Unsteady Stokes Flow Research Articles

Numerical algorithms and simulations of boundary dynamic control for optimal mixing in unsteady Stokes flows

This work develops an efficient and accurate optimization algorithm to study the optimal mixing problem driven by boundary control of unsteady Stokes flows, based on the theoretical foundation laid by Hu and Wu in a series of work. The scalar being mixed is purely advected by the flow and the control is a force exerted tangentially on the domain boundary through the Navier slip conditions. The control design has potential applications in many industrial processes such as rotating wall driven mixing, mircomixers with acoustic waves, and artificial cilia mixing.The numerical algorithms have high complexity, high accuracy demand, and high computing expense, due to the multiscale nature of the mixing problem and the optimization requirements. A crucial problem is the computation of the Gâteaux derivative of the cost functional. To this end, a hybrid approach based on variational formula and finite difference is built with high accuracy and efficiency to treat various types of control input functions. We have experimented with various optimization algorithms including the steepest descent algorithm, the conjugate gradient method and two line search options (backtracking and exact line search). We are able to identify and implement the best combinations.The numerical simulations show that the mixing efficacy is limited when only one single type of control is applied, but can be enhanced when more diverse control types and more time segmentation are utilized. The mix-norm in the optimal mixings decays exponentially. The numerical study in this work demonstrates that boundary control alone could be an effective strategy for mixing in incompressible flows.

Two-level additive Schwarz methods for three-dimensional unsteady Stokes flows in patient-specific arteries with parameterized one-dimensional central-line coarse preconditioner

We consider the numerical solution of unsteady Stokes equations in patient-specific arterial-like domains in 3D. A Stokes-like solver is a necessary component in a more sophisticated nonlinear Navier-Stokes method, for which several multilevel domain decomposition methods have been introduced recently. Because of the complex geometry, the construction and the solve of the coarse problem usually take a large percentage of the total compute time. In this paper, we introduce a parameterized one-dimensional Stokes solver defined along the centerline of the artery and use its stabilized finite element discretization to construct a coarse preconditioner. With suitable 3D-to-1D restriction and 1D-to-3D extension operators on fully unstructured meshes, a two-level additive Schwarz preconditioner can be constructed. Some numerical experiments for flows in realistic arteries are presented to show the efficiency and robustness of the new coarse preconditioner whose computational cost is considerably lower than the existing three-dimensional coarse preconditioners.

A homogenization approach to the effect of surfactant concentration and interfacial slip on the flow past viscous drops

In this paper, the collective motion of viscous drops is investigated in terms of a porous medium setting. We assume that the interface of each of the drops is fully covered with a stagnant layer of surfactant in an arbitrary unsteady Stokes flow with low surface Péclet number limit. The effect of the interfacial slip coefficient on the behavior of the flow fields is also considered. The modeling of the physical processes leads to a system of parabolic partial differential equations (PDEs) with interfacial boundary conditions. We have proposed a model in a micro–macro scale setting. We have shown the existence of a unique positive weak solution of such system. We have also obtained a homogenized (upscaled) model of the system via two-scale convergence and boundary unfolding method.

Singular geometry perturbation based method for shape-topology optimization in unsteady Stokes flow

This paper concerns the topological derivatives and its applications for solving shape and topology optimization problems in fluid mechanics. The fluid flow is governed by the unsteady incompressible Stokes equations in the two dimensional case. We derive a topological sensitivity analysis for this parabolic-type operator. The proposed approach is based on a preliminary estimate describing the variation of the velocity field caused by the presence of a small obstacle inside the fluid flow. We obtain a topological asymptotic expansion for the unsteady Stokes operator valid for a large class of shape functions and an arbitrarily shaped geometric perturbation. Then, the topological gradient is exploited for building an efficient and accurate topology optimization algorithm. Finally, we present some numerical investigations showing the efficiency of the proposed approach.

