In this paper, we consider the flow of a nematic liquid crystal in the domain exterior to a small spherical particle. We work within the framework of the $\unicode{x1D64C}$ -tensor model, taking into account the orientational elasticity of the medium. Under a suitable regime of physical parameters, the governing equations can be reduced to a system of linear partial differential equations. Our focus is on precise far-field asymptotics of the flow velocity with an emphasis on its anisotropic behaviour. We are able to analytically characterize the flow pattern and compare it with that of the classical isotropic Stokes flow. The expression for velocity away from the particle can be computed numerically or symbolically.

Stability of 3D perturbations to 2D Navier–Stokes flows with vertical dissipation

For the first time, an analytical solution has been derived for Stokes flow through a conical diffuser under the condition of partial slip. Recurrent relations are obtained that allow determination of the velocity, pressure and stream function for a certain slip length λ . The solution is analysed in the first order of decomposition with respect to a small dimensionless parameter ${\lambda }/{r}$ . It is shown that the sliding of the liquid over the surface of the cone leads to a vorticity of the flow. At zero slip length, we obtain the well-known solution to the problem of a diffuser with a no-slip boundary condition corresponding to strictly radial streamlines. To solve that problem, we use an alternative form of the general solution of the linearised, stationary, axisymmetric Navier–Stokes equations for an incompressible fluid in spherical coordinates. A previously published solution to this problem, dating back to the paper by Sampson (1891 Phil. Trans. R. Soc. A , vol. 182 , pp. 449–518), is given in terms of a stream function that leads to formulae that are difficult to apply in practice. By contrast, the new general solution is derived in the vector potential representation and is simpler to apply.

The zero viscosity limit of stochastic Navier–Stokes flows

Wall slip sensitivity and non-sphericity and orientation effects are investigated for a moving no-slip solid body immersed in a fluid above a plane slip wall with a Navier slip. The wall–particle interactions are examined for the body motion in a quiescent fluid (resistance problem) or when freely suspended in a prescribed ‘linear’ or quadratic ambient shear flow. This is achieved, assuming Stokes flows, by using a boundary method which reduces the task to the treatment of six boundary-integral equations on the body surface. For a wall slip length $\lambda$ small compared with the wall–particle gap $d$ a ‘recipe’ connecting, at $O((\lambda /d)^2),$ the results for the slip wall and another no-slip wall with gap $d+\lambda$ is established. A numerical analysis is performed for a family of inclined non-spheroidal ellipsoids, having the volume of a sphere with radius $a,$ to quantity the particle behaviour sensitivity to the normalised wall slip length $\overline {\lambda }=\lambda /a,$ the normalised wall–particle gap ${\overline {d}}=d/a$ and the particle shape and orientation (here one angle $\beta ).$ The friction coefficients for the resistance problem exhibit quite different behaviours versus the particle shape and $({\overline {d}}, \overline {\lambda },\beta ).$ Some coefficients increase in magnitude with the wall slip. The migration of the freely suspended particle can also strongly depend on $({\overline {d}}, \overline {\lambda },\beta )$ and in a non-trivial way. For sufficiently small $\overline {d}$ a non-spherical particle can move faster than in the absence of a wall for a large enough wall slip for the ambient ‘linear’ shear flow and whatever the wall slip for the ambient quadratic shear flow.

This paper is motivated by the “transfer of regularity” phenomenon for the incompressible Navier–Stokes equations (NSE) in dimension n ≥ 3 n \geq 3 ; that is, the strong solutions of NSE on R n \mathbb {R}^n can be nicely approximated by those on sufficiently large domains Ω ⊂ R n \Omega \subset \mathbb {R}^n under the no-slip boundary condition. Based on the spacetime decay estimates of mild solutions of NSE established by Miyakawa [On space-time decay properties of nonstationary incompressible Navier-Stokes flows in R n \mathbf {R}^n , Funkcial. Ekvac. 43 (2000), no. 3, 541–557], Schonbek [ L 2 L^2 decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 88 (1985), no. 3, 209–222], and others, we obtain quantitative estimates on higher-order derivatives of velocity and pressure for the incompressible Navier–Stokes flow on large domains under certain additional smallness assumptions of the Stokes system and/or the initial velocity, thus complementing the results obtained by Robinson [Using periodic boundary conditions to approximate the Navier-Stokes equations on R 3 \mathbb {R}^3 and the transfer of regularity, Nonlinearity 34 (2021), no. 11, 7683–7704] and Ożánski [Quantitative transfer of regularity of the incompressible Navier-Stokes equations from R 3 \mathbb {R}^3 to the case of a bounded domain, J. Math. Fluid Mech. 23 (2021), no. 4, Paper No. 98, 14].

