Viscous Flow Equations Research Articles

In silico modelling of the effect of pyloric intervention procedures on gastric flow and emptying in a stomach with gastroparesis.

Pyloric interventions are surgical procedures employed to increase the gastric emptying rate in gastroparesis patients. In this study, we use an in silico model to investigate the consequences of pyloric intervention on gastric flow and emptying for two phenotypes of gastroparesis: antral hypomotility and decreased gastric tone. The transpyloric pressure gradient predicted by the in silico model, based on viscous fluid flow equations, is compared against in vivo measurements. Both phenotypes exhibit a similar pre-procedural emptying rate reduction, but after pyloric surgery, antral hypomotility case with preserved gastric tone shows significant improvements in emptying rates, up to 131%, accompanied by bile reflux from the duodenum into the stomach. Conversely, severely reduced gastric tone cases exhibited a post-procedural reduction in the net emptying rate due to the relatively larger bile reflux. In cases with a combination of antral hypomotility and reduced gastric tone, post-procedural improvements were observed only when both conditions were mild. Our findings highlight the pivotal role of the relative increase in pyloric orifice diameter in determining post-operative emptying rates. The study suggests a possible explanation for the selective response of patients toward these procedures and underscores the potential of in silico modelling to generate valuable insights to inform gastric surgery.

Heat transfer analysis of MHD viscous nanofluid flow over a nonlinearly stretching sheet with heat source/sink: A numerical study

The present study analyses the magnetohydrodynamics (MHD) viscous nanofluid flow past a nonlinearly stretching sheet embedded in a porous medium under the influence of heat generation and thermal radiation. Further, variable magnetic field as well as variable permeability is used instead of constant magnetic field and permeability due to many therapeutic applications, which makes them distinctive and practically helpful. The MHD viscous flow and heat transfer equations have been generated as coupled second-order nonlinear partial differential equations using a mathematical model that resembles the physical flow problem. The partial differential equations governing the flow problems are reduced to ordinary differential equations via similarity variables. The reduced equations are then solved by Runge–Kutta–Fehlberg’s scheme with shooting method by the help of MATLAB software. The result obtained reveals that enhancing the heat generation parameter, porosity parameter, and radiation parameter upsurges the temperature of the nanofluid, whereas the magnetic parameter shows the reverse effect in the case of velocity distribution. Nusselt number decreases at the rate of 2.6147% for copper–water nanofluid when increased from 0 to 0.05. Silver nanoparticles have the highest heat transfer as compared to other nanoparticles. Because of this, silver nanoparticles are the most effective thermal conductors for increasing the effectiveness of heat storage in phase transition materials.

A mass-preserving level set method for simulating 2D/3D fluid flows with deformed interface

This research paper focuses on the development of a novel interface-capturing solver for predicting two-phase flow in 2D/3D Cartesian meshes within the framework of Eulerian approaches. The primary objective is to ensure mass conservation and accurate capture of interface topology. To achieve this, a mass-preserving level set advection equation formulated as a scalar sign-distance function is proposed. The key innovation of the Eulerian solver lies in the incorporation of a scalar speed function for robust reconstruction of the classical level set equation. The effectiveness and reliability of the proposed flow solver in solving incompressible two-phase viscous flow equations are demonstrated through several benchmark problems.

New hybrid quadrature schemes for weakly singular kernels applied to isogeometric boundary elements for 3D Stokes flow

This work proposes four novel hybrid quadrature schemes for the efficient and accurate evaluation of weakly singular boundary integrals (1/r kernel) on arbitrary smooth surfaces. Such integrals appear in boundary element analysis for several partial differential equations including the Stokes equation for viscous flow and the Helmholtz equation for acoustics. The proposed quadrature schemes apply a Duffy transform-based quadrature rule (Duffy, 1982) to surface elements containing the singularity and classical Gaussian quadrature to the remaining elements. Two of the four schemes additionally consider a special treatment for elements near to the singularity, where refined Gaussian quadrature and a new moment-fitting quadrature rule are used.The hybrid quadrature schemes are systematically studied on flat B-spline patches and on NURBS spheres considering two different sphere discretizations: An exact single-patch sphere with degenerate control points at the poles and an approximate discretization that consist of six patches with regular elements. The efficiency of the quadrature schemes is further demonstrated in boundary element analysis for Stokes flow, where steady problems with rotating and translating curved objects are investigated in convergence studies for both, mesh and quadrature refinement. Much higher convergence rates are observed for the proposed new schemes in comparison to classical schemes.

