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Highlights • Application of nonlinear dynamics to cancer ecosystems. • Chemical turbulence and strange attractor models in tumor growth, invasion and pattern formation are investigated. • Computational algorithms for detecting such structures are proposed. • Complex systems applications to cancer dynamics., Cancers are complex, adaptive ecosystems. They remain the leading cause of disease-related death among children in North America. As we approach computational oncology and Deep Learning Healthcare, our mathematical models of cancer dynamics must be revised. Recent findings support the perspective that cancer-microenvironment interactions may consist of chaotic gene expressions and turbulent protein flows during pattern formation. As such, cancer pattern formation, protein-folding and metastatic invasion are discussed herein as processes driven by chemical turbulence within the framework of complex systems theory. To conclude, cancer stem cells are presented as strange attractors of the Waddington landscape.

A model of a single reed instrument is studied in which the reed is described as an Euler-Bernoulli beam, and the air flow through the instrument is calculated using the Navier-Stokes equations. The hypothetical instrument resembles a clarinet, but is smaller than a real clarinet to keep the numerical modeling feasible on available supercomputers. This article explores the conditions under which the frequency of the reed oscillations and the emitted sound are determined by the resonant frequency of the bore of the instrument. The effect of the contact between the reed and the player's lips is also studied, and quantitative results for the air density and pressure in the mouthpiece and throughout the instrument bore are also presented.

Matrix acidizing is one of the most practical stimulation technologies for carbonate reservoirs, which effectively improve the region permeability near the wellbore. In addition to solid matrix, vugs are also very common in carbonate reservoirs. However, a few studies have been addressed with existence of vugs on carbonate acidizing process. In this work, a two-scale model is developed using dual domain method and discrete vugs model to study effect of vugs on acidizing process. Darcy equation is employed in solid matrix region. Navier Stokes equation is adopted for free flow region in vugs. The two regions are coupled by modified Beavers-Joseph-Saffman boundary condition. Numerical cases are conducted to present the effect of vug characteristics on acid-rock reaction process. The results show that acid solution has the largest effective reducing distance and the smallest breakthrough volume in circular vugs. Dominant wormhole is created when acid injection direction is parallel or vertical to the azimuth angle of vugs. Increasing amount of vugs in horizontal effectively reduces the flow distance and breakthrough volume of acid solution. Vugs with random distribution increases effective flow distance and breakthrough volume of acid solution compared to vugs with orderly distribution. Cited as: Huang, Z., Xing, H., Zhou, X., You, H. Numerical study of vug effects on acid-rock reactive flow in carbonate reservoirs. Advances in Geo-Energy Research, 2020, 4(4): 448-459, doi: 10.46690/ager.2020.04.09

Major challenges associated with the smooth production operations in the oil and gas industry that has raised technical curiosity are formation of natural gas hydrates in production facilities and flow lines which introduces significant cost to operators. Accurate modeling is therefore paramount; most existing models are based on constitutive conservation laws neglecting other dissipative energy types. To predict “if” and “where” gas hydrate would be formed in gas pipeline, the Navier-Stokes equation was modified by incorporating dissipative forces of viscosity and gravity; the equation that emerged was solved analytically to determine the hydrate formation pressure (HFP) and the position of hydrate formation along gas pipelines. The developed model, used as a quick-look tool for where and if hydrates will form revealed that when the predicted HFP is positive hydrates was formed but when it is negative hydrates were not formed. The model also showed that HFP is a function fluid composition, mass flowrate, changes in fluid and surrounding conditions and changes in elevation and direction confirming the results of earlier work done.

We consider the Cauchy problem for the three-dimensional bipolar compressible Navier-Stokes-Poisson system with unequal viscosities. Under the assumption that the H3 norm of the initial data is small but its higher order derivatives can be arbitrarily large, the global existence and uniqueness of smooth solutions are obtained by an ingenious energy method. Moreover, if additionally, the $${\dot H^{- s}}\left({{1 \over 2} \leqslant s < {3 \over 2}} \right)\,\,{\rm{or}}\,\,\dot B_{2,\infty }^{- s}\left({{1 \over 2} < s \leqslant {3 \over 2}} \right)$$ norm of the initial data is small, the optimal decay rates of solutions are also established by a regularity interpolation trick and delicate energy methods.

The boundary value problem for the three-dimensional steady viscous compressible Navier-Stokes/Allen-Cahn (NSAC) system is considered. We establish the existence of a weak solution for the adiabatic index γ > 2 with the phase function belonging to [ − 1 , 1 ] . Moreover, we give further analysis on the weak solution. More specifically, we prove that the steady NSAC system degenerates into Navier-Stokes system if the phase function is positive (negative) on Ω ‾ .

