Incompressible Navier-Stokes Fluid Research Articles

Pollution overturning instability in an incompressible fluid with a Maxwell-Cattaneo-Mariano model for the pollutant field

We develop a model for a pollutant dissolved in or dispersed in an incompressible Navier–Stokes fluid when the diffusion theory for the pollutant obeys a second order in time system of equations rather than the first order in time system obtained from Fourier’s law. A detailed analysis is performed for a layer of fluid where a pollutant is such that the top of the layer will be heavier in concentration. A detailed expression for the critical pollutant Rayleigh number is found indicating precise conditions under which a convective overturning motion will arise. The investigation is performed by a linear instability analysis, but additionally we provide a completely nonlinear energy stability analysis. Diffusion of flux is also added to a Cattaneo-like equation and this leads to surprising results.

Note on the global wellposedness of two-dimensional incompressible Navier–Stokes fluid structure interaction model

In this note we study the global wellposedness of a two-dimensional incompressible Navier–Stokes fluid structure interaction(FSI) model. The existence and uniqueness of the global strong and smooth solution to the initial boundary value problem for one class of the FSI model in the two-dimensional case, which is one model of two-dimensional incompressible Navier–Stokes equations coupled by the two-dimensional wave equation by the suitable boundary conditions at the interface boundary, are obtained for any smooth initial data under the assumption of the suitable compatibility conditions on the initial data at the interface boundary and external boundary of the domain.

Three-dimensional flows of incompressible Navier–Stokes fluids in tubes containing a sinus, with varying slip conditions at the wall

The objective of this study is to understand the formation of vortices and other flow characteristics associated with the three-dimensional motions of an incompressible Navier–Stokes fluid in tubes containing a sinusoidal extension. The study has some bearing on two conjectures that da Vinci made concerning the flow of blood through the aortic root. We investigate how the flow attributes change with increasing sinus radius and with the nature of the slip at the tube wall characterized by the parameter θ , 0 ≤ θ ≤ 1 , with θ values being 0 for free slip and 1 for the “no-slip” (adherence) condition. Two time-dependent solvers – one fully three-dimensional and the other based on the assumption that admissible flows are axially symmetric – are used to solve the equations governing the flow in a geometry associated with the inflow velocities and the other conditions closely related to flows of blood in a blood vessel containing the aortic root. Both these solvers passed a benchmark test based on steady flow in a cylinder with different slip conditions. Computing the flows systematically using both solvers, focusing first on problems with constant-in-time inflow, we have found that for the circular cylinder with constant cross-section, serving as the reference domain, the flow is steady and unidirectional for all boundary conditions that are considered, with maximal vorticity and dissipation for no-slip. For tubes with the sinus radius up to 16 mm the flow remains steady and axially symmetric, but the vorticity and the bulk dissipation in the sinus are maximal for θ between 0.6 and 0.7. For a tube with the sinus radius of 20 mm the flow remains steady and axially symmetric only for θ greater or equal than 0.9 including the case of no-slip. For θ below this value, not only does the solution become unsteady (oscillatory or even chaotic), it does not converge to the steady state and is thus different from the corresponding axially symmetric flows, and also the total dissipation and the vorticity depend on θ in a non-monotone manner. In particular, for a tube with sinus radius of 20 mm, the bulk dissipation and the vorticity are the highest for θ = 0 , i.e. for (free) slip of the fluid at the wall. For tubes with the sinus radius of 20 mm and for θ below 0.9, we computed two different solutions (one steady, axially symmetric, the other unsteady, fully three-dimensional) to three-dimensional evolutionary incompressible Navier–Stokes equations for the same set of the initial and boundary data. This last result supports the idea that, under the conditions considered, the axially symmetric solution looses its stability. Finally, we computed the problems with a pulsatile inflow. For the tube with the largest sinus radius of 20 mm, we observed that the full problem is axially symmetric for the slip parameter θ ≥ 0 . 9 , while for more significant slip, i.e. for θ < 0 . 9 , we again obtained two different solutions under the same initial and boundary conditions.

