Stokes Fluid Research Articles

Study of a variant of Stokes’ first and second problems for fluids with pressure dependent viscosities

In this paper we extend the seminal work of Stokes concerning the flow due to a suddenly accelerated plate and an oscillating plate for the Navier–Stokes fluid, to a fluid with pressure dependent viscosity. The viscosity of many fluids varies significantly with pressure, a fact recognized by Stokes, and Barus in fact conducted experiments that showed that the variation of the viscosity with pressure was exponential. Given such a tremendous variation, it would be interesting to see how this change in viscosity affects the nature of the solution to these problems. We find that the velocity field, and hence the structure of the vorticity and the shear stress at the walls for fluids with pressure dependent viscosity are markedly different from those for the classical Navier–Stokes fluid.

Comparison of Unsteady Reynolds Averaged Navier–Stokes and Large Eddy Simulation Computational Fluid Dynamics Methodologies for Air Swirl Fuel Injectors

It is well documented that various large-scale quasiperiodic flow structures, such as a precessing vortex core (PVC) and multiple vortex helical instabilities, are present in the swirling flows typical of air swirl fuel injectors. Prediction of these phenomena requires time-resolved computational methods. The focus of the present work was to compare the performance and cost implications of two computational fluid dynamics (CFD) methodologies—unsteady Reynolds averaged Navier–Stokes (URANS) using a k-ε model and large eddy simulation (LES) for such flows. The test case was a single stream radial swirler geometry, which has been the subject of extensive experimental investigation. Both approaches captured the gross (time-mean) features of strongly swirling confined flows in reasonable agreement with experiment. The temporal dynamics of the quadruple vortex pattern emanating from within the swirler and observed experimentally were successfully predicted by LES, but not by URANS. Spectral analysis of two flow configurations (with and without a central jet) revealed various coherent frequencies embedded within the broadband turbulent frequency range. LES reproduced these characteristics, in excellent agreement with experimental data, whereas URANS predicted the presence of coherent motions but at incorrect amplitudes and frequencies. For the no-jet case, LES-predicted spectral data indicated the occurrence of a PVC, which was also observed experimentally for this flow condition; the URANS solution failed to reproduce this measured trend. On the evidence of this study, although k-ε based URANS offers considerable computational savings, its inability to capture the temporal characteristics of the flows studied here sufficiently accurately suggests that only LES-based CFD, which captures the stochastic nature of the turbulence much more faithfully, is to be recommended for fuel injector flows.

A combinatorial approach to microfluidic mixing

A new computational approach to the modeling and design of efficient microfluidic mixers is demonstrated. The mixers created provide far more rapid mixing than previous designs. A set of mixer components is created and mapped using a traditional Navier–Stokes fluid solver. The maps are used to quickly model the effect each component has on the lateral distribution of species in the channel. For a mixer of a given length, all the possible combinations of components can be evaluated, and the best mixer for a given metric can be found. Although the mixers presented in this study are short (length-to-width ratios below 8), they show degrees of mixing comparable to much longer mixers found in the literature.

Multiphysics model for blood flow and drug transport with application to patient-specific coronary artery flow

A multiphysics mathematical model to simulate drug delivery in idealized and patient-specific coronary arteries is presented. Blood is modeled as an incompressible Navier–Stokes fluid, the arterial wall as a linear poroelastic medium, and the drug transport is described by a scalar advection-diffusion equation. The drug compound is released into the bloodstream, carried by the flow, deposited onto the endothelium, penetrates into the wall, and is transported within the arterial wall. NURBS-based isogeometric analysis is employed to describe the geometry and discretize the fluid-solid interaction equations.

Flows of Incompressible Fluids subject to Navier’s slip on the boundary

We establish closed form analytical solutions for the flows of a generalized fluid of complexity two, which includes the Navier–Stokes fluid, power-law fluid and the second-grade fluid as special subclasses, in special geometries under the assumption that the flows meet Navier slip conditions at the boundary. The boundary conditions allow the extremes of no-slipping and no-sticking. We also allow for different boundary conditions at different parts of the boundary, which leads to interesting consequences with regard to the solutions. The dependence of the form of the solutions on the boundary conditions is discussed in some detail.