We consider the homogenization of a coupled Stokes flow and advection–reaction–diffusion problem in a perforated domain with an evolving microstructure of size [Formula: see text]. Reactions at the boundaries of the microscopic interfaces lead to the formation of a solid layer, which is assumed to be radially symmetric with a variable, a priori unknown thickness. This results in a growth or shrinkage of the solid phase and, thus, the domain evolution is not known a priori but induced by the advection–reaction–diffusion process. The achievements of this work are the existence and uniqueness of a weak microscopic solution and the rigorous derivation of an effective model for [Formula: see text], based on [Formula: see text]-uniform a priori estimates. As a result of the passage to the limit, the processes on the macroscale are described by an advection–reaction–diffusion problem coupled to Darcy’s equation with effective coefficients (porosity, diffusivity and permeability) depending on local cell problems. These local problems are formulated on cells that depend on the macroscopic position and evolve in time. In particular, the evolution of these cells depends on the macroscopic concentration. Thus, the cell problems (respectively the effective coefficients) are coupled to the macroscopic unknowns and vice versa, leading to a strongly coupled micro–macro model. Homogenization results have been obtained recently for the case of reactive–diffusive transport coupled with microscopic domain evolution, but in the absence of advective transport. We extend these models by including the advective transport, which is driven by the Stokes equations in the a priori unknown evolving pore domain.

Flows are essential to transport resources over large distances. As soon as diffusion becomes time-limiting, flows are needed. Flows are key for the function of multiple human organs, from the blood vasculature to the lungs, the digestive tract, the lymphatic system, and many more. While physics governs the flow dynamics, biology’s response to flows governs the flow network architecture. We start with the fluid physics of Stokes flow, the prerequisite to describe the flows in biological flow networks. Then we explore how the network adaptation dynamics of biological flow networks reorganize network architecture to minimize flow dissipation or homogenize transport, storing memories of past flows along the way.

Stokes flow describes the motion of a Newtonian, incompressible fluid in regimes where inertial effects are negligible compared to viscous forces. In the case of axisymmetric flow and by employing a stream function ψ, the governing equation reduces to E4ψ=0, where E2 is a second order elliptic partial differential operator and E4=E2○E2. In this work, we derive the general solution of the equation E4ψ=0 in the parabolic coordinate system, which is given as series expansions of specific combinations of mixed-order Bessel and modified Bessel functions of first and second kinds or as a polynomial. This analytical framework is applied to study the Stokes flow around a rigid paraboloid solving a boundary value problem, accordingly. A truncated error analysis is conducted, proving that using at least eight terms of the obtained series expansion, the absolute error is less than 10−5. Sample streamlines are depicted with respect to the order of truncation. The obtained stream function expansion allows for the computation of key hydrodynamic quantities, such as the velocity and the pressure fields, offering insights into the behavior of the particular flow. It may also serve as a foundation for further investigations of problems in a wide range of areas, from chemical engineering to biomedicine.

We study the two-dimensional steady-state creeping flow in a converging–diverging channel gap formed by two immobile rollers of identical radius. For this purpose, we analyse the Stokes equation in the streamfunction formulation, i.e. the biharmonic equation, which has homogeneous and particular solutions in the roll-adapted bipolar coordinate system. The analysis of existing works, investigating the particular solutions allowing arbitrary velocities at the two rollers, is extended by an investigation of homogeneous solutions. These can be reduced to an algebraic eigenvalue problem, whereby the associated discrete but infinite eigenvalue spectrum generates symmetric and asymmetric eigenfunctions with respect to the centre line between the rollers. These represent nested viscous vortex structures, which form a counter-rotating chain of vortices for the smallest unsymmetrical eigenvalue. With increasing eigenvalue, increasingly complex finger-like structures with more and more layered vortices are formed, which continuously form more free stagnation points. In the symmetrical case, all structures are mirror-symmetrical to the centre line and with increasing eigenvalues, finger-like nested vortex structures are also formed. As the gap height in the pressure gap decreases, the vortex density increases, i.e. the number of vortices per unit length increases, or the length scales of the vortices decrease. At the same time the rate of decay between subsequent vortices increases and reaches and asymptotic limit as the gap vanishes.