Fluid structure computational model of simulating mitral valve motion in a contracting left ventricle

Background:Fluid structure interaction simulations h hold promise in studying normal and abnormal cardiac function, including the effect of fluid dynamics on mitral valve (MV) leaflet motion. The goal of this study was to develop a 3D fluid structure interaction computational model to simulate bileaflet MV when interacting with blood motion in left ventricle (LV). Methods:The model consists of ideal geometric-shaped MV leaflets and the LV, with MV dimensions based on human anatomical measurements. An experimentally-based hyperelastic isotropic material was used to model the mechanical behaviour of the MV leaflets, with chordae tendineae and papillary muscle tips also incorporated. LV myocardial tissue was prescribed using a transverse isotropic hyperelastic formulation. Incompressible Navier–Stokes fluid formulations were used to govern the blood motion, and the Arbitrary Lagrangian Eulerian (ALE) method was employed to determine the mesh deformation of the fluid and solid domains due to trans-valvular pressure on MV boundaries and the resulting leaflet movement. Results:The LV–MV generic model was able to reproduce physiological MV leaflet opening and closing profiles resulting from the time-varying atrial and ventricular pressures, as well as simulating normal and prolapsed MV states. Additionally, the model was able to simulate blood flow patterns after insertion of a prosthetic MV with and without left ventricular outflow tract flow obstruction. In the MV–LV normal model, the regurgitant blood flow fraction was 10.1 %, with no abnormality in cardiac function according to the mitral regurgitation severity grades reported by the American Society of Echocardiography. Conclusion:Our simulation approach provides insights into intraventricular fluid dynamics in a contracting LV with normal and prolapsed MV function, as well as aiding in the understanding of possible complications after transcatheter MV implantation prior to clinical trials.

Global weak solutions in a three-dimensional coral fertilization model involving the Navier–Stokes equations and arbitrary porous medium type diffusion

A degenerate coral fertilization model concerning the evolution of gametes densities, of signal concentration and of an incompressible Navier–Stokes fluid velocity is investigated in this paper. It is proved that an associated initial–boundary problem with no-flux, Dirichlet boundary conditions in a three-dimensional smoothly bounded domain admits a globally defined weak solution for any appropriately regular initial data whenever the nonlinear diffusion parameter m>1.

Boundary Conditions and Polymeric Drag Reduction for the Navier–Stokes Equations

Reducing wall drag in turbulent pipe and channel flows is an issue of great practical importance. In engineering applications, end-functionalized polymer chains are often employed as agents to reduce drag. These are polymers which are floating in the solvent and attach (either by adsorption or through irreversible chemical binding) at one of their chain ends to the substrate (wall). We propose a PDE model to study this setup in the simple setting where the solvent is a viscous incompressible Navier–Stokes fluid occupying the bulk of a smooth domain \(\Omega \subset {\mathbb {R}}^d\), and the wall-grafted polymer is in the so-called mushroom regime (inter-polymer spacing on the order of the typical polymer length). The microscopic description of the polymer enters into the macroscopic description of the fluid motion through a dynamical boundary condition on the wall-tangential stress of the fluid, something akin to (but distinct from) a history-dependent slip-length. We establish the global well-posedness of strong solutions in two-spatial dimensions and prove that the inviscid limit to the strong Euler solution holds with a rate. Moreover, the wall-friction factor \(\langle f\rangle \) and the global energy dissipation \(\langle \varepsilon \rangle \) vanish inversely proportional to the Reynolds number \(\mathbf{Re } \). This scaling corresponds to Poiseuille’s law for the friction factor \(\langle f\rangle \sim 1/\mathbf{Re } \) for laminar flow and thereby quantifies drag reduction in our setting. These results are in stark contrast to those available for physical boundaries without polymer additives modeled by, for example, no-slip conditions, where no such results are generally known even in two-dimensions.