Unsteady Stokes Flow Past a Shear Free Sphere

A method of solution for the problem of an arbitrary unsteady Stokes flow in the presence of a shear free sphere is discussed. The corresponding Faxén [2] relations for a shear-free sphere are derived. Some previously known results are derived as limiting cases and are detailed in an example.

Study of MHD Mixed Convention Flow and Heat Transfer Through Porous Media

In this present paper, the effects of chemical reaction on two-dimensional mixed convection flow and heat transfer of an electrically conducting micropolar fluid in a porous medium between two parallel plates with Soret and Dufour have been considered. The reduced flow field equations are solved using the quasilinearization method. The effects of various parameters such as Hartmann number, inverse Darcy's parameter, Schmidt number, Prandtl number, chemical reaction rate, Soret and Dufour numbers on the velocity components, microrotation, temperature distribution, and concentration are studied in detail and presented in the form of graphs. The flow through porous boundaries has many applications in science and technology such as water waves over a shallow beach, mechanics of the cochlea in the human ear, aerodynamic heating, flow of blood in the arteries, and petroleum industry. Several authors have studied theoretically the laminar flow in porous channels. Berman [1] considered the viscous fluid and analyzed the flow characteristics when it passed through the porous walls. Later the same problem for different permeability was studied by Terril and Shrestha [2]. The theory of micropolar fluids was introduced by Eringen [3] which are considered as an extension of generalized viscous fluids with microstructure. Examples for micropolar fluids include lubricants, colloidal suspensions, porous rocks, aerogels, polymer blends, and microemulsions. The same Berman problem with micropolar fluid was discussed by Sastry and Rama Mohan Rao [4]. The flow and heat transfer of micropolar fluid between two porous parallel plates was analyzed by Ojjela and Naresh Kumar [5]. Srinivasacharya et al. [6] obtained an analytical solution for the unsteady Stokes flow of micropolar fluid between two parallel plates. The effect of buoyancy parameter on flow and heat transfer of micropolar fluid between two vertical parallel plates was investigated by Maiti [7].

Hydrodynamic Green functions: paradoxes in unsteady Stokes conditions and infinite propagation velocity in incompressible viscous models

We present a simple representation of the hydrodynamic Green functions grounded on the free propagation of a vector field without any constraints (such as incompressibility) coupled with a gradient gauge in order to enforce these constraints. This approach involves the solution of two scalar problems: a couple of Poisson equations in the case of the Stokes regime, and a system of diffusion/Poisson equations for unsteady Stokes flows. The explicit and closed-form expression of the Green function for unsteady Stokes flow is developed. The relevance of this approach resides in its conceptual simplicity and it enables us to focus on the intrinsic singularities (Stokesian paradoxes) associated with the propagation of the stresses in incompressible flows under unsteady Stokes conditions, determining the occurrence of power-law tails in the velocity profile arbitrarily far away from the location of the impulsive force.

A particle-based method using the mesh-constrained discrete point approach for two-dimensional Stokes flows

Meshless methods inherently do not require mesh topologies and are practically used for solving continuum equations. However, these methods generally tend to have a higher computational load than conventional mesh-based methods because calculation stencils for spatial discretization become large. In this study, a novel approach for the use of compact stencils in meshless methods is proposed, called the mesh-constrained discrete point (MCD) approach. The MCD approach introduces a Cartesian mesh system to the background of a domain. And the approach rigorously constrains the distribution of discrete points (DPs) in each mesh by solving a dynamic problem with nonlinear constraints. This can avoid the heterogeneity of the DP distribution at the mesh-size level and impose compact stencils with a fixed degree of freedom for derivative evaluations. A fundamental formulation for arrangements of DPs and an application to unsteady Stokes flows are presented in this paper. Numerical tests were performed for the distribution of DPs and flow problems in co-axial and eccentric circular channels. The proposed MCD approach achieved a reasonable distribution of DPs independently of the spatial resolution with a few iterations in pre-processing. Additionally, solutions using the obtained DP distributions in Stokes flow problems were in good agreement with theoretical and reference solutions. The results also confirmed that the numerical accuracies of velocity and pressure achieved the expected convergence order, even when compact stencils were used.

