in the coordinate system attached to the undisturbed fluid before the impact. The integral hodograph method is applied to derive the complex potential and the complex velocity, both defined in a parameter plane. The boundary-value problem is reduced to a system of the Fredholm integral equations of the first kind. One of them is in the velocity magnitude at the free surface, and the other is in the velocity direction at the bottom surface. The velocity field, the impulsive pressure on the body surface, and the added mass are computed in a wide range of submergence depths for various cross-sectional shapes of the body, such as a flat plate, a circular cylinder, and a rectangle. The associated added masses are found depending on the submergence depth. As the submergence depth tends to infinity, the added mass tends to the value corresponding to that in an unbounded fluid domain. The upward and downward impacts are shown to generate similar magnitudes of the velocity on the free surface and added mass coefficients but with the opposite velocity directions. The obtained solution may be considered as the first-order approximation when solving the problem by the method of small-time series. The presence of the free surface does not change the structure of the flow near the body, determining the added mass. Therefore, the obtained results are expected to reflect practical situations similar to those occurring in the case of the unbounded fluid domain.

Ship hydrodynamics in a constant waterway have been well studied. With a coordinate system fixed on the moving ship, the boundary value problem (BVP) is usually treated as a steady one. However, the hydrodynamics of a ship moving in confined waterways with abrupt changes in width or depth are complex due to their unsteady nature. The studies on such unsteady problems are insufficient. In the engineering practice, the hydrodynamic unsteadiness can be fully reflected by a scenario when a ship enters a lock. Prior studies have predominantly focused on predicting the ships' hydrodynamic forces without considering the unsteady terms on the free-surface boundary conditions. Obviously, such steady or quasi-steady methods overlooked the crucial unsteady phenomena on free water surface. To address this gap, the present study introduces a novel three-level difference scheme to discretize the free surface condition, preserving unsteady terms while maintaining temporal continuity of cells on the free surface. With the implementation of such fully unsteady BVP, we observed some interesting unsteady free surface motions, which were not well documented in the existing literature. To verify our new observations, as well as to validate the numerical method proposed in this study, two physical model tests were designed and conducted in a towing tank: a submerged ellipsoid enters a deep lock at relatively high speeds, and a box enters a shallow and narrow lock at very low speeds. The discussion is highlighted on the unsteady waves in front of the moving bodies, as well as the unsteady resistance induced by such waves.

Under certain conditions of liquid flow through rotating channels, the Coriolis force can induce a free surface to be formed. This problem is of practical importance in a Coriolis wear tester, which is used for determining the sliding wear coefficient of wear materials in slurry handling equipment. A deforming Galerkin finite element method is presented for predicting two‐dimensional turbulent free surface mean flow in rotating channels. Reynolds‐averaged Navier–Stokes (RANS) equations are cast into weak(algebraic) form using primitive variables (velocity and pressure). Eddy viscosity is determined via a mixing length model. Velocity is interpolated biquadratically, while pressure is interpolated bilinearly. The kinematic condition is used to form the Galerkin residual for the free surface. The free surface is represented by Hermite polynomials of zeroeth order for continuity of position and slope. Combined Newton's iteration is used to simultaneously solve for the free surface and the field variables. Results of velocity and pressure fields, as well as the free surface are shown to converge with mesh‐size refinement. There is excellent respect for mass conservation. Results are presented for various values of Rossby number (Ro) and height‐based Reynolds number (ReH). Parameter continuation in Ro and ReH space is used to compute solutions at higher values of flow rate and angular velocity. Copyright © 2002 John Wiley & Sons, Ltd.

Uniqueness of solutions to the SPH integral formulation of the hydrostatic problem, and the convergence of such solution to the exact linear pressure field, are theoretically demonstrated in this paper using Fourier analytical techniques. This problem involves the truncation of the kernel when the Dirichlet boundary condition (BC) on the pressure is imposed at the free surface. Certain hypotheses are assumed, the most important being that the variations of the pressure field occur in length scales of the order of the smoothing length, h. The theoretical analysis is complemented with numerical tests. In addition to the BC at the free surface, the numerical solution requires truncating the infinite subdomain below it, imposing a Neumann BC for the pressure. The consistency and convergence of the numerical solution of the truncated equation with these BCs are sought herein with a global approach, as opposed to previous studies which exclusively assessed it based on the class of the flow extensions. In these numerical tests, and consistently with the theoretical results, the convergence to the exact solution is shown numerically for discretizations with an inter-particle distance to h ratio of order one. However, when this ratio goes to zero as h also goes to zero, it is shown that length scales shorter than h appear in the solution, and that convergence is lost. The conclusions are important for SPH practitioners as setting that ratio to be of order one is a standard practice to lower the computational time.

The effect of the slip boundary condition on the Newtonian die-swell free surface issue is the main objective of this study. The major aspect of this article is the determination of the free surface position using the free surface location methodology under the slip boundary condition. A Taylor Galerkin/pressure-correction (TG/PC) finite element method is used to solve this issue numerically. Furthermore, the system of equations that governs such a problem includes a time-dependent momentum equation and a continuity equation for mass conservation. These equations are shown here in an axisymmetric frame with Newtonian flow. The free-surface location is calculated using the Phan-Thien (dh/dt) approach in conjunction with slip boundary impact. This work focuses on the effect of slip boundary condition on swelling ratio for Newtonian representation. The current findings reveal that the slip boundary condition has a considerable impact on the swelling ratio of the fluid, causing a decrease in the fluid swell. Furthermore, the behaviors of solution components are presented.