A benchmark problem to evaluate implementational issues for three-dimensional flows of incompressible fluids subject to slip boundary conditions

We consider flows of an incompressible Navier–Stokes fluid in a tubular domain with Navier’s slip boundary condition imposed on the impermeable wall. We focus on several implementational issues associated with this type of boundary conditions within the framework of the standard Taylor-Hood mixed finite element method and present the computational results for flows in a tubular domain of finite length with one inlet and one outlet. In particular, we present the details regarding variants of the Nitsche method concerning the incorporation of the impermeability condition on the wall. We also find that the manner in which the normal to the boundary is numerically implemented influences the nature of the computational results. As a benchmark, we set up steady flows in a tube of finite length and compare the computational results with the analytical solutions. Finally, we identify various quantities of interest, such as the dissipation, wall shear stress, vorticity, pressure drop, and provide their precise mathematical definitions. We document how well these quantities are computationally approximated in the case of the benchmark.Although the geometry of the benchmark is simple, the correct computational results require careful selection of numerical methods and surprisingly non-trivial computational resources. Our goal is to test, using the setting with a known analytical solution, a robust computational tool that would be suitable for computations on real complex geometries that have relevance to problems in engineering and medicine. The model parameters in our computations are chosen based on flows in large arteries.

Gravity dual of Navier-Stokes equation in a rotating frame through parallel transport

The fluid-gravity correspondence documents a precise mathematical map between a class of dynamical spacetime solutions of the Einstein field equations of gravity and the dynamics of its corresponding dual fluid flows governed by the Navier-Stokes (NS) equations of hydrodynamics. This striking connection has been explored in several dynamics-based approaches and has surfaced in various forms over the past four decades. In a recent construction, it has been shown that the manifold properties of geometric duals are in fact intimately connected to the dynamics of incompressible fluids, thus bypassing the conventional on-shell standpoints. Following such a prescription, we construct the geometrical description that effectively captures the dynamics of an incompressible NS fluid with respect to a uniformly rotating frame. We propose the gravitational dual(s) described by bulk metric(s) in $(p+2)$-dimensions such that the equations of parallel transport of an appropriately defined bulk velocity vector field when projected onto an induced timelike hypersurface require that the incompressible NS equation of a fluid relative to a uniformly rotating frame be satisfied at the relevant perturbative order in $(p+1)$-dimensions. We argue that free fluid flows on manifold(s) described by the proposed metric(s) can be effectively considered as an equivalent theory of non-relativistic viscous fluid dynamics with respect to (w.r.t) a uniform rotating frame. We also present suggestive insights as to how space-time rotation parameters encode information pertaining to the inertial effects in the corresponding fluid dual.

Reference map technique for incompressible fluid–structure interaction

Author(s): Rycroft, CH; Wu, CH; Yu, Y; Kamrin, K | Abstract: We present a general simulation approach for fluid-solid interactions based on the fully Eulerian reference map technique. The approach permits the modelling of one or more finitely deformable continuum solid bodies interacting with a fluid and with each other. A key advantage of this approach is its ease of use, as the solid and fluid are discretized on the same fixed grid, which greatly simplifies the coupling between the phases. We use the method to study a number of illustrative examples involving an incompressible Navier-Stokes fluid interacting with multiple neo-Hookean solids. Our method has several useful features including the ability to model solids with sharp corners and the ability to model actuated solids. The latter permits the simulation of active media such as swimmers, which we demonstrate. The method is validated favourably in the flag-flapping geometry, for which a number of experimental, numerical and analytical studies have been performed. We extend the flapping analysis beyond the thin-flag limit, revealing an additional destabilization mechanism to induce flapping.

Validation of the MRT-LBM for three-dimensional free-surface flows: an investigation of the weak compressibility in dam-break benchmarks

ABSTRACTFree-surface flow problems occur in numerous disaster simulations, such as tsunami inland penetration in urban areas. Simulation models for these problems must be non-hydrostatic, three-dimensional and highly resolved because of the strong non-linearity and higher-order physical phenomena. However, three-dimensional large-scale tsunami simulations based on conventional computational fluid dynamics (CFD) solving the Navier-Stokes equations are challenging due to the pressure Poisson equation in incompressible Navier-Stokes fluid modeling. The lattice Boltzmann method (LBM) is an alternative simulation tool that is attracting attention as a fully explicit and efficient approach. The LBM does not have to iteratively solve the pressure Poisson equation and is therefore considered to have an advantage over other methods when executing high-performance three-dimensional tsunami simulations. In the current study, we developed a fully explicit free-surface model using the LBM in which the more advanced multiple-relaxation-time (MRT) collision model is used, along with the piecewise linear interface calculation (PLIC) approach. Through classic dam-break problems, we validated the appropriate parameter settings, including the weak compressibility, for tsunami simulations. The benchmark tests showed that our model accurately simulates the three-dimensional dam-break flows and controls the compressibility drop in the second-order value of the Mach number.

