Stokes Flow Research Articles

Impact of microorganisms on Blasius–Rayleigh–Stokes flow and heat transfer of nanofluid: A numerical study

AbstractIn this article, we present a numerical study of unsteady flow and heat transfer in a viscous‐based nanofluid, examining the impact of microorganisms on a moving surface emerging from a moving slot. This gives rise to a new type of boundary layer flow, which can resemble that over a stretching or shrinking surface depending on the motion of the slot. To solve the partial differential equations governing the bio‐convectional flow, we employ the Blasius–Rayleigh–Stokes variable and transform the equations into a corrected similar form. Our numerical solutions reveal the presence of dual solutions within a specific range of the moving slot parameter. Our findings have important implications for understanding the behavior of nanofluids and microorganisms in dynamic flow environments.

Sedimentation of short-lived fluid-solid suspensions

In this study, we propose a particle sedimentation/aggradation law for homogeneous concentrated suspensions. This law, valid in the Stokes flow regime, can be used to describe the sedimentation process observed in short-lived rapid flows, developed at high Reynolds numbers, that can be described as low-viscosity quasi-parallel flows traveling at constant velocity and that progressively sediment during the dominant phase of transport to leave a triangular or trapezoidal deposit of constant slope. The particle aggradation velocity can thus be predicted from the product of the mean flow velocity and the deposit slope and turns out to be roughly similar to that measured from static suspensions of the same concentration, provided that the flow Reynolds number, based on the mean flow velocity, the fluid properties, and the particle size, remains inferior to a few hundred, such as the mixture agitation cannot disturb the sedimentation process. These important results provide the possibility of describing the depositional dynamics during the final stage of extreme events, as well as to infer the mixture rheology, from physical parameters that can be easily measured in the field.

Odd viscous flow past a sphere at low but non-zero Reynolds numbers

The measurement of lift on symmetrically shaped obstacles immersed in low Reynolds number flow is the quintessential way to signal odd viscosity. For flow past cylinders, such a lift force does not arise if incompressibility and no-slip boundary conditions are fulfilled, whereas for spheres, a lift force has been found in Stokes flow, which is valid for cases where the Reynolds numbers are negligible and convection can be ignored. When considering the role of convection at low but non-zero Reynolds numbers, two hurdles arise, the Whitehead paradox and the breaking of axial symmetry, which are overcome by the method of matched asymptotic expansions and the Lorentz reciprocal theorem, respectively. We also consider the case where axial symmetry is preserved because the translation of the sphere is aligned with the axis of chirality of odd viscosity. We find that while lift vanishes, the interplay between odd viscosity and convection gives rise to a stream-induced torque.

Multiscale preconditioning of Stokes flow in complex porous geometries

Fluid flow through porous media is central to many subsurface (e.g., CO2 storage) and industrial (e.g., fuel cell) applications. The optimization of design and operational protocols, and quantifying the associated uncertainties, requires fluid-dynamics simulations inside the microscale void space of porous samples. This often results in large and ill-conditioned linear(ized) systems that require iterative solvers, for which preconditioning is key to ensure rapid convergence. We present robust and efficient preconditioners for the accelerated solution of saddle-point systems arising from the discretization of the Stokes equation on geometrically complex porous microstructures. They are based on the recently proposed pore-level multiscale method (PLMM) and the more established reduced-order method called the pore network model (PNM). The four preconditioners presented are the monolithic PLMM, monolithic PNM, block PLMM, and block PNM. Compared to existing block preconditioners, accelerated by the algebraic multigrid method, we show our preconditioners are far more robust and efficient. The monolithic PLMM is an algebraic reformulation of the original PLMM, which renders it portable and amenable to non-intrusive implementation in existing software. Similarly, the monolithic PNM is an algebraization of PNM, allowing it to be used as an accelerator of direct numerical simulations (DNS). This bestows PNM with the, heretofore absent, ability to estimate and control prediction errors. The monolithic PLMM/PNM can also be used as approximate solvers that yield globally flux-conservative solutions, usable in many practical settings. We systematically test all preconditioners on 2D/3D geometries and show the monolithic PLMM outperforms all others. All preconditioners can be built and applied on parallel machines.