Unique continuation property near a corner and its fluid-structure controllability consequences

We study a non standard unique continuation property for the biharmonic spectral prob- lem Δ 2 w = −λΔw in a 2D corner with homogeneous Dirichlet boundary conditions and a supplemen- tary third order boundary condition on one side of the corner. We prove that if the corner has an angle 0 <θ 0 < 2π, θ0 � π and θ0 � π/2, a unique continuation property holds. Approximate controllability of a 2-D linear fluid-structure problem follows from this property, with a control acting on the elastic side of a corner in a domain containing a Stokes fluid. The proof of the main result is based in a power series expansion of the eigenfunctions near the corner, the resolution of a coupled infinite set of finite dimensional linear systems, and a result of Kozlov, Kondratiev and Mazya, concerning the absence of strong zeros for the biharmonic operator (Math. USSR Izvestiya 34 (1990) 337-353). We also show how the same methodology used here can be adapted to exclude domains with corners to have a local version of the Schiffer property for the Laplace operator. Mathematics Subject Classification. 35B60, 35B37.

A two‐dimensional prototype multi‐physics model of the right ventricle of the heart

AbstractThe heart operates as a delicate function of the interactions among its electro‐chemical, structural and fluid components. The objective of this paper is to describe the computational modelling of the interaction between simplified forms of the electro‐chemical, structural and fluid behaviour. It is because the objective here is to evaluate the interactions among the phenomena that the specific models of the electro‐chemical, fluid and structural behaviour are kept as simple as possible. The electro‐chemical model is drawn from that of Clayton and Holden, while the blood is assumed to be a conventional slightly compressible Navier–Stokes fluid and the heart wall is a simple elastic structure modelled via the static equations. The model is implemented within a multi‐physics finite volume solver environment using an unstructured mesh approach, which enables the implementation of models for each of the specific phenomenon and their interactions. The model is two dimensional, but shows how the electric field drives a cross section of the wall so that it pumps the blood in and out of the right ventricle geometry. Copyright © 2008 John Wiley &amp; Sons, Ltd.

An improved penalty method for power-law Stokes problems

For the numerical approximation of fluid flow phenomena, it is often highly desirable to decouple the equations defining conservation of momentum and conservation of mass by using a penalty function method. The current penalty function methods for power-law Stokes fluids converge at a sublinear rate with respect to the penalty parameter. In this article, we show theoretically and numerically that a linear penalty function approximation to a power-law Stokes problem yields a higher-order accuracy over the known nonlinear penalty method. Theoretically, finite element approximation of the linear penalty function method is shown to satisfy an improved order of approximation with respect to the penalty parameter. The numerical experiments presented in the paper support the theoretical results and satisfy a linear order of approximation.

On MHD transient flow of a Maxwell fluid in a porous medium and rotating frame

Abstract Analytic solution for unsteady magnetohydrodynamic (MHD) flow is constructed in a rotating non-Newtonian fluid through a porous medium. Constitutive equations for a Maxwell fluid have been taken into consideration. The hydromagnetic flow in the uniformly rotating fluid is generated by a suddenly moved infinite plate in its own plane. Analytic solution of the governing flow problem is obtained by means of the Fourier sine transform. It is shown that the obtained solution satisfies both the associate partial differential equation and the initial and boundary conditions. The solution for a Navier–Stokes fluid is recovered if λ → 0 . The steady state solution is also obtained for t → ∞ .