This work presents a Reynolds-averaged Navier Stokes–based computational fluid dynamics methodology for the calculation of pressure drop and mass flow rate distribution in a material test reactor f...

We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn–Hilliard–Navier–Stokes phase- field model, prove its energy stability, and carry out error analysis for the corresponding Cahn–Hilliard–Stokes model only. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase-field variable, chemical potential, velocity and pressure in different discrete norms for the Cahn–Hilliard–Stokes phase-field model. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of our scheme.

The goal of the paper is to study the convergence of finite volume approximations of the Navier–Stokes–Fourier system describing the motion of compressible, viscous and heat-conducting fluids. The numerical flux uses upwinding with an additional numerical diffusion of order $\mathcal O(h^{ \varepsilon +1})$, $0

In the present research article, we implemented two well-known analytical techniques to solve fractional-order multi-dimensional Navier–Stokes equation. The proposed methods are the modification of Adomian decomposition method and variational iteration method by using natural transformation. Furthermore, some illustrative examples are presented to confirm the validity of the suggested methods. The solutions graphs and tables are constructed for both fractional and integer-order problems. It is investigated that the suggested techniques have the identical solutions of the problems. The solution comparison via graphs and tables have also supported the greater accuracy and higher rate of convergence of the present methods.

We revisit the classical work of Huang-Matsumura-Xin [9] for contact wave of the one-dimensional compressible Navier-Stokes system under the zero mass condition. The large-time asymptotic stability of a contact wave pattern with a uniform convergence rate ( 1 + t ) − 1 4 was proved by Huang-Matsumura-Xin. In this paper, Huang-Matsumura-Xin's convergence rate is improved to ( 1 + t ) − 5 8 ln 1 2 ⁡ ( 2 + t ) by using a new Poincare type inequality and a detailed energy analysis. Our work is motivated by a problem arising in [9] .

In this paper we study the numerical approximation of the stochastic Cahn–Hilliard–Navier–Stokes system on a bounded polygonal domain of $\mathbb{R}^{d}$, $d=2,3$. We propose and analyze an algorithm based on the finite element method and a semiimplicit Euler scheme in time for a fully discretization. We prove that the proposed numerical scheme satisfies the discrete mass conservative law, has finite energies and constructs a weak martingale solution of the stochastic Cahn–Hilliard–Navier–Stokes system when the discretization step (both in time and in space) tends to zero.

In this study, the double Laplace Adomian decomposition method and the triple Laplace Adomian decomposition method are employed to solve one- and two-dimensional time-fractional Navier–Stokes problems, respectively. In order to examine the applicability of these methods some examples are provided. The presented results confirm that the proposed methods are very effective in the search of exact and approximate solutions for the problems. Numerical simulation is used to sketch the exact and approximate solution.

In this paper, we investigate Cauchy problem for the two-dimensional incompressible chemotaxis-Navier-Stokes equations with the lower fractional diffusion { ∂ t n + u ⋅ ∇ n − Δ n = − ∇ ⋅ ( n ∇ c ) + λ n − μ n 2 , ∂ t c + u ⋅ ∇ c − Δ c = − c n , ∂ t u + u ⋅ ∇ u + Λ 2 α u + ∇ P = − n ∇ ϕ , where Λ : = ( − Δ ) 1 2 and α ∈ [ 1 2 , 1 ] . We obtain the global-in-time existence and uniqueness of weak solution to the equations for a class of large initial data by making use of the coupled structure of system and damping effect of the logistic source, and developing the L 4 3 ( R 2 ) estimate for vorticity.

We propose a method for solving the Navier–Stokes and Euler equations by introducing new unknowns and transforming the defining equations, which allows us to reduce the problem to simpler mathematical problems.

The main concern of this paper is to investigate the asymptotic stability of stationary solution to the compressible Navier–Stokes–Poisson equations with the classical Boltzmann relation in a half ...

We consider a quadratic optimal control problem for the 3D Navier-Stokes-Voigt equations with periodic inputs. We prove the existence of optimal solutions, then establish necessary and sufficient o...

We consider the Navier-Stokes initial boundary value problem in 3D-exterior domains with non-decaying initial data. We investigate the existence of weak solutions defined for all t > 0 for large non-decaying initial data. To achieve the result, we extend the half-space technique developed by Maremonti and Shimizu (2018) [26] to the case of an exterior domain.