Formula omitted]-solutions for the stationary Navier–Stokes equations in the exterior of a rotating obstacle

Consider the stationary motion of an incompressible Navier–Stokes fluid around a rotating body R3∖Ω which is also moving in the direction of the axis of rotation with nonzero constant velocity −ke1. We assume that the angular velocity ω=|ω|e1 is also constant and the external force is given by f=divF. Then the motion is described by a variant of the stationary Navier–Stokes equations with the velocity ke1 at infinity. Our main result is the existence of at least one solution u satisfying u−ke1∈L3(Ω) for arbitrarily large F∈L3∕2(Ω). The uniqueness is also proved by assuming that |ω|+|k|+‖F‖L3∕2(Ω) is sufficiently small in comparison with the viscosity ν. Moreover, we establish several regularity results to obtain an existence theorem for weak solutions u satisfying ∇u∈L3∕2(Ω) and u−ke1∈L3(Ω).

Numerical measurements of scaling relations in two-dimensional conformal fluid turbulence

We present measurements of relativistic scaling relations in (2+1)-dimensional conformal fluid turbulence from direct numerical simulations, in the weakly compressible regime. These relations were analytically derived previously in [1] for a relativistic fluid; this work is a continuation of that study, providing further analytical insights together with numerical experiments to test the scaling relations and extract other important features characterizing the turbulent behavior. We first explicitly demonstrate that the non-relativistic limit of these scaling relations reduce to known results from the statistical theory of incompressible Navier-Stokes turbulence. In simulations of the inverse-cascade range, we find the relevant relativistic scaling relation is satisfied to a high degree of ac-curacy. We observe that the non-relativistic versions of this scaling relation underperform the relativistic one in both an absolute and relative sense, with a progressive degradation as the rms Mach number increases from 0.14 to 0.19. In the direct-cascade range, the two relevant relativistic scaling relations are satisfied with a lower degree of accuracy in a simulation with rms Mach number 0.11. We elucidate the poorer agreement with further simulations of an incompressible Navier-Stokes fluid. Finally, as has been observed in the incompressible Navier-Stokes case, we show that the energy spectrum in the inverse-cascade of the conformal fluid exhibits k−2 scaling rather than the Kolmogorov/Kraichnan expectation of k−5/3, and that it is not necessarily associated with compressive effects. We comment on the implications for a recent calculation of the fractal dimension of a turbulent (3 + 1)-dimensional AdS black brane.

Microscopic statistical description of incompressible Navier-Stokes granular fluids

Based on the recently established Master kinetic equation and related Master constant H-theorem which describe the statistical behavior of the Boltzmann-Sinai classical dynamical system for smooth and hard spherical particles, the problem is posed of determining a microscopic statistical description holding for an incompressible Navier-Stokes fluid. The goal is reached by introducing a suitable mean-field interaction in the Master kinetic equation. The resulting Modified Master Kinetic Equation (MMKE) is proved to warrant at the same time the condition of mass-density incompressibility and the validity of the Navier-Stokes fluid equation. In addition, it is shown that the conservation of the Boltzmann-Shannon entropy can similarly be warranted. Applications to the plane Couette and Poiseuille flows are considered showing that they can be regarded as final decaying states for suitable non-stationary flows. As a result, it is shown that an arbitrary initial stochastic 1-body PDF evolving in time by means of MMKE necessarily exhibits the phenomenon of Decay to Kinetic Equilibrium (DKE), whereby the same 1-body PDF asymptotically relaxes to a stationary and spatially uniform Maxwellian PDF.

Asymptotic behavior of the incompressible Navier-Stokes fluid with degree of freedom in porous medium

The authors study the asymptotic behavior of the incompressible Navier-Stokes fluid with degree of freedom in the porous medium in ℝ n with n = 2 or 3. They derive the Darcy law as e, the character size of the hole, tends to zero. Moreover, the authors obtain the expression of the degree of freedom from the homogenized model.