Rotational dynamics of a superhelix towed in a Stokes fluid

Motivated by the intriguing motility of spirochetes (helically shaped bacteria that screw through viscous fluids due to the action of internal periplasmic flagella), we examine the fundamental fluid dynamics of superhelices translating and rotating in a Stokes fluid. A superhelical structure may be thought of as a helix whose axial centerline is not straight, but also a helix. We examine the particular case in which these two superimposed helices have different handedness, and employ a combination of experimental, analytic, and computational methods to determine the rotational velocity of superhelical bodies being towed through a very viscous fluid. We find that the direction and rate of the rotation of the body is a result of competition between the two superimposed helices; for small axial helix amplitude, the body dynamics is controlled by the short-pitched helix, while there is a crossover at larger amplitude to control by the axial helix. We find far better, and excellent, agreement of our experimental results with numerical computations based upon the method of Regularized Stokeslets than upon the predictions of classical resistive force theory.

Solutions of slow Brinkman flows using the method of fundamental solutions

AbstractThis paper develops the method of fundamental solutions (MFS) as a meshless numerical method to obtain solutions of two‐ and three‐dimensional slow Brinkman‐extended Darcy's flows. The solutions of the steady Brinkman equations are obtained by utilizing the boundary collocation method as well as the expansion of the fundamental solutions, which are derived by using the Hörmander operator decomposition technique. All the velocities, their partial derivatives, the pressure, and the stresses corresponding to the fundamental solutions are addressed explicitly in tensor forms. Two‐ and three‐dimensional Brinkman problems with Dirchlet and Robin boundary conditions are carried out to validate the proposed numerical schemes. Then, the method is applied to solve a peanut‐shaped problem and a joint flow of Stokes and Brinkman fluids. In the spirits of MFS, the proposed numerical scheme is free from singularities and numerical integrations and it also does not require any domain discretization. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Optimal control of mixing in Stokes fluid flows

Motivated by the problem of microfluidic mixing, optimal control of advective mixing in Stokes fluid flows is considered. The velocity field is assumed to be induced by a finite set of spatially distributed force fields that can be modulated arbitrarily with time, and a passive material is advected by the flow. To quantify the degree of mixedness of a density field, we use a Sobolev space norm of negative index. We frame a finite-time optimal control problem for which we aim to find the modulation that achieves the best mixing for a fixed value of the action (the time integral of the kinetic energy of the fluid body) per unit mass. We derive the first-order necessary conditions for optimality that can be expressed as a two-point boundary value problem (TPBVP) and discuss some elementary properties that the optimal controls must satisfy. A conjugate gradient descent method is used to solve the optimal control problem and we present numerical results for two problems involving arrays of vortices. A comparison of the mixing performance shows that optimal aperiodic inputs give better results than sinusoidal inputs with the same energy.

On the unsteady Poiseuille flow in a pipe

We show that in an unsteady Poiseuille flow of a Navier–Stokes fluid in an infinite straight pipe of constant cross-section, σ, the flow rate, F(t), and the axial pressure drop, q(t), are related, at each time t, by a linear Volterra integral equation of the second type, where the kernel depends only upon t and σ. One significant consequence of this result is that it allows us to prove that the inverse parabolic problem of finding a Poiseuille flow corresponding to a given F(t) is equivalent to the resolution of the classical initial-boundary value problem for the heat equation.

Generalized analytic functions in 3d stokes flows

A class of generalized analytic functions, defined by a special case of the Carleman system that arises in three-dimensional asymmetric problems of hydrodynamics of Stokes flows, stationary electromagnetic fields in conductive materials, etc., has been considered. Hilbert formulae, establishing relationships between the real and imaginary parts of a generalized analytic function from this class, have been derived for the domains exterior to the contour of bi-spheres and torus in the meridional cross-section plane. This special case of the Carleman system has been reduced to an infinite three-diagonal system of algebraic equations with respect to the coefficients in series representations of the real and imaginary parts. For torus, the system has been solved by means of the Fourier transform, while for bi-spheres, it has been solved by an algebraic method. As examples, analytical expressions for the pressure in the problems of the steady axially symmetric motion of rigid bi-spheres and torus in a Stokes fluid have been derived based on the corresponding Hilbert formulae.