Two-dimensional fluid–structure interaction (FSI) simulations were conducted to study the influence of the supraglottal tract on intraglottal pressure during vocal fold closing in human phonation. The FSI framework couples a three-mass vocal fold model, an incompressible Navier–Stokes flow solver, and a linear perturbed compressible acoustic solver. Four configurations were simulated under varying subglottal pressures, combining divergent and nearly straight vocal fold closing patterns with and without a supraglottal tract. Comparison of straight closing configurations with and without a supraglottal tract revealed the isolated effect of supraglottal inertance, while the analysis of divergent closing configurations showed the combined effects of supraglottal inertance and intraglottal flow separation. Parameters, including the glottal opening, glottal angle, medial surface wall pressure, which directly reflects how intraglottal pressure is converted to the driving force of the vocal fold motion, and intraglottal flow dynamics, were analyzed. The results show that the supraglottal tract influences intraglottal pressure, as well as the medial surface wall pressure, through two primary mechanisms. First, it reduces medial surface wall pressure through inertance effects during flow deceleration. Second, in the divergent closing configurations, it further decreases medial surface wall pressure by intensifying the flow separation effect. This enhanced flow separation effect is because the supraglottal tract increases peak jet velocity during glottis closing, which strengthens the shear layer and enhances flow circulation after flow separation. Consequently, it leads to more negative glottal pressure, particularly under high subglottal pressures.

Parabolic-scalings on large-time behavior of the incompressible Navier–Stokes flow

A Study on the Impact of Magnetic Field Inclination and Concentration on Unsteady MHD Stokes Flow of a Dusty Fluid through Moving Channel of Riga Plates

ABSTRACTWall shear stress (WSS) is a crucial hemodynamic quantity extensively studied in cardiovascular research, yet its numerical computation is not straightforward. This work compares WSS results obtained from two different finite element discretizations, quantifies the differences between continuous and discontinuous stresses, and introduces a modified variationally consistent method for WSS evaluation through the formulation of a boundary‐flux problem. Two benchmark problems are considered: a 2D Stokes flow on a unit square and a 3D Poiseuille flow through a cylindrical pipe. These are followed by investigations of steady‐state Navier–Stokes flow in two image‐based, patient‐specific aneurysms. The study focuses on P1/P1 stabilized and Taylor–Hood P2/P1 mixed finite elements for velocity and pressure. WSS is computed using either the proposed boundary‐flux method or as a projection of tangential traction onto first order Lagrange (P1), discontinuous Galerkin first order (DG‐1), or discontinuous Galerkin zero order (DG‐0) space. For the P1/P1 stabilized element, the boundary‐flux and P1 projection methods yielded equivalent results. With the P2/P1 element, the boundary‐flux evaluation demonstrated faster convergence in the Poiseuille flow example but showed increased sensitivity to pressure field inaccuracies in image‐based geometries compared to the projection method. Furthermore, a paradoxical degradation in WSS accuracy was observed when combining the P2/P1 element with fine boundary‐layer meshes on a cylindrical geometry, an effect attributed to inherent geometric approximation errors. In aneurysm geometries, the P2/P1 element exhibited superior robustness to mesh size when evaluating average WSS and low shear area (LSA), outperforming the P1/P1 stabilized element. Projecting discontinuous finite element functions into continuous spaces can introduce artifacts, such as the Gibbs phenomenon. Consequently, it is crucial to carefully select the finite element space for boundary stress calculations, not only in applications involving WSS computations for aneurysms.

This paper introduces a novel framework for designing optimal airfoils for crosswind kites which are tethered flying systems used to harness high-altitude wind energy. The improved geometric parameter method, a state-of-the-art airfoil design approach, is employed here for the first time in the context of airborne wind energy airfoil design. The nondominated sorting genetic algorithm is used as the optimization method, and XFOIL is adopted to obtain aerodynamic lift and drag coefficients of the airfoils. Pareto-optimal fronts and the corresponding optimal airfoil profiles at various maximum thickness ratios are obtained for a baseline system that neglects three-dimensional flow effects and tether drag. For the first time in the literature, the effects of induced drag due to finite aspect ratio kites on the optimal airfoils are examined. Additionally, the effects of including the tether drag on the optimal solutions are explored. It is found that when the induced drag is included optimal airfoils feature a cusped trailing edge. On the other hand, when the tether drag is considered, the optimal airfoils are found in shape to be reminiscent of flapped airfoils, suggesting a multi-element airfoil design. Finally, unlike most studies in the literature, the present work conducts post-optimization Reynolds-averaged Navier–Stokes flow simulations to gain deeper insights into the aerodynamic performance of the optimized airfoils and to provide comparisons with XFOIL results.

