Motion of an array of drops through a cylindrical tube

1998 ◽  
Vol 358 ◽  
pp. 1-28 ◽  
Author(s):  
C. COULLIETTE ◽  
C. POZRIKIDIS
Keyword(s):  
Stokes Flow ◽  
Apparent Viscosity ◽  
Numerical Solutions ◽  
Initial Period ◽  
Mathematical Formulation ◽  
Axisymmetric Flow ◽  
Circular Cross Section ◽  
Cylindrical Tube ◽  
Boundary Integral ◽  
Tube Radius

We study the pressure-driven transient motion of a periodic file of deformable liquid drops through a cylindrical tube with circular cross-section, at vanishing Reynolds number. The investigations are based on numerical solutions of the equations of Stokes flow obtained by the boundary-integral method. It is assumed that the viscosity and density of the drops are equal to those of the suspending fluid, and the interfaces have constant tension. The mathematical formulation uses the periodic Green's function of the equations of Stokes flow in a domain that is bounded externally by a cylindrical tube, which is computed by tabulation and interpolation. The surface of each drop is discretized into quadratic triangular elements that form an unstructured interfacial grid, and the tangential velocity of the grid-points is adjusted so that the mesh remains regular for an extended but limited period of time. The results illustrate the nature of drop motion and deformation, and thereby extend previous studies for axisymmetric flow and small-drop small-deformation theories. It is found that when the capillary number is sufficiently small, the drops start deforming from a spherical shape, and then reach slowly evolving quasi-steady shapes. In all cases, the drops migrate radially toward the centreline after an initial period of rapid deformation. The apparent viscosity of the periodic suspension is expressed in terms of the effective pressure gradient necessary to drive the flow at constant flow rate. For a fixed period of separation, the apparent viscosity of a non-axisymmetric file is found to be higher than that of an axisymmetric file. In the case of non-axisymmetric motion, the apparent viscosity reaches a minimum at a certain ratio of the drop separation to tube radius. Drops with large effective radii to tube radius ratios develop slipper shapes, similar to those assumed by red blood cells in flow through capillaries, but only for capillary numbers in excess of a critical value.

Flow of a spherical capsule in a pore with circular or square cross-section

2011 ◽  
Vol 705 ◽  
pp. 176-194 ◽  
Author(s):  
X.-Q. Hu ◽  
A.-V. Salsac ◽  
D. Barthès-Biesel
Keyword(s):  
Cross Section ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Cylindrical Tube ◽  
Boundary Integral ◽  
Size Ratio ◽  
Constitutive Law ◽  
Boundary Integral Method ◽  
Flow Strength ◽  
Capsule Deformation

AbstractThe motion and deformation of a spherical elastic capsule freely flowing in a pore of comparable dimension is studied. The thin capsule membrane has a neo-Hookean shear softening constitutive law. The three-dimensional fluid–structure interactions are modelled by coupling a boundary integral method (for the internal and external fluid motion) with a finite element method (for the membrane deformation). In a cylindrical tube with a circular cross-section, the confinement effect of the channel walls leads to compression of the capsule in the hoop direction. The membrane then tends to buckle and to fold as observed experimentally. The capsule deformation is three-dimensional but can be fairly well approximated by an axisymmetric model that ignores the folds. In a microfluidic pore with a square cross-section, the capsule deformation is fully three-dimensional. For the same size ratio and flow rate, a capsule is more deformed in a circular than in a square cross-section pore. We provide new graphs of the deformation parameters and capsule velocity as a function of flow strength and size ratio in a square section pore. We show how these graphs can be used to analyse experimental data on the deformation of artificial capsules in such channels.