The gravity-driven migration of a drop in an unsteady Stokes flow

The motion of a spherical droplet through a fluid driven by gravity is revisited theoretically. Analytical expressions for the translational velocity of the droplet as well as its internal and external flows coupled at their interface and described by the unsteady Stokes equations are derived combining two-point Padé and Padé-type approximants with the Laplace transform. These expressions, involving scaled complementary error functions, are fast and globally convergent to the exact solutions of the equations. The migration of the droplet depends on three non-dimensional parameters, two of which determine the flow patterns inside and outside, and the last is related to the timescale. The analytical solutions reveal that there is a diffusive shear layer of thickness O(t) on either side of the interface in the early stage. Several plots are made to compare migration velocities of the droplet for different parameter sets and changes in the flow field with time.

A new approach to the complex Helmholtz equation with applications to diffusion wave fields, impedance spectroscopy and unsteady Stokes flow

Abstract A new transform pair representing solutions to the complex Helmholtz equation in a convex 2D polygon is derived using the theory of Bessel’s functions and Green’s second identity. The derivation is a direct extension of that given by Crowdy (2015, A transform method for Laplace’s equation in multiply connected circular domains. IMA J. Appl. Math., 80, 1902–1931) for ‘Fourier–Mellin transform’ pairs associated with Laplace’s equation in various domain geometries. It is shown how the new transform pair fits into the collection of ideas known as the Fokas transform where the key step in solving any given boundary value problem is the analysis of a global relation. Here we contextualize those global relations from the point of view of ‘reciprocal theorems’, which are familiar tools in the study of the effective properties of physical systems. A survey of the many uses of this new transform approach to the complex Helmholtz equation in applications is given. This includes calculation of effective impedance in electrochemical impedance spectroscopy and in other spectroscopy methods in diffusion wave field theory, application to the 3$\omega $ method for measuring thermal conductivity and to unsteady Stokes flow. A theoretical connection between this analysis of the global relations and Lorentz reciprocity in mathematical physics is also pointed out.

An immersed boundary projection method for incompressible interface simulations in 3D flows

In this paper, we extend our previous immersed boundary (IB) projection method for 2D inextensible vesicle in unsteady Stokes flow (Ong and Lai, 2020 [23]) to 3D incompressible interface as a prototype of vesicles. In spite of similar numerical algorithm to the 2D case, the present 3D numerical implementation is far from straightforward. An incompressible interface immersed in Newtonian fluid must ensure both conservative constraints associated with local fluid volume and local interfacial area. To accomplish this, we triangulate the incompressible interface, whereby the unknown elastic tension is defined on the triangles, while the associated tension force term is defined at vertices (Lagrangian markers). Consistent approximations of surface differential operators on a triangulated interface are developed accordingly. An extra scalar field is introduced in the IB projection approach so that the fluid pressure and interfacial elastic tension are efficiently solved in a divide-and-conquer manner. As a result, the local interfacial and volumetric incompressibilities can be ideally satisfied simultaneously. A series of numerical tests on the present scheme is performed to verify the robustness and applicability of the method. We first carry out the convergence study of the solution variables defined on the fluid and interface. We then study the shear-induced deformations of incompressible interface for various initial configurations. At last, the bending force and the penalty force for improved volume conservation are further incorporated into our method to simulate an oblate vesicle dynamics in quiescent flow.

Unsteady Stokes Flow of a Viscous Fluid in a Parallel Plate Channel through a Porous Medium

Unsteady Stokes Flow of a Viscous Fluid in a Parallel Plate Channel through a Porous Medium

Unsteady motion of a perfectly slipping sphere.