An optical method for the measurement of the surface topography, height and normal, and the velocity in free surface flows is presented. This method of surface reconstruction is based on the analysis of the apparent displacement between a reference and refracted images of a laser sheet through an interface viewed from two viewpoints. First, the validation of the method is performed in a wave tank by comparison with resistive wave probe measurements. This technique is then applied in a towing tank to measure the flow in the wake of a ship model. The results highlight correspondences between the free surface shape and the 2D velocity field beneath surface in the wake of the ship model. The instantaneous measurement of free surface allows a reconstruction of the global wave field pattern behind the ship. These results, obtained with a non-intrusive technique and with a high spatial resolution, allow an accurate analysis of the flow behind the ship. Tests were conducted for different Froude numbers in a range between 0.24 and 0.52. The results highlight the effect of the Froude number on established waves system.

In order to describe the transition between slip and rigid boundary conditions, a model with a thin interphasial viscoelastic layer is introduced. By changing the relaxation time (ωτ), the transition from a viscous liquid to a solid layer is observed. When the interface layer is in the liquid state with very small shear nodulus the shear stress components on the boundary vanish, resulting in a slip boundary condition. On the other hand, after the layer is solidified, the two semispaces become welded with the transition of shear and longitudinal stresses to one another. It is shown that the ratio of a hydrodynamic boundary layer thickness to the interface layer thickness plays a major role in this transition. A simple approximate equation for the reflection coefficient from the interface is derived and compared to the exact solutions for different transitional stages of the boundary condition. Theoretical predictions are compared to experimental results.

An experimental and analytical investigation was conducted to determine the free surface shapes of circular jets impinging normal to sharp-edged disks in zero gravity. Experiments conducted in a zero gravity drop tower yielded three distinct flow patterns which were classified in terms of the relative effects of surface tension and inertial forces. An order of magnitude analysis was conducted indicating regions where viscous forces were not significant when computing free surface shapes. The free surface analysis was simplified by transforming the governing potential flow equations and boundary conditions into the inverse plane. The resulting nonlinear equations were solved numerically and comparisons were made with the experimental data for the inertia dominated regime.

This experimental investigation into the nature of free-surface flows was to study the effects of surfactants on the wake of a surface-piercing cylinder. A better understanding of the process of vorticity generation and conversion at a free surface due to the absence or presence of surfactants has been gained. Surfactants, or surface contaminants, have the tendency to reduce the surface tension proportionally to the respective concentration at the free surface. Thus when surfactant concentration varies across a free surface, surface tension gradients occur and this results in shear stresses, thus altering the boundary condition at the free surface. A low Reynolds number wake behind a surface-piercing cylinder was chosen as the field of study, using digital particle image velocimetry (DPIV) to map the velocity and vorticity field for three orthogonal cross-sections of the flow. Reynolds numbers ranged from 350 to 460 and the Froude number was kept below 1.0. In addition, a new technique was used to simultaneously map the free surface deformation. Shadowgraph imaging of the free surface was also used to gain a better understanding of the flow. It was found that, depending on the surface condition, the connection of the shedding vortex filaments in the wake of the cylinder was greatly altered with the propensity for surface tension gradients to redirect the vorticity near the free surface to that of the surface-parallel component. This result has an impact on the understanding of turbulent flows in the vicinity of a free surface with varying surface conditions.

A three-dimensional non-hydrostatic coupled model for free surface – Subsurface variable – Density flows

We propose an approach for temporally coherent patch‐based texture synthesis on the free surface of fluids. Our approach is applied as a post‐process, using the surface and velocity field from any fluid simulator. We apply the texture from the exemplar through multiple local mesh patches fitted to the surface and mapped to the exemplar. Our patches are constructed from the fluid free surface by taking a subsection of the free surface mesh. As such, they are initially very well adapted to the fluid's surface, and can later deform according to the free surface velocity field, allowing a greater ability to represent surface motion than rigid or 2D grid‐based patches. From one frame to the next, the patch centers and surrounding patch vertices are advected according to the velocity field. We seek to maintain a Poisson disk distribution of patches, and following advection, the Poisson disk criterion determines where to add new patches and which patches should e flagged for removal. The removal considers the local number of patches: in regions containing too many patches, we accelerate the temporal removal. This reduces the number of patches while still meeting the Poisson disk criterion. Reducing areas with too many patches speeds up the computation and avoids patch‐blending artifacts. The final step of our approach creates the overall texture in an atlas where each texel is computed from the patches using a contrast‐preserving blending function. Our tests show that the approach works well on free surfaces undergoing significant deformation and topological changes. Furthermore, we show that our approach provides good results for many fluid simulation scenarios, and with many texture exemplars. We also confirm that the optical flow from the resulting texture matches the fluid velocity field. Overall, our approach compares favorably against recent work in this area.