Liquid toroidal drop in compressional Stokes flow

2015 ◽  
Vol 785 ◽  
pp. 372-400 ◽  
Author(s):  
Michael Zabarankin ◽  
Olga M. Lavrenteva ◽  
Avinoam Nir
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Stokes Flow ◽  
Capillary Number ◽  
Circular Cross Section ◽  
Stationary Solutions ◽  
Boundary Integral ◽  
Normal Velocity ◽  
Dynamic Simulations ◽  
Outer Flow

The deformation of an immiscible toroidal drop embedded in axisymmetric compressional Stokes flow is analysed via the boundary integral formulation in the case of equal viscosity. Numerical simulations are performed for the drop having initially the shape of a torus with circular cross-section. The quasi-stationary dynamic simulations reveal that, when the viscous forces, proportional to the intensity of the flow, are relatively weak compared with the surface tension (the ratio of these forces is characterized by the capillary number,$Ca$), three different scenarios of drop evolution are possible: indefinite expansion of the liquid torus, contraction to the centre and a stationary toroidal shape. When the intensity of the flow is low, the stationary shapes are shown to be close to circular tori. Once the outer flow strengthens, the cross-section of the stationary torus assumes first an elliptic and then an egg-like shape. For the capillary number greater than a critical value,$Ca_{cr}$, toroidal stationary shapes were not found. Remarkably,$Ca_{cr}$is close to the critical capillary number found previously for a simply connected drop flattened in compressional flow. Thus, a new example of non-uniqueness of stationary drop shape in viscous flow is obtained. Approximate stationary solutions in the form of tori with circular and elliptic cross-sections are obtained by minimizing the normal velocity over the drop interface. They are shown to be in good agreement with the stationary shapes from quasi-dynamic simulations for the corresponding intervals of the capillary number.

scholarly journals Investigation of the periodic axisymmetric flow of a viscoelastic fluid through a cylindrical tube

2020 ◽  
pp. 49-52
Author(s):  
J. Braude ◽  
N. Kizilova
Keyword(s):  
Viscous Incompressible Fluid ◽  
Pressure Fluctuations ◽  
Axisymmetric Flow ◽  
Circular Cross Section ◽  
Cylindrical Tube ◽  
Second Order ◽  
Rheological Parameters ◽  
Flow Parameters ◽  
Periodic Pressure ◽  
Relaxation Coefficients

A generalized Womersley model of a nonstationary axisymmetric flow of a viscous incompressible fluid through a tube of circular cross-section to periodic pressure fluctuations at the inlet of the tube is obtained due for the case of a fluid with complicated rheology. The rheological parameters of the fluid are viscosity and four relaxation coefficients for strains and stresses of the first and second order. Such rheology is proper to the non-Newtonian viscoelastic fluids with mesostructure, namely technical and biological micro/ nanofluids. It was shown that with the increase of the relaxation coefficients of the first/second order the flow rate, the average and maximum velocities decrease/increase, accordingly. Simultaneous changes in these parameters can lead to complex changes in the velocity profile, especially for higher harmonics. The studied regularities can explain the deviations of the flow parameters of different micro/nanofluids from the values predicted by the classical Womersley solution for a homogeneous Newtonian fluid, which does not take into account viscous dissipation during the rearrangement of the fluid mesostructure.

scholarly journals Compressible integral representation of rotational and axisymmetric rocket flow

2016 ◽  
Vol 809 ◽  
pp. 213-239 ◽  
Author(s):  
M. Akiki ◽  
J. Majdalani
Keyword(s):  
Numerical Simulations ◽  
Closed Form ◽  
Mass Flux ◽  
Mathematical Formulation ◽  
Circular Cross Section ◽  
Ideal Gas ◽  
Injection Rate ◽  
Dimensional Solution ◽  
Abel Transformation ◽  
Injection Speed

This work focuses on the development of a semi-analytical model that is appropriate for the rotational, steady, inviscid, and compressible motion of an ideal gas, which is accelerated uniformly along the length of a right-cylindrical rocket chamber. By overcoming some of the difficulties encountered in previous work on the subject, the present analysis leads to an improved mathematical formulation, which enables us to retrieve an exact solution for the pressure field. Considering a slender porous chamber of circular cross-section, the method that we follow reduces the problem’s mass, momentum, energy, ideal gas, and isentropic relations to a single integral equation that is amenable to a direct numerical evaluation. Then, using an Abel transformation, exact closed-form representations of the pressure distribution are obtained for particular values of the specific heat ratio. Throughout this effort, Saint-Robert’s power law is used to link the pressure to the mass injection rate at the wall. This allows us to compare the results associated with the axisymmetric chamber configuration to two closed-form analytical solutions developed under either one- or two-dimensional, isentropic flow conditions. The comparison is carried out assuming, first, a uniformly distributed mass flux and, second, a constant radial injection speed along the simulated propellant grain. Our amended formulation is consequently shown to agree with a one-dimensional solution obtained for the case of uniform wall mass flux, as well as numerical simulations and asymptotic approximations for a constant wall injection speed. The numerical simulations include three particular models: a strictly inviscid solver, which closely agrees with the present formulation, and both $k$–$\unicode[STIX]{x1D714}$ and Spalart–Allmaras computations.

