On pressure-driven Hele–Shaw flow of power-law fluids

The thrust vector control of a rocket engine by disturbing the supersonic flow in its nozzle is used for missile development for various purposes in different countries. Disturbance of the supersonic flow in the jet engine nozzle can be caused by various obstacles on the nozzle wall: solid obstacle, liquid or gas jet, combinations of solid obstacle with injected jets. The simplest and most effective way to create a disturbance is to disturb it by setting a solid cylindrical obstacle on the nozzle wall. The high efficiency is explained by the lack of the working fluid consumption on board the aircraft to create a control force, or its minimum amount necessary to protect the obstacle from the high-temperature oncoming gas flow in the rocket engine nozzle. This paper presents the study results of gas flow simulation with cylindrical obstacle perturbation on the wall of the Laval rocket engine nozzle in its subsonic and supersonic parts. The optimal placement in the nozzle is determined to obtain the maximum lateral control force. As a result of research, it was found that the perturbation of a supersonic flow in a rocket engine nozzle by a cylindrical obstacle has practically the same character when its position changes along the length of the nozzle. In the subsonic part of the nozzle in the median plane, the perturbed pressure on the wall has a positive sign, and on the obstacle wall its sign-alternating. When an obstacle is in the subsonic part of the nozzle, the integral value of the lateral force is negative in comparison with positive for the supersonic part.

In this paper, we report new insight into a symmetry-breaking phenomenon that occurs for turbulent flow in periodic porous media composed of cylindrical solid obstacles with circular cross section. We have used large eddy simulation to investigate the symmetry-breaking phenomenon by varying the porosity (0.57–0.99) and the pore scale Reynolds number (37–1000). Asymmetrical flow distribution is observed in the intermediate porosity flow regime for values of porosities between 0.8 and 0.9, which is characterized by the formation of alternating low and high velocity flow channels above and below the solid obstacles. These channels are parallel to the direction of the flow. Correspondingly, the microscale vortices formed behind the solid obstacles exhibit a bias in the shedding direction. The transition from symmetric to asymmetric flow occurs in between the Reynolds numbers of 37 (laminar) and 100 (turbulent). A Hopf bifurcation resulting in unsteady oscillatory laminar flow marks the origin of a secondary flow instability arising from the interaction of the shear layers around the solid obstacle. When turbulence emerges, stochastic phase difference in the vortex wake oscillations caused by the secondary flow instability results in flow symmetry breaking. We note that symmetry breaking does not occur for cylindrical solid obstacles with square cross section due to the presence of sharp vertices in the solid obstacle surface. At the macroscale level, symmetry-breaking results in residual transverse drag force components acting on the solid obstacle surfaces. Symmetry-breaking promotes attached flow on the solid obstacle surface, which is potentially beneficial for improving transport properties at the solid obstacle surface such as convection heat flux.

Hele-Shaw cells are used to model creeping flow through porous media (where Darcy's law is valid). The effects of inertia on flow about obstructions in a Hele-Shaw cell can be calculated by a perturbation method if one can determine a solution to Laplace's equation. Results of a computer solution for flow about circular, square and elliptical obstructions are presented These results show that for a modified presented These results show that for a modified Reynolds number of less than 1, the inertia terms are small; and for values of less than 3, the average streamline predicts the ideal flow. Therefore, the analogy might be used for studying flow in porous media up to a modified Reynolds number of at least 3. Introduction The nature of fluid flow in porous media is of interest in the fields of soil mechanics, ground water flow, petroleum production, filtration and flow, in packed beds. Because it is very difficult to study the phenomenological behavior of flow in porous media, homologs and analogs are used to study flow characteristics. A Hele-Shaw model, made of two closely spaced plates - usually glass - is often used as an analogy to two-dimensional flow in porous media. Hele-Shaw showed experimentally that the streamline configuration for creeping flow around an obstacle located between two closely spaced parallel plates is the same as for two-dimensional parallel plates is the same as for two-dimensional ideal flow about the same obstacle. Stokes verified these observations mathematically. The usual equation of motion for flow in porous media is Darcy's law. The form of the mathematical statement of Darcy's law is identical, within a multiplicative constant, to the expression for the average velocity over the place gap in the plane of a Hele-Shaw model. These models may be used to describe flow in both homogeneous and heterogeneous porous media.