Local Controllability for a “Swimming” Model

We study the local controllability of a mathematical model of an abstract object which “swims” in the two-dimensional (2D) nonstationary Stokes fluid. We assume that this object consists of finitely many subsequently connected small sets (“thick points”), each of which can act upon any of the adjacent sets in a rotation fashion with the purpose of generating its fish- or snake-like motion. We regard the magnitudes of the respective rotation forces, entering the system’s equations as coefficients, as multiplicative (or bilinear) controls. The structural integrity of the object is maintained by the elastic forces acting between the aforementioned adjacent sets according to Hooke’s law. Models like this are of a interest in biology and engineering applications dealing with propulsion systems in fluids.

Efficient Numerical Solution of Cahn–Hilliard–Navier–Stokes Fluids in 2D

A finite element discretization of the variable density Cahn-Hilliard-Navier-Stokes system is presented. This system is then decoupled with the Cahn-Hilliard part solved via the nonlinear multigrid method proposed by the authors in [J. Comput. Phys., 212 (2006), pp. 288-304] and the Navier-Stokes part via a multigrid preconditioned GMRES scheme. This new solver is examined numerically and is shown to be efficient with respect to mesh refinement and robust with respect to the problem parameters.

Hilbert formulas for 𝑟-analytic functions and the Stokes flow about a biconvex lens

The so-called r r -analytic functions are a subclass of p p -analytic functions and are defined by the generalized Cauchy-Riemann system with p ( r , z ) = r p(r,z)=r . In the system of toroidal coordinates, the real and imaginary parts of an r r -analytic function are represented by Mehler-Fock integrals with densities, which are assumed to be meromorphic functions. Hilbert formulas, establishing relationships between those functions, are derived for the domain exterior to the contour of a biconvex lens in the meridional cross-section plane. The derivation extends the framework of the theory of Riemann boundary-value problems, suggested in our previous work, to solving the three-contour problem for the case of meromorphic functions with a finite number of simple poles. For numerical calculations, Mehler-Fock integrals with Hilbert formulas reduce to the form of regular integrals. The 3D problem of the axially symmetric steady motion of a rigid biconvex lens-shaped body in a Stokes fluid is solved, and the Hilbert formula for the real part of an r r -analytic function is used to express the pressure in the fluid via the vorticity analytically. As an illustration, streamlines and isobars about the body, the vorticity and pressure at the contour of the body and the drag force exerted on the body by the fluid are calculated.

Steady-state solutions for some simple flows of generalized Burgers fluids

The steady-state solutions for three types of unsteady oscillating flows of generalized Burgers fluids are determined by means of the Fourier sine transforms. These solutions are also presented in equivalent forms in terms of elementary functions exp, sine, cosine, hyperbolic sine and hyperbolic cosine. The similar solutions for Burgers, Oldroyd-B, Maxwell, Second grade and Navier–Stokes fluids can be also obtained as limiting cases of our solutions.

Starting solutions for the motion of a second grade fluid due to longitudinal and torsional oscillations of a circular cylinder

The starting solutions corresponding to the motions of a second grade fluid due to the longitudinal and torsional oscillations of a circular cylinder are determined by means of the finite Hankel transforms. These solutions, presented as sum of the steady-state and transient solutions tend to those for a Navier–Stokes fluid as a limiting case. The steady-state solutions are also presented in simpler forms, in terms of the modified Bessel functions of the first and second kind.

Two-dimensional local null controllability of a rigid structure in a Navier–Stokes fluid

We consider the two-dimensional motion of a rigid structure immersed in an incompressible fluid governed by Navier–Stokes equations. The control force acts on a fixed subset of the fluid domain. We prove that our system is null controllable; that is, for small initial data, the system can be driven at rest and the structure can be driven to the origin at a given T > 0 . The result holds for a structure symmetric with respect to the center of mass and for initial conditions satisfying strong compatibility conditions. To cite this article: M. Boulakia, A. Osses, C. R. Acad. Sci. Paris, Ser. I 343 (2006).