An Improved Assembling Algorithm in Boundary Elements With Galerkin Weighting Applied to Three-Dimensional Stokes Flows

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
Sofia Sarraf ◽  
Ezequiel López ◽  
Laura Battaglia ◽  
Gustavo Ríos Rodríguez ◽  
Jorge D'Elía
Keyword(s):  
Boundary Element Method ◽  
Linear System ◽  
Boundary Element ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Computational Time ◽  
Creeping Flows ◽  
Order Of Magnitude ◽  
Element Method

In the boundary element method (BEM), the Galerkin weighting technique allows to obtain numerical solutions of a boundary integral equation (BIE), giving the Galerkin boundary element method (GBEM). In three-dimensional (3D) spatial domains, the nested double surface integration of GBEM leads to a significantly larger computational time for assembling the linear system than with the standard collocation method. In practice, the computational time is roughly an order of magnitude larger, thus limiting the use of GBEM in 3D engineering problems. The standard approach for reducing the computational time of the linear system assembling is to skip integrations whenever possible. In this work, a modified assembling algorithm for the element matrices in GBEM is proposed for solving integral kernels that depend on the exterior unit normal. This algorithm is based on kernels symmetries at the element level and not on the flow nor in the mesh. It is applied to a BIE that models external creeping flows around 3D closed bodies using second-order kernels, and it is implemented using OpenMP. For these BIEs, the modified algorithm is on average 32% faster than the original one.

Physical basis of the dependence of blood viscosity on tube radius

1960 ◽  
Vol 198 (6) ◽  
pp. 1193-1200 ◽  
Author(s):  
Robert H. Haynes
Keyword(s):  
Flow Rate ◽  
Blood Viscosity ◽  
Apparent Viscosity ◽  
Hind Limb ◽  
Marginal Zone ◽  
Tube Wall ◽  
Effective Diameter ◽  
Tube Radius ◽  
Vascular Bed ◽  
A Cell

Two theories are applied to measurements of the decrease in apparent viscosity of blood in narrow tubes (Fahraeus-Lindqvist effect). First, the effect may be attributed to the presence of unsheared laminae in the fluid (sigma phenomenon), and it was found that the thickness of such laminae must vary between 3.5 µ at 10% hematocrit and 34 µ at 80%. Alternatively, the effect may be caused by a cell-free marginal zone adjacent to the tube wall, which would have to be 6 µ thick at 10% hematocrit and 1.5 µ at 80%. There is a slight suggestion in the data that the effect may be reversed as the flow rate approaches zero (i.e. the apparent viscosity rises in small tubes). Finally, a method is proposed for calculating the effective diameter of a vascular bed, and it was found to be 55 µ for a dog's hind limb.

scholarly journals Mechanical Quadrature Methods and Extrapolation for Solving Nonlinear Problems in Elasticity

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Pan Cheng ◽  
Ling Zhang
Keyword(s):  
Asymptotic Expansion ◽  
Nonlinear Boundary ◽  
Numerical Solutions ◽  
Boundary Integral Equations ◽  
Logarithmic Singularity ◽  
Nonlinear Problems ◽  
Boundary Integral ◽  
Convergence Theory ◽  
Mechanical Quadrature ◽  
Quadrature Methods

This paper will study the high accuracy numerical solutions for elastic equations with nonlinear boundary value conditions. The equations will be converted into nonlinear boundary integral equations by the potential theory, in which logarithmic singularity and Cauchy singularity are calculated simultaneously. Mechanical quadrature methods (MQMs) are presented to solve the nonlinear equations where the accuracy of the solutions is of three orders. According to the asymptotical compact convergence theory, the errors with odd powers asymptotic expansion are obtained. Following the asymptotic expansion, the accuracy of the solutions can be improved to five orders with the Richardson extrapolation. Some results are shown regarding these approximations for problems by the numerical example.

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