Some Results on the Navier–Stokes Equations in Thin 3D Domains

The sensitivity of regional air quality model to various lateral and top boundary conditions is studied at 2 scales: a 60 km domain covering the whole USA and a 12 km domain over northeastern USA. Three global models (MOZART‐NCAR, MOZART‐GFDL and RAQMS) are used to drive the STEM‐2K3 regional model with time‐varied lateral and top boundary conditions (BCs). The regional simulations with different global BCs are examined using ICARTT aircraft measurements performed in the summer of 2004, and the simulations are shown to be sensitive to the boundary conditions from the global models, especially for relatively long‐lived species, like CO and O3. Differences in the mean CO concentrations from three different global‐model boundary conditions are as large as 40 ppbv, and the effects of the BCs on CO are shown to be important throughout the troposphere, even near surface. Top boundary conditions show strong effect on O3 predictions above 4 km. Over certain model grids, the model's sensitivity to BCs is found to depend not only on the distance from the domain's top and lateral boundaries, downwind/upwind situation, but also on regional emissions and species properties. The near‐surface prediction over polluted area is usually not as sensitive to the variation of BCs, but to the magnitude of their background concentrations. We also test the sensitivity of model to temporal and spatial variations of the BCs by comparing the simulations with time‐varied BCs to the corresponding simulations with time‐mean and profile BCs. Removing the time variation of BCs leads to a significant bias on the variation prediction and sometime causes the bias in predicted mean values. The effect of model resolution on the BC sensitivity is also studied.

In the design of a building envelope, there is the issue of heat flow through the partitions. In the heat flow process, we distinguish steady and dynamic states in which heat fluxes need to be obtained as part of building physics calculations. This article describes the issue of determining the size of those heat fluxes. The search for the temperature field in a two-dimensional problem is common in building physics and heat exchange in general. Both numerical and analytical methods can be used to obtain a solution. Two methods were dealt with, the first of which was used to obtain the solution in the steady state and the other in the transient. In the steady state a method of initial functions, the basics of which were given by W.Z. Vlasov and A.Y. Lur’e was adopted. Originally MIF was used for analysis of the loads of a flat elastic medium. Since then it was used for solving concrete beams, plates and composite materials problems. Polynomial half-reverse solutions are used in the theory of a continuous medium. Here solutions were obtained by the direct method. As a result, polynomial forms of the considered temperature field were obtained. A Cartesian coordinate system and rectangular shape of the plate were assumed. The problem is governed by the Laplace equation in the steady state and Poisson in the transient state. Boundary conditions in the form of temperature (τ(x), t(y)) or/and flux (p(x), q(y)) can be provided. In the steady state the solution T(x, y) was assumed in the form of an infinite power series developed in relation to the variable y with coefficients Cn depending on x. The assumed solution was substituted into the Fourier equation and after expanding into the Taylor series the boundary condition for y = 0 and y = h was taken into account. From this condition the coefficient Cn can be calculated and, therefore, a closed solution for the temperature field in the plate.

The analysis of heat transfer problems can be highly complex due to factors such as temperature, position, and time. Most heat transfers are typically two-dimensional as conduction is often negligible in the third dimension. Two-dimensional heat conduction problems can be solved analytically or numerically. In steady-state conditions, the Laplace equation can be applied to solve two-dimensional heat conduction problems analytically, in which the separation of variables method is used to solve the Laplace equation under fixed boundary conditions to determine the temperature at a specific point. The Laplace equation plays a significant role in the solution of heat transfer problems, as it demonstrates the behavior of linear and non-linear equations in the computational fluid dynamics domain. Despite their inability to provide exact results at any point, numerical methods are superior to analytical methods when handling complex geometries with various boundary conditions. This project involves the development of a computational code using MATLAB to solve two-dimensional steady-state heat conduction problems using Gauss-Seidel iterations. Comparing analytical solutions from Excel with numerical solutions from MATLAB and ANSYS, specifically the developed MATLAB code, revealed an accuracy level of 99.902% for the Laplace equation. An analysis of the produced code from MATLAB found that it could solve two-dimensional steady-state heat conduction across different combinations of materials while allowing users to specify initial and boundary conditions to produce a contour plot similar to ANSYS.