An integral expression for the translational velocity of a perfectly slipping spherical particle under a time-dependent applied force in unsteady Stokes flow is derived. For example, when the ratio of particle density to fluid density is small, our analysis pertains to an inviscid bubble in a viscous fluid. We determine an explicit form of the particle velocity under an impulsive force, wherefrom the velocity autocorrelation function and mean-squared displacement of a perfectly slipping sphere undergoing Brownian motion are obtained. The above results are contrasted against the time-dependent diffusion of a rigid sphere with no hydrodynamic slip. Finally, the thermal force power spectral density is analytically calculated for a diffusing sphere with arbitrary slip length. We suggest this quantity to be suitable to infer slip length from the measurement of nondiffusive Brownian motion.

Analysis of unsteady two dimensional MHD flow between two parallel plates under angular velocity with one plate moving uniformly and the other plate at rest and uniform suction at the stationary plate

In this paper we discuss the study of Unsteady magnetohydrodynamic stokes flow between two parallel plates under angular velocity. Differential equation and transform techniques are used to attain the solution of the Mathematical equations and discussed the MHD Stokes flow through two parallel plates with one in uniform motion and the other plate at rest and uniform suction at the stationary plate. The velocity profile for the flow of the fluid and pressure gradient are discussed for different values of Hartmann number with angular velocity M 1. And the effects of velocity potential are investigated and discussed graphically.

Unsteady Stokes Flow through a Porous Pipe with Periodic Suction and Injection with Slip Conditions

The problem of unsteady Stokes flow of certain Newtonian fluids in a circular pipe of uniform cross section is discussed. The pipe is uniformly porous. The unsteady Navier-Stokes equations for the system in cylindrical polar coordinates have been solved analytically to obtain a complete description of the flow. The solution of the flow equations subject to the slip boundary conditions leads to the detailed expressions for axial and radial components of velocity and the pressure distribution depending on position coordinates and time. As a special case we have presented the situation when no-slip boundary conditions are implemented. The velocity profile is analyzed for different values of the flow parameters like Womersley number, slip length and time.

Flow of Micropolar Fluid Between Two Parallel Plates with Different Periodic Suction and Injection

The unsteady stokes flow of incompressible micropolar fluid between two porous plates is considered. The lower plate is subjected to periodic suction and different periodic injection is applied at the upper plate. Stream function for the flow is obtained and the variation of velocity function f  & g with  is shown graphically. The effects of the dimensionless parameters p, frequency parameter pt , micropolarity parameter pl and the microrotation parameter pj on the velocity functions f  and microrotation velocity function g are discussed and shown through the graphs.

The unsteady drag of a translating spherical drop with a viscoelastic membrane at small Reynolds number

This manuscript quantifies the hydrodynamic drag of a spherical droplet translating in unsteady Stokes flow when the droplet has a thin, complex membrane. All forces (e.g., Stokes drag, Basset forces, unsteady memory forces, etc.) have the same functional form as the drag of a clean spherical droplet, as long as one replaces the interior viscosity with an effective (frequency-dependent) one that depends on the viscoelastic moduli of the membrane. The shear elastic moduli of the membrane do not modify the unsteady drag of the droplet – only the dilatational modes matter. We demonstrate how this result can be obtained using simple symmetry/scaling arguments. The results are written for any linearly viscoelastic membrane, but we in particular examine the cases of a purely viscous membrane, purely elastic membrane, one-mode Maxwell membrane, and one-mode Kelvin–Voigt membrane. For the latter two models, we quantify how the membrane relaxation time alters the time-dependent motion of the droplet.