Multivalent biomacromolecules including multi-domain and intrinsically disordered proteins form biomolecular condensates via reversible phase transitions. Condensates are viscoelastic materials that display composition-specific rheological properties and responses to mechanical forces. Graph-based descriptions of microstructures can be combined with computational rheometry to model the outcomes of passive and active mechanical measurements. We consider two types of network models for microstructures. In the Jeffreys model, each edge in the network is a Jeffreys element. In the Stokes-Maxwell model, each edge is a Maxwell element that is embedded in an incompressible viscous fluid that can undergo Stokes flow. We describe results from comparative assessments of the two models for individual elements, ordered lattices, random geometric graphs, structured graphs, and graphs for condensates that are extracted from coarse-grained simulations of disordered proteins. Results from deformation and relaxation tests and flow field analysis reveal how distinct length and time scales contribute to the responses of different types of networks. No single test provides definitive assessments of the connections between material properties and microstructures. Instead, a range of active and passive rheometric tests are essential for distinguishing the responses of different types of networks. Our work establishes computational rheometry as a framework for bridging disparate length and timescales to assess how molecular-scale interactions and dynamics give rise to viscoelastic responses on the mesoscale.

Abstract We focus on the numerical analysis of a polygonal discontinuous Galerkin scheme for the simulation of the exchange of fluid between a deformable saturated poroelastic structure and an adjacent free-flow channel. We specifically address wave phenomena described by the low-frequency Biot model in the poroelastic region and unsteady Stokes flow in the open channel, possibly an isolated cavity or a connected fracture system. The coupling at the interface between the two regions is realized by means of transmission conditions expressing conservation laws. The spatial discretization hinges on the weak form of the two-displacement poroelasticity system and a stress formulation of the Stokes equation with weakly imposed symmetry. We present a complete stability analysis for the proposed semi-discrete formulation and derive a-priori hp-error estimates.

Consistency enforcement for the iterative solution of weak Galerkin finite element approximation of Stokes flow

Abstract A biharmonic stream function equation is solved analytically in axisymmetric toroidal coordinates for Stokes flow in a sessile droplet with a surface tension gradient and diffusive evaporation on the surface. Four different types of streamlines are defined and utilized to predict particle deposition either at the center or at the contact line of the droplet. The impact of UV light radius, droplet's contact angle, and humidity level on the flow pattern is investigated. When the relative humidity is high, the formation of Moffatt eddies is observed. However, the eddy size and the number of eddies decrease with an increase in UV light radius or droplet contact angle. As humidity decreases, the Moffatt eddies are restricted to a specific region, decrease in number and size, and ultimately disappear. Under lower humidity, the Marangoni flow near the contact line is dominated by capillary action. The toroidal streamlines are broken from outside and turned into transporting streamline to deposit more particles on the contact line. The region near the droplet center, where the surface tension gradient is significant, is always dominated by toroidal streamlines.

A new arbitrary Lagrangian–Eulerian (ALE) formulation for Navier–Stokes flow on self-evolving surfaces is presented. It is based on a general curvilinear surface parameterisation that describes the motion of the ALE frame. Its in-plane part becomes fully arbitrary, while its out-of-plane part follows the material motion of the surface. This allows for the description of flows on deforming surfaces using only surface meshes. The unknown fields are the fluid density or pressure, the fluid velocity and the surface motion, where the latter two share the same normal velocity. The corresponding field equations are the continuity equation or area-incompressibility constraint, the surface Navier–Stokes equations and suitable surface mesh equations. Particularly advantageous are mesh equations based on membrane elasticity. The presentation focuses on the coupled set of strong and weak form equations, and presents several manufactured steady and transient solutions. These solutions are used together with numerical simulations to illustrate and discuss the properties of the proposed new ALE formulation. They also serve as basis for the development and verification of corresponding computational methods. The new formulation allows for a detailed study of fluidic membranes such as soap films, capillary menisci and lipid bilayers.