Well-posedness and stability for the two-phase periodic quasistationary Stokes flow

The two-phase horizontally periodic quasistationary Stokes flow in \mathbb{R}^{2} , describing the motion of two immiscible fluids with equal viscosities that are separated by a sharp interface, parameterized as the graph of a function f=f(t) , is considered in the general case when both gravity and surface tension effects are included. Using potential theory, the moving boundary problem is formulated as a fully nonlinear and nonlocal parabolic problem for the function f . Based on abstract parabolic theory, it is shown that the problem is well-posed in all subcritical spaces \mathrm{H}^{r}(\mathbb{S}) , with r\in(3/2,2) . Moreover, the stability properties of the flat equilibria are analyzed in dependence on the physical properties of the fluids.

TetraFEM: Numerical Solution of Partial Differential Equations Using Tensor Train Finite Element Method

In this paper, we present a methodology for the numerical solving of partial differential equations in 2D geometries with piecewise smooth boundaries via finite element method (FEM) using a Quantized Tensor Train (QTT) format. During the calculations, all the operators and data are assembled and represented in a compressed tensor format. We introduce an efficient assembly procedure of FEM matrices in the QTT format for curvilinear domains. The features of our approach include efficiency in terms of memory consumption and potential expansion to quantum computers. We demonstrate the correctness and advantages of the method by solving a number of problems, including nonlinear incompressible Navier–Stokes flow, in differently shaped domains.

Time Evolution of the Navier–Stokes Flow in Far-Field

Time Evolution of the Navier–Stokes Flow in Far-Field

The method of fundamental solutions for multi-particle Stokes flows: Application to a ring-like array of spheres

A method is presented for calculating Stokes flow around multiple particles of arbitrary shape. It uses the Method of Fundamental Solutions (MFS) applied to single particles, combined with an iterative scheme to resolve the many particle-particle hydrodynamic interactions; an approach that is reminiscent of the Method of Reflections. The attractive features of the proposed method are inherited from the MFS — simplicity and accuracy — while providing orders of magnitude computational speed-up for large particle systems. The method is verified through a series of test cases, including those involving strong lubrication forces and non-spherical particles. Unlike applications of the Method of Reflections reported in the literature, the iterative scheme we propose (a block Gauss-Seidel approach to solving a particle-particle interaction matrix) converges for all the cases we consider, for both resistance and mobility problems. The scheme is applied to the study of Stokes flow around ring-like arrays of spheres. We show that the relationship between globally applied velocity or force to the response of individual spheres can be described by just 5 coefficients (or 9 in total) for any given configuration. The results indicate that for 7–10 spheres in sedimentation, there exists a certain spacing that produces steady-state translation of the ring, independent of its orientation. For large numbers of spheres, slender-body theory can be applied to the problem, providing remarkably close agreement to the numerical results over a wide range of parameters.

Boundary Element Method for Viscous Flow through Out-phase Slip-patterned Microchannel under the Influence of Inclined Magnetic Field

Abstract In this paper, we investigate the impact of an inclined magnetic field of uniform intensity on viscous, incompressible pressure-driven Stokes flow through a slip-patterned, rectangular microchannel using the boundary element method based on the stream function-vorticity variables approach. The present investigation focuses only on the out-phase slip patterning of the microchannel walls. We address two scenarios of slip patterning, specifically large and fine slip patterning, which are determined by the periodicity of the patterning. We utilized the no-slip and Navier’s slip boundary conditions in an alternative manner on the walls. The Stokes equations govern the viscous flow through a microchannel. We assume a very small magnetic Reynold’s number to eliminate the equation of induced magnetic field in the present study. We analyzed the impact of considered dimensionless hydrodynamic parameters, including the Hartman number (Ha), inclination angle (θ) of the magnetic field, and the slip length (ls ) on fluid dynamics. In the case of fine slip, we observed significant variations in both velocity and pressure gradient, in contrast to large slip patterning. Fine slip patterning significantly increases the shear stress at slip regimes, while large slip periodicity significantly reduces it at no-slip regimes. The present investigation has several notable implications, such as regulation and advancement of mixing and heat transmission within microfluidic systems.