Study on Numerical Simulation Approach to Interaction Effects between Ships

To forecast the hydrodynamic interference between ships accurately and intuitively so as to study the ship maneuvering considering the effect of ship-ship interaction, it is significant for navigation safety in restricted water. This paper adopted an easy way to fully simulate interference between the ships. By taking advantage of the Computational Fluid Dynamics approach, the governing equations of viscous flow and the equations of motion of the ships are solved. The ship-ship motion induced by the hydrodynamic forces is numerically simulated so as to study the characteristic of the interaction effects between ships. To verify the validity of the present numerical methods, the numerical results about the influences on the hydrodynamic force and moment are illustrated and analyzed. The simulated results are acknowledged by the experienced ship operators.

Calculation model for lubrication of bearings with unconventional support surface profile and fusible shaft surface coating

This research work uses the true viscous fluid flow equation for the thin layer, the continuity equation, and the equation describing the radius of the fused shaft coated with a metal alloy taking into account the mechanical energy dissipation rate formula to find the asymptotic and the exact self-similar solution for the zero and first-order approximation for the radial sliding bearing with the support profile adapted to friction and lightalloy coating for incomplete filling of the working gap. We obtained the analytical dependencies for the fused surface radius of metal coating and the speed and pressure field. Besides, we determined the key operating parameters of the reviewed friction couple, the carrying capacity, and the friction force. The authors assess the impact of the parameters associated with the fusing of the coating of the friction-adapted support profile and the length of the loaded area on the carrying capacity and the friction force.

Symmetry actions and brackets for adjoint-symmetries. II: Physical examples

AbstractSymmetries and adjoint-symmetries are two fundamental (coordinate-free) structures of PDE systems. Recent work has developed several new algebraic aspects of adjoint-symmetries: three fundamental actions of symmetries on adjoint-symmetries; a Lie bracket on the set of adjoint-symmetries given by the range of a symmetry action; a generalised Noether (pre-symplectic) operator constructed from any non-variational adjoint-symmetry. These results are illustrated here by considering five examples of physically interesting nonlinear PDE systems – nonlinear reaction-diffusion equations, Navier-Stokes equations for compressible viscous fluid flow, surface-gravity water wave equations, coupled solitary wave equations and a nonlinear acoustic equation.

Effect of viscous flux flow on magnetization and ac susceptibility in type-II superconductors

In this paper, the effect of viscous flux flow on magnetization and ac susceptibility in type-II superconductors is studied. The magnetization curves and ac susceptibility was further investigated by numerically solving both the Bean model and the viscous flux flow equations. In the unified picture, the fundamental ac susceptibilities of the slab as a function of temperature under different viscous flux flow velocities is derived, and some unique characteristics are reproduced. The results indicate that the critical state model can be well described by the viscous flux flow equation, which has a significant impact on magnetization and ac susceptibility.

Generalized boundary dilatation flux on a flexible wall

In this paper, by applying theoretical method to the governing equations of compressible viscous flow, we derive the theoretical formula of the boundary dilatation flux (BDF) on a flexible wall, which generalizes the most recent work of Mao et al. (Acta Mechanica Sinica 38 (2022) 321583) for a stationary wall. Different boundary sources of dilatation are explicitly identified, revealing not only the boundary generation mechanisms of vortex sound and entropy sound, but also some additional sources due to the surface vorticity, surface angular velocity, surface acceleration and surface curvature. In particular, the generation mechanism of dilatation at boundary due to the coupled divergence terms is highlighted, namely, the product of the surface velocity divergence (∇∂B·U) and the vorticity-induced skin friction divergence (∇∂B·τω). The former is attributed to the surface flexibility while the latter characterizes the footprints of near-wall coherent structures. Therefore, by properly designing the surface velocity distribution, the dilatation generation at the boundary could be controlled for practical purpose in near-wall compressible viscous flows.