The focus of this paper is a numerical simulation study of the flow dynamics in a periodic porous medium to analyse the physics of a symmetry-breaking phenomenon, which causes a deviation in the direction of the macroscale flow from that of the applied pressure gradient. The phenomenon is prominent in the range of porosity from 0.43 to 0.72 for circular solid obstacles. It is the result of the flow instabilities formed when the surface forces on the solid obstacles compete with the inertial force of the fluid flow in the turbulent regime. We report the origin and mechanism of the symmetry-breaking phenomenon in periodic porous media. Large-eddy simulation (LES) is used to simulate turbulent flow in a homogeneous porous medium consisting of a periodic, square lattice arrangement of cylindrical solid obstacles. Direct numerical simulation is used to simulate the transient stages during symmetry breakdown and also to validate the LES method. Quantitative and qualitative observations are made from the following approaches: (1) macroscale momentum budget and (2) two- and three-dimensional flow visualization. The phenomenon draws its roots from the amplification of a flow instability that emerges from the vortex shedding process. The symmetry-breaking phenomenon is a pitchfork bifurcation that can exhibit multiple modes depending on the local vortex shedding process. The phenomenon is observed to be sensitive to the porosity, solid obstacle shape and Reynolds number. It is a source of macroscale turbulence anisotropy in porous media for symmetric solid-obstacle geometries. In the macroscale, the principal axis of the Reynolds stress tensor is not aligned with any of the geometric axes of symmetry, nor with the direction of flow. Thus, symmetry breaking in porous media involves unresolved flow physics that should be taken into consideration while modelling flow inhomogeneity in the macroscale.

Antiplane shear deformations of a cylindrical body, with a single displacement field parallel to the generators of the cylinder and independent of the axial coordinate, are one of the simplest classes of deformations that solids can undergo. They may be viewed as complementary to the more familiar plane deformations. Antiplane (or longitudinal) shear deformations have been the subject of the considerable recent interest in nonlinear elasticity theory for homogeneous isotropic solids. In contrast, for the linear theory of isotropic elasticity, such deformations are usually not extensively discussed. The purpose of the present paper is to demonstrate that for inhomogeneous anisotropic linearly elastic solids the antiplane shear problem does provide a particularly tractable and illuminating setting within which effects of anisotropy and inhomogeneity may be examined. We consider infinitesimal antiplane shear deformations of an inhomogeneous anisotropic linearly elastic cylinder subject to prescribed surface tractions on its lateral boundary whose only nonzero component is axial and which does not vary in the axial direction. In the absence of body forces, not all arbitrary anisotropic cylinders will sustain an antiplane shear deformation under such tractions. Necessary and sufficient conditions on the elastic moduli are obtained which do allow an antiplane shear. The resulting boundary value problems governing the axial displacement are formulated. The most general elastic symmetry consistent with an antiplane shear is described. There are at most 15 independent elastic coefficients associated with such a material. In general, there is a normal axial stress present, which can be written as a linear combination of the two dominant shear stresses. For a material with the cylindrical cross-section a plane of elastic symmetry (monoclinic, with 13 moduli), the normal stress is no longer present. For homogeneous materials, it is shown how the governing boundary value problem can be transformed to an equivalent isotropic problem for a transformed cross-sectional domain. Applications to the issue of assessing the influence of anisotropy and inhomogeneity on the decay of Saint-Venant end effects are described.

Turbulent flow in a homogeneous porous medium was investigated through the use of numerical methods by employing the Reynolds Averaged Navier-Stokes (RANS) modeling technique. The focus of our research was to study how microscopic vortices in porous media flow influence the heat transfer from the solid obstacles comprising the porous medium to the fluid. A Representative Elementary Volume (REV) with 4 × 4 cylindrical obstacles and periodic boundary conditions was used to represent the infinite porous medium structure. Our hypothesis is that the rate of heat transfer between the obstacle surface and the fluid (qavg) is strongly influenced by the size of the contact area between the vortices and the solid obstacles in the porous medium (Avc). This is because vortices are regions with low velocity that form an insulating layer on the surface of the obstacles. Factors such as the porosity (φ), Pore Scale Reynolds number (Rep), and obstacle shape of the porous medium were investigated. All three of these factors have different influences on the contact area Avc, and, by extension, the overall heat transfer rate qavg. Under the same Pore Scale Reynolds number (Rep), our results suggest that a higher overall heat transfer rate is exhibited for smaller contact areas between the vortices and the obstacle surface. Although the size of the contact area, Avc, is affected by Rep, the direct influence of Rep on the overall heat transfer rate qavg is much stronger, and exceeds the effect of Avc on qavg. The Pore Scale Reynolds number, Rep, and the mean Nusselt number, Num, have a seemingly logarithmic relationship.