We study the stabilization of stationary solutions to Navier-Stokes-Voigt equations by finite-dimensional feedback control scheme introduced by Azouani and Titi (Evol. Equ. Control Theory 3, 579–594 2014). The designed feedback control scheme is based on the finite number of determining parameters (degrees of freedom), namely, finite number of determining Fourier modes, determining nodes, and volume elements.

The main purpose of this work is to explore a general cell–fluid model which is based on a mixture theory formulation that accounts for the interplay between oxytactically (chemotaxis toward gradient in oxygen) moving bacteria cells in water and the buoyance forces caused by the difference in density between cells and fluid. The model involves two mass balance and two general momentum balance equations, respectively, for the cell and fluid phase, combined with a convection–diffusion–reaction equation for oxygen. In particular, the momentum balance equations include interaction terms which describe the cell–fluid drag force effect. Hence, the model is an extension of the classical Navier–Stokes equation in two different ways: (i) inclusion of two phases (cell and fluid) instead of one; (ii) inclusion of a chemotactic transport mechanism. The model can be understood as a natural generalization of the much studied chemotaxis-Stokes model explored by [I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA 102 (2005) 2277–2282]. First, we explore the model for parameters in a range which ensures that it lies close to the previously studied chemotaxis-Stokes model (essentially very low cell volume fraction). Main observations are (i) formation of sinking finger-shaped plumes and (ii) convergence to plumes that possibly can be stationary (i.e. persist over time). The general cell–fluid model provides new insight into the role played by the cell–fluid interaction term which controls the competition between gravity segregation and chemotaxis effect on the formation of cell plumes. Second, we explore cases with large cell volume fraction (far beyond the regime captured by the chemotaxis-Stokes model), which gives rise to rich pattern-formation behavior. The general cell–fluid model opens for exploring a hierarchy of different “submodels”. Hence, it seems to be an interesting model for further investigations of various, general cell–fluid spatio-temporal evolution dynamics, both from an experimental and mathematical point of view.

The goal of this article is to study the feedback control for non-stationary three-dimensional Navier–Stokes–Voigt equations. Based on the existence, uniqueness, and boundedness result of the weak solutions to the equations, we obtain the existence of solutions to the feedback control system. An existence result for an optimal control problem is also given. We illustrate our main result with an evolutionary hemivariational inequality.

The diffusion of cells in a viscous incompressible fluid (e.g. water) may be viewed like movement in a porous medium and there is a bidirectorial oxygen exchange between water and their surrounding air in thin fluid layers near the air–water contact surface. This leads to the following chemotaxis-Navier–Stokes system with nonlinear diffusion: [Formula: see text] endowed with the inhomogeneous boundary conditions [Formula: see text] and the initial data [Formula: see text] in [Formula: see text], where the incoming oxygen [Formula: see text] is non-negative, and the outgoing oxygen molecule is modeled by [Formula: see text] with positive coefficient [Formula: see text]. In this paper, we investigate the asymptotic dynamics of the above system in a bounded domain [Formula: see text] with the smooth boundary [Formula: see text]. We will show that arbitrary porous medium diffusion mechanism [Formula: see text] can inhibit the singularity formation. In the incoming oxygen-free case, we further prove that the solution will stabilize to the unique mass-preserving spatial equilibrium [Formula: see text] in the sense that as [Formula: see text], [Formula: see text] hold uniformly with respect to [Formula: see text], where [Formula: see text].

In this paper, we establish the existence and uniqueness of both mild(/variational) solutions and weak (in the sense of PDE) solutions of coupled system of 2D stochastic Chemotaxis-Navier-Stokes equations. The mild/variational solution is obtained through introducing a new method of cutting off the stochastic system and using a fixed point argument in a carefully constructed Banach space. To get the weak solution we first prove the existence of a martingale weak solution and then we show that the pathwise uniqueness holds for the martingale solution.

In this paper, we study a three-dimensional chemotaxis-Navier–Stokes system which characterizes the fertilization process of coral. It is proved that in the context of the nonlinear diffusion of cells with the index the corresponding initial-boundary problem is globally solvable in the weak sense.

In this paper, the two-dimensional incompressible chemotaxis fluid with logical source is studied as following: nt+u⋅∇n=Δn−∇⋅(n∇c)+n−n2,ct+u⋅∇c=Δc−nc,ut+u⋅∇u+∇P=Δu−n∇φ,∇⋅u=0. By taking advantage of...