Random vortex dynamics via functional stochastic differential equations

In this paper, we present a novel, closed, three-dimensional random vortex dynamics system, which is equivalent to the Navier–Stokes equations for incompressible viscous fluid flows. The new random vortex dynamics system consists of a stochastic differential equation which is, in contrast with the two-dimensional random vortex dynamics equations, coupled to a finite-dimensional ordinary functional differential equation. This new random vortex system paves the way for devising new numerical schemes (random vortex methods) for solving three-dimensional incompressible fluid flow equations by Monte Carlo simulations. In order to derive the three-dimensional random vortex dynamics equations, we have developed two powerful tools: the first is the duality of the conditional distributions of a couple of Taylor diffusions, which provides a path space version of integration by parts; the second is a forward type Feynman–Kac formula representing solutions to nonlinear parabolic equations in terms of functional integration. These technical tools and the underlying ideas are likely to be useful in treating other nonlinear problems.

An efficient explicit jump high-order compact immersed interface approach for transient incompressible viscous flows

In the present work, we propose a novel hybrid explicit jump immersed interface approach in conjunction with a higher-order compact scheme for simulating transient complex flows governed by the streamfunction-vorticity (ψ-ζ) formulation of the Navier–Stokes (N–S) equations for incompressible viscous flows. A new strategy has been adopted for the jump conditions at the irregular points across the interface using Lagrangian interpolation on a Cartesian grid. This approach, which starts with the discretization of parabolic equations with discontinuities in the solutions, source terms, and the coefficients across the interface, can easily be accommodated into simulating flow past bluff bodies immersed in the flow. The superiority of the approach is reflected by the reduced magnitude and faster decay of the errors in comparison to other existing methods. It is seen to handle several fluid flow problems having practical implications in the real world very efficiently, including flows involving multiple and moving bodies. This includes the flow past a stationary circular and a twenty-four edge cactus cylinder, flows past two tandem cylinders, where in one situation both are fixed and in another, one of them is oscillating transversely with variable amplitude in time. To the best of our knowledge, the last two examples have been tackled for the first time by such an approach employing the ψ-ζ formulation in finite difference set-up. The extreme closeness of our computed solutions with the existing numerical and experimental results exemplifies the accuracy and the robustness of the proposed approach.

Sparse Polynomial Approximations for Affine Parametric Saddle Point Problems

In this work we study convergence properties of sparse polynomial approximations for a class of affine parametric saddle point problems, including the Stokes equations for viscous incompressible flow, mixed formulation of diffusion equations for groundwater flow, time-harmonic Maxwell equations for electromagnetics, etc. Due to the lack of knowledge or intrinsic randomness, the (viscosity, diffusivity, permeability, permittivity, etc.) coefficients of such problems are uncertain and can often be represented or approximated by high- or countably infinite-dimensional random parameters equipped with suitable probability distributions, and the coefficients affinely depend on a series of either globally or locally supported basis functions, e.g., Karhunen–Loève expansion, piecewise polynomials, or adaptive wavelet approximations. We consider sparse polynomial approximations of the parametric solutions, in particular sparse Taylor approximations, and study their convergence properties for these parametric problems. Under suitable sparsity assumptions on the parametrization of the random coefficients, we show the algebraic convergence rates O(N−r) for the sparse polynomial approximations of the parametric solutions based on the results for affine parametric elliptic PDEs (Cohen, A. et al.: Anal. Appl. 9, 11–47, 2011), (Bachmayr, M., et al.: ESAIM Math. Model. Numer. Anal. 51, 321–339, 2017), (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015), (Chkifa, A., et al.: J. Math. Pures Appl. 103, 400–428, 2015), (Chkifa, A., et al.: ESAIM Math. Model. Numer. Anal. 47, 253–280, 2013), (Cohen, A., Migliorati, G.: Contemp. Comput. Math., 233–282, 2018), with the rate r depending only on a sparsity parameter in the parametrization, not on the number of active parameter dimensions or the number of polynomial terms N. We note that parametric saddle point problems were considered in (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015, Section 2.2) with the anticipation that the same results on the approximation of the solution map obtained for elliptic PDEs can be extended to more general saddle point problems. In this paper, we consider a general formulation of saddle point problems, different from that presented in (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015, Section 2.2), and obtain convergence rates for the two variables, e.g., velocity and pressure in Stokes equations, which are different for the case of locally supported basis functions.