Stokes flow with Tresca boundary condition: a posteriori error analysis

Stokes flow with Tresca boundary condition: a posteriori error analysis

Divergence-free cut finite element methods for Stokes flow

We develop two unfitted finite element methods for the Stokes equations based on Hdiv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{H}^{{{\\,\ extrm{div}\\,}}}$$\\end{document}-conforming finite elements. Both cut finite element methods exhibit optimal convergence order for the velocity, pointwise divergence-free velocity fields, and well-posed linear systems, independently of the position of the boundary relative to the computational mesh. The first method is a cut finite element discretization of the Stokes equations based on the Brezzi–Douglas–Marini (BDM) elements and involves interior penalty terms to enforce tangential continuity of the velocity at interior edges in the mesh. The second method is a cut finite element discretization of a three-field formulation of the Stokes problem involving the vorticity, velocity, and pressure and uses the Raviart–Thomas (RT) space for the velocity. We present mixed ghost penalty stabilization terms for both methods so that the resulting discrete problems are stable and the divergence-free property of the Hdiv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{H}^{{{\\,\ extrm{div}\\,}}}$$\\end{document}-conforming elements is preserved also on unfitted meshes. In both methods boundary conditions are imposed weakly. We show that imposing Dirichlet boundary conditions weakly introduces additional challenges; (1) The divergence-free property of the RT and the BDM finite elements may be lost depending on how the normal component of the velocity field at the boundary is imposed. (2) Pressure robustness is affected by how well the boundary condition is satisfied and may not hold even if the incompressibility condition holds pointwise. We study two approaches of weakly imposing the normal component of the velocity at the boundary; we either use a penalty parameter and Nitsche’s method or a Lagrange multiplier method. We show that appropriate conditions on the velocity space has to be imposed when Nitsche’s method or penalty is used. Pressure robustness can hold with both approaches by reducing the error at the boundary but the price we pay is seen in the condition numbers of the resulting linear systems, independent of if the mesh is fitted or unfitted to the boundary.

A priori and a posteriori error analysis of a mixed DG method for the three-field quasi-Newtonian Stokes flow

Abstract In this paper we propose and analyse a new mixed-type DG method for the three-field quasi-Newtonian Stokes flow. The scheme is based on the introduction of the stress and strain tensor as further unknowns as well as the elimination of the pressure variable by means of the incompressibility constraint. As such, the resulting system involves three unknowns: the stress, the strain tensor and the velocity. All these three unknowns are approximated using discontinuous piecewise polynomials, which offers flexibility for enforcing the symmetry of the stress and the strain tensor. The unique solvability and a comprehensive convergence error analysis for all the variables are performed. All the variables are proved to converge optimally. Adaptive mesh refinement guided by a posteriori error estimator is computationally efficient, especially for problems involving singularity. In line of this mechanism we derive a residual-type a posteriori error estimator, which constitutes the second main contribution of the paper. In particular, we employ the elliptic reconstruction in conjunction with the Helmholtz decomposition to derive the a posteriori error estimator, which avoids using the averaging operator. Several numerical experiments are carried out to verify the theoretical findings.