This paper addresses several problems associated to local energy solutions (in the sense of Lemari\'e-Rieusset) to the Navier-Stokes equations with initial data which is sufficiently small at large or small scales as measured using truncated Morrey-type quantities, namely: (1) global existence for a class of data including the critical $L^2$-based Morrey space; (2) initial and eventual regularity of local energy solutions to the Navier-Stokes equations with initial data sufficiently small at small or large scales; (3) small-large uniqueness of local energy solutions for data in the critical $L^2$-based Morrey space. A number of interesting corollaries are included, including eventual regularity in familiar Lebesgue, Lorentz, and Morrey spaces, a new local generalized Von Wahl uniqueness criteria, as well as regularity and uniqueness for local energy solutions with small discretely self-similar data.

In this article, we prove a regularity result for weak solutions away from singular set of stationary Navier-Stokes systems with subquadratic growth under controllable growth condition. The proof is based on the $$\mathcal{A}$$ -harmonic approximation technique. In this article, we extend the result of Shuhong Chen and Zhong Tan [7] and Giaquinta and Modica [18] to the stationary Navier-Stokes system with subquadratic growth.

The current study numerically investigates the flow instability under several part-load conditions in a centrifugal pump with a straight inlet pipe to explore the underlying relationship between a positive slope phenomenon and internal flow using a partially averaged Navier–Stokes model. The model was validated by comparing the hydraulic performance and averaged flow in the impeller between the numerical results and experimental data of a tested pump. The internal flows in pumps have been intensively investigated based on Batchelor vortex family, Rayleigh–Taylor criterion, entropy generation rate, and energy equation to analyze the flow instability from different aspects. The simulation results using partially averaged Navier–Stokes model are acceptable due to the good agreement with the experimental data for the tested pump. No matter the geometry of the inlet pipe, the pre-swirling flows in the inlet pipe are in the convective instability region. Under the part-load condition of φ = 0.5 φbep, the axial vorticity coefficient is affected by the geometry of the inlet pipe. However, under the part-load condition with rotating stall, e.g. φ = 0.78 φbep, the flow in the inlet pipe is affected by the unstable flow in the pump impeller. For the pump with a straight inlet pipe, the vortex inside the blade-to-blade passage is in a stable state according to Rayleigh–Taylor criterion under the condition of φ = 0.5 φbep. However, the vortex in the blade-to-blade passage is in an unstable state due to centrifugal instability under those operation conditions with rotating stall cells in the impeller, and the dominant oscillations are dependent on the propagation of rotating stall cells. Finally, head loss analysis based on energy equations elucidates that turbulent kinetic energy production term is predominant in the head loss in pump impeller. The present results are helpful for better understanding of the unstable flows and positive slope phenomenon for centrifugal pumps.

This paper proves a Serrin’s type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density ϱ and velocity field u satisfy $$\Vert\nabla\varrho\Vert_{L^{\infty}(0,T;W^{1,q})}+\Vert u\Vert_{L^{s}(0,T;L_{\omega}^{r})} 3 and any (r, s) satisfying 2/s + 3/r ⩽ 1, 3 < r ⩽ ∞, then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over [0, T]. Here L denotes the weak Lr space.

The work of Manuel Quezada de Luna was supported primarily by an appointment to the Postgraduate Research Participation Program at the U.S. Army Engineer Research and Development Center, Coastal and Hydraulics Laboratory (ERDC-CHL) administrated by the Oak Ridge Institute for Science and Education through an interagency agreement between the U.S. Department of Energy and ERDC. Haydel Collins and Chris Kees were supported by the ERDC University program and the ERDC Future Investment Fund. Permission was granted by the Chief of Engineers, US Army Corps of Engineers, to publish this information.

The present paper reports numerical simulation of turbulent flow over a square cylinder using a novel scale resolving computational fluid dynamics technique named Partially-Averaged Navier–Stokes (PANS), which bridges Reynolds-Averaged Navier–Stokes (RANS) with Direct Numerical Simulation (DNS) in a seamless manner. All stream-wise and wall normal mean velocity components, turbulent stresses behavior have been computed along the flow (streamwise) as well as in transverse (wall normal) direction. The measurement locations are chosen based on the previous studies so that results could be compared. However, the Reynolds number ( Re) of the flow is maintained at 21,400 and K– ω turbulence model is considered for the present case. All the computations are performed in OpenFOAM framework using a finite volume solver. Additionally, turbulent kinetic energy variations are presented over a wide range of measurement planes in order to explain the energy transfer process in highly unsteady turbulent flow field. The fluctuating root mean square velocities in the streamwise as well as in the wall normal direction have been discussed in the present work. It has been found that Partially-Averaged Navier–Stokes (PANS) model is capable of capturing the properties of highly unsteady turbulent flows and gives better results than Reynolds-Averaged Navier–Stokes (RANS). The results obtained using Partially-Averaged Navier–Stokes (PANS) are quite comparable with Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) data available in literature. The partially-averaged Navier–Stokes results are compared with our simulated Reynolds-Averaged Navier–Stokes (RANS) results, available experimental as well as numerical results in literature and it is found to be good in agreement.