RANS-based numerical simulation of shock wave/turbulent boundary layer interaction induced by a blunted fin normal to a flat plate

Results of refined RANS-based numerical simulation of shock wave/turbulent boundary layer interaction induced by a blunted fin mounted on a flat plate are presented. The RANS equations for compressible viscous gas flow closed by a (varied) eddy-viscosity turbulence model were solved using the finite volume method implemented in the unstructured-grid parallelized in-house code SINF/Flag-S developed at the SPbPU. Detailed numerical data concerning time-averaged characteristics of the separation region and the shock wave/turbulent boundary layer interaction are acquired, including identification of the horse-shoe vortices and their influence on the pressure and shear stress distributions on the streamlined surfaces. A thorough comparison with measurements presented in literature were conducted. Methodical data on the influence of grid resolution and turbulence model on the predicted structure of the viscous-inviscid interaction are obtained and recommendations for achieving a high-quality resolution of the viscous effects and the complex shock-wave pattern of the flow are worked out.

Theoretical and experimental investigation on the seismic performance of a novel variable-damping viscous fluid damper

This paper presents a novel three-stage variable-damping-coefficient viscous fluid damper (VCVFD) and investigates its effectiveness in mitigating the seismic structural response. The concept and design of the displacement-dependent VCVFD were based on the equations for one-dimensional viscous flow in the damper orifice. A series of performance tests on VCVFDs were conducted to investigate the mechanical behaviour of the dampers with different design parameters under various loading conditions. The damping force increases more significantly than in the conventional viscous fluid damper (VFD) when the operating displacement is larger, and the expected effect of variable damping is achieved. A smaller gap height is more beneficial to energy dissipation in the VCVFD. Moreover, the hysteresis loop of the VCVFD was consistent with the theoretical results. To validate the effectiveness of the VCVFD in structural control, shaking table tests of two-storey steel frames equipped with VCVFDs under different seismic intensities were conducted. VCVFD can slightly increase the natural frequencies of the steel frame and significantly increase the structural damping ratio. The effectiveness of the VCVFD in mitigating the structural acceleration under low earthquake intensities was not prominent, but as the earthquake intensity increased, the acceleration amplification factor decreased. The added damping contributed by VCVFDs can significantly reduce the displacement response. It is concluded that with reasonable damping structural design, the VCVFD can achieve an almost similar mitigation effect as VFD in low-intensity earthquakes, whereas the VCVFD can provide higher damping under high-intensity earthquakes.

Enhanced anisotropic block-based adaptive mesh refinement for three-dimensional inviscid and viscous compressible flows

A parallel anisotropic block-based adaptive mesh refinement (AMR) algorithm is proposed for the prediction of physically complex flow problems having disparate spatial and temporal scales and exhibiting highly anisotropic features on three-dimensional multi-block body-fitted hexahedral meshes with non-uniform grid blocks. The proposed AMR scheme makes use of a binary tree hierarchical data structure to permit anisotropic refinement of grid blocks in a preferred coordinate direction as dictated by appropriately selected physics-based refinement criteria. The anisotropic coarsening of the grid blocks in a manner that is independent of the refinement history allows the mesh to rapidly re-adapt for evolving unsteady flow applications. Moreover, the proposed anisotropic AMR scheme adopts a non-uniform representation of the cells within each block by directly using the neighboring cells as the ghost cells, even at a grid resolution change. This affords a number of computational advantages especially related to evaluating the solution in the ghost cells as well as to ensuring flux conservation at block interfaces. A modified upwind-based finite-volume spatial discretization scheme is applied in conjunction with the AMR scheme to the solution of Euler and Navier-Stokes equations for inviscid and viscous compressible gaseous flows. Steady-state and time-varying flow problems are considered on anisotropic adapted meshes. The potential flexibility and efficiency of the proposed enhanced anisotropic AMR scheme are demonstrated for the simulation of a number of representative flows of varying complexity.