Torque-driven superhydrophobic cylinders swim in circles

A superhydrophobic cylinder is a circular cylinder with a boundary pattern of alternating shear-free menisci and solid portions. This article derives analytical solutions for the torque-driven Stokes flow around a superhydrophobic cylinder that is free to move, or ‘swim’, subject to the constraint that it is free of net force. The mathematical approach is a natural generalization of one used to solve for transverse shear flow over a superhydrophobic surface comprising a periodic array of no-shear slots (Crowdy 2011 Phys. Fluids 23 (doi: 10.1063/1.3605575 )). First, the torque-driven Stokes flow around a superhydrophobic cylinder with two unequal no-shear menisci and two equal solid portions is solved in detail. It is then shown how that analysis generalizes, in a natural way, to a cylinder with M no-shear menisci between M solid portions with full exposition given of the case of M = 3 equal solid portions. A further generalization to flow around a superhydrophobic cylinder with an N -fold rotationally symmetric patterning with M menisci in each 2 π / N sector is made. Torque-driven superhydrophobic cylinders with a no-shear pattern devoid of any rotational symmetry are shown to swim on circular trajectories with formulas for the radius and cylinder angular velocity on these trajectories derived here as functions of the no-shear pattern geometry. The M = 1 case of a ‘Janus’ superhydrophobic cylinder having one meniscus and one solid portion can be solved completely explicitly.

Particle response to oscillatory flows at finite Reynolds numbers

The response of spherical particles to oscillatory fluid flow forcing at finite Reynolds numbers exhibits significant deviations from classical analytical predictions due to nonlinear convective contributions. This study employs finite element simulations to explore the long-term (stationary) behavior of such particles across a wide range of conditions, including various external and particle Reynolds numbers, Strouhal numbers, and fluid-to-particle density ratios. Key contributions of this work include determining the range of validity of Tchen's equation of motion for infinitesimal and finite Reynolds numbers and correlating particle response for a wide range of density ratios and flow conditions at high frequency oscillations. This work introduces a modified form of the history drag term in a newly proposed Lagrangian equation of motion. The new equation incorporates a parameter-dependent fractional-order derivative tailored to accommodate nonlinearities due to convective effects. These novel correlations not only extend the operational range of existing model equations but also provide accurate estimates of particle response under a range of external flow conditions, as validated by comparison with numerical solutions of the Navier–Stokes flow around the particles.

Stokes flow past an array of circular cylinders through slip-patterned microchannel using boundary element method

Two-dimensional viscous incompressible, pressure-driven, creeping flow at low Reynold's number (Re ≪ 1) around a series of circular cylinders in a slip-patterned rectangular microchannel is investigated numerically by using the boundary element method (BEM) based on a non-primitive variables approach. The non-primitive variables approach refers to the combination of stream function and vorticity variables. The Stokes equations are used to govern the flow of creeping fluid through a microchannel. We consider the alteration of the slip on both the upper and lower surfaces of the microchannel maintain the same phase (i.e., in-phase configuration). Here, the slip boundary condition refers to Navier's slip boundary condition. We considered both small as well as large patterned slip on both surfaces of the microchannel. Moreover, we have assumed that a number of cylinders of equal diameter are present in the in-line configuration in the path of flow. We studied streamlines, velocity profiles, pressure gradients, and the shear stresses with varied slip-length, and the radius of the cylinder, to get a complete comprehension of flow dynamics. We observed that the velocity and shear stress profiles exhibit significant variability in the case of fine slip patterning. Additionally, the proposed investigation holds several potential applications, such as drug capsule delivery systems, hemodynamics, bio-MEMS technology, and so forth.

Predicting the Representative Elementary Volume by determining the evolution law of the convergence cone