The existence of a weak solution for the Navier–Stokes-alpha model with temperature-dependent viscosity is proved using the topological approximation method for the study of hydrodynamic problems.

SUMMARYWe introduce SPECFEM2D-DG, an open-source, time-domain, hybrid Galerkin software modelling the propagation of seismic and acoustic waves in coupled solid–fluid systems. For the solid part, the visco-elastic system from the routinely used SPECFEM2D software is used to simulate linear seismic waves subject to attenuation. For the fluid part, SPECFEM2D-DG includes two extensions to the acoustic part of SPECFEM2D, both relying on the Navier–Stokes equations to model high-frequency acoustics, infrasound and gravity waves in complex atmospheres. The first fluid extension, SPECFEM2D-DG-FNS, was introduced in 2017 by Brissaud, Martin, Garcia, and Komatitsch; it features a nonlinear Full Navier–Stokes (FNS) approach discretized with a discontinuous Galerkin numerical scheme. In this contribution, we focus only on introducing a second fluid extension, SPECFEM2D-DG-LNS, based on the same numerical method but rather relying on the Linear Navier–Stokes (LNS) equations. The three main modules of SPECFEM2D-DG all use the spectral element method (SEM). For both fluid extensions (FNS and LNS), two-way mechanical coupling conditions preserve the Riemann problem solution at the fluid–solid interface. Absorbing outer boundary conditions (ABCs) derived from the perfectly matched layers’ approach is proposed for the LNS extension. The SEM approach supports complex topographies and unstructured meshes. The LNS equations allow the use of range-dependent atmospheric models, known to be crucial for the propagation of infrasound at regional scales. The LNS extension is verified using the method of manufactured solutions, and convergence is numerically characterized. The mechanical coupling conditions at the fluid–solid interface (between the LNS and elastodynamics systems of equations) are verified against theoretical reflection-transmission coefficients. The ABCs in the LNS extension are tested and prove to yield satisfactory energy dissipation. In an example case study, we model infrasonic waves caused by quakes occurring under various topographies; we characterize the acoustic scattering conditions as well as the apparent acoustic radiation pattern. Finally, we discuss the example case and conclude by describing the capabilities of this software. SPECFEM2D-DG is open-source and is freely available online on GitHub.

A new partially averaged Navier–Stokes (PANS) closure is derived based on the k−kL (KSKL) model. The aim of this new model is to incorporate the desirable features of the KSKL model, compared to the k−ω shear stress transport model, into the PANS framework. These features include reduced eddy-viscosity levels, a lower dependency on the cell height at the wall, well-defined boundary conditions, and improved iterative convergence. As well as the new model derivation, the paper demonstrates that these desirable features are indeed maintained, for a range of modeled-to-total turbulence kinetic energy ratios (fk), and even for multiphase flow.

The three main approaches to exploring fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. Numerical simulations and theoretical solutions are based on the continuity equation and Navier-Stokes equations (NSE) that govern experimental observations of fluid dynamics. Theoretical solutions can offer huge advantages over numerical solutions and experiments in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, theoretical solutions have been limited by the prized NSE problem that seeks a physically consistent solution than what classical potential theory (CPT) offers. Therefore, the current author embarked on a doctoral research on the refinement of CPT. He introduced the Refined Potential Theory (RPT) that provides the Kwasu function as a physically consistent solution to the NSE problem. The Kwasu function is a viscous scalar potential function that captures known and observable unsteady features of experimentally observed wall bounded flows including flow separation, wake formation, vortex shedding, compressibility effects, turbulence and Reynolds-number-dependence. It is appropriately defined to combine the properties of a three-dimensional potential function to satisfy the inertia terms of the NSE and the features of a stream function to satisfy the continuity equation, the viscous vorticity equation and the viscous terms of the NSE. RPT has been verified and validated against experimental and numerical results of incompressible unsteady sub-critical Reynolds number flows on stationary finite circular cylinder, sphere and spheroid. It is concluded that the Kwasu function is a physically consistent and closed-form analytical solution to the NSE problem.