Very weak solutions of the stationary Navier–Stokes equations for an incompressible fluid past obstacles

We consider the stationary motion of an incompressible Navier–Stokes fluid past obstacles in R3, subject to the given boundary velocity vb, external force f=divF and nonzero constant vector ke1 at infinity. Our main result is the existence of at least one very weak solution v in ke1+L3(Ω) for arbitrary large F∈L3/2(Ω)+L12/7(Ω) provided that the flux of vb−ke1 on the boundary of each body is sufficiently small with respect to the viscosity ν. The uniqueness of very weak solutions is proved by assuming that F and vb−ke1 are suitably small. Moreover, we establish weak and strong regularity results for very weak solutions. In particular, our existence and regularity results enable us to prove the existence of a weak solution v satisfying ∇v∈L3/2(Ω).

On three-dimensional incompressible Navier-Stokes fluid on cantor sets in spherical Cantor type co-ordinate system

This paper addresses the systems of the incompressible Navier-Stokes equations on Cantor sets without the external force involving the fractal heat-conduction problem vial local fractional derivative. The spherical Cantor type co-ordinate method is used to transfer the incompressible Navier-Stokes equation from the Cantorian co-ordinate system into the spherical Cantor type co-ordinate system.

Flow of an Eyring-Powell Model Fluid between Coaxial Cylinders with Variable Viscosity

We consider the flow of Eyring-Powell model fluid in the annulus between two cylinders whose viscosity depends upon the temperature. We consider the steady flow in the annulus due to the motion of inner cylinder and constant pressure gradient. In the problem considered the flow is found to be remarkedly different from that for the incompressible Navier-Stokes fluid with constant viscosity. An analytical solution of the nonlinear problem is obtained using homotopy analysis method. The behavior of pertinent parameters is analyzed and depicted through graphs.

A note on the modeling of incompressible fluids with material moduli dependent on the mean normal stress

In incompressible materials, both fluids and solids, a part of the stress is not prescribed by constitutive specification, that is, the part of the stress is not determined in terms of kinematical quantities, temperature, et cetera. This “indeterminate” part of the stress is variously referred to as the “constraint stress”, the “reaction stress” or the “Lagrange multiplier” enforcing the constraint. In the case of an incompressible Navier–Stokes fluid, the part of the stress, that is a consequence of the constraint, also happens to coincide with the mean value of the stress which is referred to as the “mechanical pressure”. However, in general non-Newtonian fluids this is not the case, and, unfortunately, in view of the widespread use of the Navier–Stokes equation, the terminology “pressure” is used interchangeably for both the part of the stress that is not constitutively specified and the mean value of the stress, leading to considerable confusion with regard to important issues concerning non-Newtonian fluids. Recognizing the distinction between the mean value of the stress and the part of the stress that is not constitutively specified becomes critical in materials whose moduli depend on the mean value of the stress. An example of the same concerns the viscosity, which depending on whether it is a function of the indeterminate part of the stress or the mean value of the stress could lead to different flow characteristics. In this short note we discuss an error that is a consequence of not recognizing the distinction between these different quantities but misidentifying them as being the same, the mechanical “pressure”.

Level Set Method for Topological Optimization of the Naiver-Stokes Fluid Flow

This paper presents a general algorithm for topological optimization of the incompressible Navier-Stokes fluid flow based on a level set method. This is a direct extension of our previous work on Stokes flow of such problems. First we obtain the shape sensitivity analysis using the material derivative concept and adjoint variable technique, and then we couple the shape sensitivity analysis result into the level set function as the advection velocity. Since the level set method is implemented in an Euleran framework, the computational cost of the proposed algorithm is moderate. A Benchmark example is provided to illustrate the efficiency and validity of this method.

Black holes as incompressible fluids on the sphere

We consider finite deformations of the p+2-dimensional Schwarzschild geometry which obey the vacuum Einstein equation, preserve the mean curvature and induced conformal metric on a sphere a distance $\lambda$ from the horizon and are regular on the future horizon. We show perturbatively that in the limit $\lambda$ approaches 0 the deformations are given by solutions of the nonlinear incompressible Navier-Stokes equation on the p-sphere. This relation provides a link between global existence for p-dimensional incompressible Navier-Stokes fluids and a novel form of cosmic censorship in p+2-dimensional general relativity.