Near-wall Lamb vector and its temporal–spatial evolution in the viscous sublayer of wall-bounded flows

This paper gives the relation between the Lamb vector and fundamental surface quantities in the vicinity of a no-slip flat wall using the Taylor-series expansion solution of the Navier–Stokes equations for incompressible viscous flow. The wall-normal component of the Lamb vector is dominated by the boundary enstrophy at the first order and the boundary enstrophy flux at the second order. The tangential Lamb vector is contributed by the terms related to both the boundary vorticity divergence and the skin friction divergence. Then, the derived relation is validated in the three-dimensional nonorthogonal Hiemenz flow (an attachment line flow) and a single-phase turbulent channel flow simulated using the lattice Boltzmann method. For this Hiemenz flow where the boundary vorticity divergence vanishes, the skin friction divergence-related term dominates the distribution of the near-wall tangential Lamb vector. In the turbulent channel flow, both the skin friction divergence and boundary vorticity divergence-related terms have significant contributions to the streamwise component of the Lamb vector, which are associated with the strong wall-normal velocity events (SWNVEs) in the viscous sublayer. In contrast, the boundary vorticity divergence-related terms have the main contribution to the spanwise component of the Lamb vector. Furthermore, the temporal–spatial evolution of the kinetic energy of the Lamb vector (half of the inner product of the Lamb vector denoted by J) is studied. In the turbulent channel flow, the SWNVEs are the main contributors to the temporal–spatial evolution of J in the viscous sublayer. This evolution is dominated by the viscous dissipation effect due to the gradient of the Lamb vector and the coupling effect between a viscous source term and the Lamb vector. The relations presented in this paper could be useful in understanding the physical mechanisms of the initial formation and evolution of the Lamb vector in the viscous sublayer of wall-bounded turbulence.

Regions of Reduced Total Enthalpy in the Near Wake of a Body in a Viscous Gas Flow

—The influence of the body shape on gas cooling (decrease in the stagnation temperature) in regions of reduced total enthalpy in the wake is studied. The problem is simulated by numerically solving the Navier–Stokes equations for a two-dimensional viscous perfect gas flow around thermally insulated bodies at the Reynolds number {text{Re}} = 1000 and the Mach number {{{text{M}}}_{infty }} = 0.4. Elliptic-cross-section cylinders and a pair of side-by-side circular cylinders located across the flow are considered. The accuracy of the results of simplified models for describing the phenomenon based on the velocity field in the developed wake is discussed. Some examples that demonstrate that neglecting the total enthalpy redistribution during the vortex formation process in such models can lead to considerable deviations from the data of numerical calculations in vortex cores are given.

Incompressible limit on thin domains with periodic boundary condition

We consider convergence to the incompressible Navier–Stokes equations/Euler equations for compressible viscous flows on periodic thin domain with general initial data. Compared with Caggio et al (2020 Nonlinearity 33 840–863), where similar problems are considered in the whole space, we have to deal with the persistence of acoustic waves. We prove that the weak solutions in the 3D domain converge weakly/strongly to the solutions of the 2D incompressible Navier–Stokes equations/Euler equations when the Mach number ɛ goes to 0 as well as the thickness δ/the viscosity goes to 0.

An Integral of the Two-Dimensional Stationary Viscous Fluid Flow Equations

This study demonstrates that the equations for a two-dimensional steady flow of a viscous fluid contain an integral of motion, i.e., they have some function whose gradient is zero in the solutions of hydrodynamic equations A system of partial differential equations has been derived to calculate this function. It has been established that, at a certain integral of motion, the calculation of hydrodynamic fields is reduced to the solution of a system consisting of three first-order partial differential equations with a characteristic manifold.

A comparative study on wave-deck interactions of T-type and box girder decks under regular waves

Extreme wave forces have damaged massive coastal bridges in the past decades. To the best knowledge of the authors, a direct comparison of dynamic wave-deck interactions between T-type and Box girder decks under the same wave conditions has not been studied. This paper establishes a numerical model solving the Navier-Stokes equations for viscous and incompressible free surface flows, which is verified by existing experimental data. A total of 54 numerical experiments based on this model are then carried out to investigate the dynamic responses of two bridge decks under various wave conditions, and the influences of hydrodynamic parameters on the wave-deck interaction are studied. Two peak values are found in the curve of lateral displacement versus dimensionless submersion depth for the T-type girder deck, while one peak value is found in the curve for the Box girder deck. The relations between lateral displacement and period for two decks are both uncertain, which highly depends on the submersion depth and girder type. The lateral displacement is positively correlated with the wave height for two bridge decks, which is only valid when the horizontal wave force in the specified wave height is large enough to overcome the frictional force. This study provides researchers an insight into the difference of instantaneous lateral displacement between two most commonly used cross-section shapes of bridge deck, and helps to further understand the fragility of T-type and Box girder decks under extreme waves.