In order to characterise a rock formation prior to subsurface operations, it is required to find a microscale rock volume for which the homogenised property does not fluctuate when the size of the sample is increased; the Representative Elementary Volume (REV). Its determination usually comes at the cost of a large number of simulations, making it overall a computationally expensive process. Therefore, many scientific studies have been dedicated to optimising the process of finding REV. Using statistical numerical methods, it is shown that the fluctuation of the effective property corresponds overall to a cone-like shape convergence. We suggest determining the generic evolution law of the cone of convergence, which can be used to predict the size of the REV and the effective physical property. This study is based on simulations of Stokes flow through idealised microstructures from which the permeability is upscaled. By tracing and plotting the convergence of permeability for multiple samples, the full cone of convergence appears. The cone shows exponential growth and decay, converging towards the effective permeability of the microstructure. By fitting a log-normal distribution on the collected data points, we show that the generic evolution law of the cone of convergence can always be described with two parameters, independently of the porosity. We show that the determined law of the cone also applies to real microstructures, despite the presence of natural heterogeneities. The new method allows us to reduce the computational costs of finding all characteristics related to REV by simulating several subsamples rather than the full-sized sample, unlocking thereby high-resolution samples which are often too computationally expensive. The use of a statistical model provides quantification of the precision level we can obtain on the REV determination.

Almost periodic motions and their stability of the non-autonomous Oseen–Navier–Stokes flows

Almost periodic motions and their stability of the non-autonomous Oseen–Navier–Stokes flows

Series solution and its extension for the nonlinear flow response of soft hair beds

In nature and engineering applications, flexible fiber beds covering biological surfaces can play a role in reducing resistance. These fibers deform under the action of fluids, and this deformation affects the fluid flow state, forming a complex fluid–solid interaction phenomenon. To quantitatively analyze these issues, the physical model is simplified. We focus on the deformation of a soft hair bed caused by Stokes flow. Additionally, we study the deformation of a single hair under Stokes flow in greater detail. The deformation problem of an elastic single fiber in a channel caused by Stokes flow can be described by a nonlinear integral equation. We have obtained a new series solution, which has been compared with the previous perturbation method to verify the accuracy and effectiveness of the series solution. Meanwhile, we have further provided an extended form of flexible fiber deformation through experimental fitting. This fluid–solid interaction problem involves multiple fields and is very important in many natural and engineering systems. The research in this paper can not only help us better understand complex phenomena in nature but also delve into the interaction mechanism between fluids and solids, providing a theoretical basis for future scientific research and engineering applications.

The effect of slip parameter in an axisymmetric oscillatory Stokes flow

A general solution of Stokes equations for the problem of an axisymmetric oscillatory flow of an incompressible, viscous fluid past a sphere satisfying general boundary conditions is obtained. The behavior of the magnitude of drag is observed with the variation of the slip parameter. Some more interesting behaviors are detailed, and several existing results pertaining to steady flows and flows with rigid and shear free boundary conditions are recovered as special cases. The corresponding results are discussed for four different axisymmetric oscillatory Stokes flows, namely, uniform flow, flows generated due to a dipole, a source, and a Stokeslet. A few interesting streamline patterns like formation, elongation, and disappearance of viscous eddies in the vicinity of the sphere with a periodic reversal of the flow are observed at different frequencies for different values of the slip parameter.

A symmetric multigrid-preconditioned Krylov subspace solver for Stokes equations

Numerical solution of discrete PDEs corresponding to saddle point problems is highly relevant to physical systems such as Stokes flow. However, scaling up numerical solvers for such systems is often met with challenges in efficiency and convergence. Multigrid is an approach with excellent applicability to elliptic problems such as the Stokes equations, and can be a solution to such challenges of scalability and efficiency. The degree of success of such methods, however, is highly contingent on the design of key components of the multigrid scheme, including the hierarchy of discretizations, and the relaxation scheme used. Additionally, in many practical cases, it may be more effective to use a multigrid scheme as a preconditioner to an iterative Krylov subspace solver, as opposed to striving for maximum efficacy of the relaxation scheme in all foreseeable settings. In this paper, we propose an efficient symmetric multigrid preconditioner for the Stokes Equations on a staggered finite difference discretization. Our contribution is focused on crafting a preconditioner that (a) is symmetric indefinite, matching the property of the Stokes system itself, (b) is appropriate for preconditioning the SQMR iterative scheme [1], and (c) has the requisite symmetry properties to be used in this context. In addition, our design is efficient in terms of computational cost and facilitates scaling to large domains.