Effects of surfactant transport on electrodeformation of a viscous drop

In this work we quantify the effects of surfactant transport on the deformation of a viscous drop under a DC electric field. We study how convective and diffusive transport of surfactants at drop surfaces influence the equilibrium and dynamic deformation of a leaky dielectric drop and a conducting drop. Focusing on the prolate drop shape (elongates along the electric field), we show the differences in equilibrium deformation and flow circulation between a leaky dielectric drop and a conducting drop. We quantify the drop electrodeformation via its dependence on the interior flow circulation and the dominant surfactant transport regime (characterized by the surface Péclet number Pe_{s}). For a leaky dielectric drop with dominant surfactant diffusion (Pe_{s}≪1), equator-to-pole (pole-to-equator) circulation yields smaller (larger) equilibrium deformation with increasing surfactant coverage, compared to a clean drop. However, when convection dominates (Pe_{s}≫1), the equilibrium drop deformation increases (decreases) with larger surfactant coverage for equator-to-pole (pole-to-equator) circulation. Larger equilibrium drop deformation is found for a leaky dielectric drop than a conducting drop when the interior flow is from equator to pole. For an interior flow from pole to equator, we identify cases where larger deformation is found for a conducting interior fluid. Finally, we study the effect of the surfactant transport on the dynamic evolution of drop shape. We found the drop undergoes an overshoot in the early deformation phase, before settling to its equilibrium shape-similar to the overshoot observed for unsteady Stokes flow.

The singularity method in unsteady Stokes flow: hydrodynamic force and torque around a sphere in time-dependent flows

The equations for the hydrodynamic force and torque acting on a sphere in unsteady Stokes equations under different flow conditions are solved analytically by means of the singularity method. This analytical technique is based on the combination of suitable singularity solutions (also called fundamental solutions) such as primary Stokeslets, potential dipoles, or higher-order singularities, to construct the flow field. The different flows considered here include four examples: (1) a rotating sphere in a viscous flow, (2) a stationary sphere in a time-dependent shear flow, (3) a sphere with free rotation in a simple shear flow, as well as (4) a stationary sphere in a time-dependent axisymmetric parabolic flow. Our paradigm is to derive the fundamental solutions in unsteady Stokes flows and to express the solutions as a convolution integral in time using the time–space fundamental solutions. Next the Laplace transform is used to determine the strength of the distributed singularities that induce the velocity field around a stationary or rotating sphere. Then we use the computed strength of the singularities to derive hydrodynamic force and torque. In particular, for the problem of a stationary sphere in unsteady axisymmetric parabolic flow, our solution for the time-dependent force acting on the sphere consists of five force components – the well-known quasi-steady Stokes drag, the added mass term, the Basset historic (memory) force, and two additional memory forces. The first additional memory force due to the rate change of velocity, we find, is similar to the result obtained by Lawrence & Weinbaum (J. Fluid Mech., vol. 171, 1986, pp. 209–218) for the ostensibly unrelated setting of a slightly deformed translating spheroid. The second additional memory force comes from the effect of the rate change of acceleration and is found for the first time in this study to the best of our knowledge.

An Approximating Control Design for Optimal Mixing by Stokes Flows

We consider an approximating control design for optimal mixing of a non-dissipative scalar field $$\theta $$ in an unsteady Stokes flow. The objective of our approach is to achieve optimal mixing at a given final time $$T>0$$ , via the active control of the flow velocity v through boundary inputs. Due to zero diffusivity of the scalar field $$\theta $$ , establishing the well-posedness of its Gâteaux derivative requires $$\sup _{t\in [0,T]}\Vert \nabla \theta \Vert _{L^2}<\infty $$ , which in turn demands the flow velocity field to satisfy the condition $$\int ^{T}_{0}\Vert \nabla v\Vert _{L^{\infty }(\Omega )}\, dt<\infty $$ . This condition results in the need to penalize the time derivative of the boundary control in the cost functional. Consequently, the optimality system becomes difficult to solve (Hu in Appl Math Optim 78(1):201–217, 2018). Our current approximating approach provides a more transparent optimality system, with the set of admissible controls square integrable in space-time. This is achieved by first introducing a small diffusivity to the scalar equation and then establishing a rigorous analysis of convergence of the approximating control problem to the original one as the diffusivity approaches to zero. Uniqueness of the optimal solution is obtained for the two dimensional case.