The pattern formation of a radially growing interface in a Hele–Shaw cell is studied. In contrast to the previous studies based on the Young–Laplace equation, a boundary condition is employed that included the effect of the wetting layer of the displacing fluid in the cell. Under this boundary condition, a weakly nonlinear analysis is carried out and the time evolution of the interface is numerically calculated. These results, compared to those obtained in the previous studies, suggest that our model reflects the nonlinear features of the viscous fingering phenomena more precisely.

The spreading of a passive tracer that is convected back and forth inside a porous medium depends both on the random characteristics of the medium and on the presence of stagnation points. We single out the effect of the latter in the present study of hydrodynamic dispersion in the creeping (low Reynolds number) high P\'eclet number flow around the single stagnation point on a cylindrical obstacle in a Hele-Shaw cell [U. Oxaal, E. G. Flekko/y, and J. Feder, Phys. Rev. Lett. 72, 3514 (1994)]. Employing both experiments and lattice Boltzmann simulations we analyze the dispersive spreading of a single tracer line, which is initially perpendicular to the flow direction and then convected back and forth around the cylinder. The lattice Boltzmann model used is a modification of the recently introduced two-dimensional lattice bathnagar-Gross-Krook model for miscible fluid dynamics [E. G. Flekko/y, Phys. Rev. E 47, 4247 (1993)]. It includes the full three-dimensional viscous interaction in the Hele-Shaw cell, and, in the case of steady state flow, it allows for a freely tunable Reynolds number. The diffusive behavior of the system is explored extensively and excellent agreement between simulations and experiment is observed. A method to determine very small molecular diffusion coefficients D, which relies on the combination of results from experiment and simulation, is proposed. It is demonstrated that there is good agreement between the result of this method and independent measurement that are carried out in the present case of relatively large D values.

We revisit the radial viscous fingering problem in a Hele-Shaw cell, and consider the action of viscous stresses originated from velocity gradients normal to the fluid-fluid interface. The evolution of the interface during linear and weakly nonlinear stages is described analytically through a mode-coupling approach. We find that the introduction of normal stresses influences the stability and the ultimate morphology of the emerging patterns. Although at early stages normal stresses tend to stabilize the interface, they act to favor the development of tip-splitting phenomena at the weakly nonlinear regime. We have also verified that finger competition events are only significantly affected by normal stresses for circumstances involving the development of a large number of interfacial fingers.

A linear two-dimensional boundary value problem, that describes steady-state surface and internal waves due to the forward motion of a body in a fluid consisting of two superposed layers with different densities, is considered. The body is fully submerged and intersects the interface between the two layers. Two well-posed formulations of the problem are proposed in which, along with the Laplace equation, boundary conditions, coupling conditions on the interface, and conditions at infinity, a pair of supplementary conditions are imposed at the points where the body contour intersects the interface. In one of the well-posed formulations (where the differences between the horizontal momentum components are given at the intersection points), the existence of the unique solution is proved for all values of the parameters except for a certain (possibly empty) nowhere dense set of values.

The removal of the background magnetic field is a critical step in generating phase images and quantitative susceptibility maps, which have recently been receiving increasing attention. Although it is known that the background field satisfies Laplace's equation, the boundary values of the background field for the region of interest have not been explicitly addressed in the existing methods, and they are not directly available from MRI measurements. In this paper, we assume simple boundary conditions and remove the background field by explicitly solving the boundary value problems of Laplace's or Poisson's equation. The proposed Laplacian boundary value (LBV) method for background field removal retains data near the boundary and is computationally efficient. Tests on a numerical phantom and an experimental phantom showed that LBV was more accurate than two existing methods.