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The pressure-driven displacement flow of a less viscous fluid initially occupying an axisymmetric pipe by a highly viscous fluid is investigated. The fluids consist of two species diffusing at different rates. The fluids are assumed to be Newtonian, incompressible with the same density, but of different viscosity modelled as an exponential function of the concentration of both the species. A parametric study investigating the effects of diffusivity ratio, log-mobility ratios of the slower and faster diffusing species and Reynolds number on the flow dynamics is conducted. Our results demonstrate the presence of instability patterns due to double-diffusive effect in situations when a less viscous fluid displaces a highly viscous fluid. These instabilities are qualitatively different from those observed in planar channel. The intensity of these instabilities increases with increasing the values of diffusivity ratio. It is demonstrated that a highly viscous stenosis region is created near the entrance of the pipe due to double-diffusive effect, providing a favourable condition to start the instability. In addition to this, because of double-diffusive effect locally at some portion of the pipe, the less viscous fluid becomes the displacing fluid, which promotes the development of instability.

Structural properties of regions of uniform streamwise momentum in turbulent boundary layers are examined using experimental databases obtained from particle image velocimetry. This investigation employs a large range of Reynolds numbers, spanning more than an order of magnitude ($Re_{{\it\tau}}=10^{3}{-}10^{4}$), enabling us to provide a detailed description of uniform momentum zones as a function of Reynolds number. Our analysis starts by examining the identification criterion used by Adrian et al. (J. Fluid Mech., vol. 422, 2000, pp. 1–54) to report the presence of uniform momentum zones in turbulent boundary layers. This criterion is then applied to show that a zonal-like structural arrangement is prevalent in all datasets examined, emphasising its importance in the structural organisation. Streamwise velocity fluctuations within the zones are observed to be small but they are bounded by distinct step changes in streamwise momentum which indicate that shear layers of intense vorticity separate each zone. A log-linear increase in the number of these zones with increasing Reynolds number is revealed, together with an increase in the thicknesses of zones with increasing distance from the wall. These results support a hierarchical length-scale distribution of coherent structures, which generate zonal-like organisation within turbulent boundary layers. Interpretation of these findings is aided by employing synthetic velocity fields generated using a model based on the attached eddy hypothesis, which is described in the work of Perry and co-workers. Comparisons between the model and experimental results show that a hierarchy of self-similar structures leads to population densities and length-scale distributions of uniform momentum zones that closely adhere to those observed experimentally in this study.

The boundary value problem for the stationary magnetohydrodynamics model of a viscous heatconducting fluid considered under inhomogeneous mixed boundary conditions for an electromagnetic field and the temperature and Dirichlet condition for the velocity is investigated. This problem describes the flow of an electricaland heat-conducting liquid in a bounded three-dimensional domain the boundary of which consists of several parts with different thermoand electrophysical properties. Sufficient conditions imposed on the initial data to provide for global solvability of the problem and local uniqueness of its solution are established.

When modeling gas–solid fluidized beds with Eulerian–Eulerian approach, the predicted hydrodynamics is significantly affected by the solid phase wall boundary condition. The specularity coefficient φ in Johnson and Jackson wall boundary condition ranges from 0 for perfect specular collisions to 1 for perfectly diffuse collisions. Normally, it is reported that no-slip wall boundary condition is obtained for φ = 1. However, in most reports without applying Johnson and Jackson wall boundary condition, the no-slip wall boundary condition was also claimed to be used for the solid phase analogous to the gas phase. Therefore, the dynamic segregation process of binary particles was simulated to investigate whether φ = 1 is equivalent with no-slip wall boundary condition. The results reveal the significant differences between φ = 1 and no-slip wall boundary condition, which indicate that the no-slip wall boundary condition may not be suitable for describing φ = 1 wall boundary condition.

We present a new energy-stable open boundary condition, and an associated numerical algorithm, for simulating incompressible flows with outflow/open boundaries. This open boundary condition ensures the energy stability of the system, even when strong vortices or backflows occur at the outflow boundary. Under certain situations it can be reduced to a form that can be analogized to the usual convective boundary condition. One prominent feature of this boundary condition is that it provides a control over the velocity on the outflow/open boundary. This is not available with the other energy-stable open boundary conditions from previous works. Our numerical algorithm treats the proposed open boundary condition based on a rotational velocity-correction type strategy. It gives rise to a Robin-type condition for the discrete pressure and a Robin-type condition for the discrete velocity on the outflow/open boundary, respectively at the pressure and the velocity sub-steps. We present extensive numerical experiments on a canonical wake flow and a jet flow in open domain to test the effectiveness and performance of the method developed herein. Simulation results are compared with the experimental data as well as with other previous simulations to demonstrate the accuracy of the current method. Long-time simulations are performed for a range of Reynolds numbers, at which strong vortices and backflows occur at the outflow/open boundaries. The results show that our method is effective in overcoming the backflow instability, and that it allows for the vortices to discharge from the domain in a fairly natural fashion even at high Reynolds numbers., Comment: 38 pages, 11 figures, 4 tables

In this paper, the free convective flow of magnetohydrodynamic fluid through a channel with time periodic boundary condition is investigated by taking the effects of Joule dissipation into consideration. Based on simplifying assumptions, the coupled governing equations are reduced to a set of nonlinear boundary valued problem. Approximate solutions are obtained by using semi-analytical Adomian decomposition method. The effect of pertinent parameters on the fluid velocity, temperature distribution, Nusselt number and skin friction are presented graphically and discussed. The result of the computation shows that an increase in the magnetic field intensity has significant influence on the fluid flow.

This article deals with artificial boundaries which you encounter when a large spatial domain is confined to a smaller computational domain. Such an artificial boundary condition should not preferably interact with the fluid at all. Standard boundary conditions, e.g., a pressure or velocity condition, result in unphysical reflections. So far, existing artificial boundary conditions for the lattice Boltzmann method (LBM) are transferred from macroscopic formulations.In this work we propose novel discrete artificial boundary conditions (ABCs) which are tailored on the LBM's mesoscopic level. They are derived directly for the chosen LBM with the aim of higher accuracy. We describe the idea of discrete ABCs in a three velocity (D1Q3) model governing the Navier-Stokes equations in one dimension. Numerical results finally demonstrate the superiority of our new boundary condition in terms of accuracy compared to previously used ABCs.

In this paper, a solid–gas–liquid phase double slip model for flat plate was established by correcting the boundary conditions with the boundary layer slip velocity. According to Karman boundary layer momentum theorem, the velocity distributions both in the solid–gas boundary layer and in the gas–liquid boundary layer were deduced. The boundary layer thickness, displacement thickness and momentum loss thickness, as well as friction coefficient were also obtained, which offer an understanding for the laminar boundary layer characteristic. The theoretical results indicate that the flow field velocity near the solid surface can be increased by the micro-bubbles on the solid surface. In the case of low Reynolds number, the boundary slip phenomenon occurs in the bubble surface boundary layer, which can lead to the resistance reduction effect. The bubble with higher height offers a longer boundary ship length, and also a more significant resistance reduction. Experimental research was also made by establishing a holographic optical experiment system and the results show that there is higher flow velocity distribution in the bubble surface boundary layer.

The bounce-back boundary condition in the lattice Boltzmann method distorts curved or inclined boundaries by forcing them to conform to a rectangular grid. This paper proposes a modification that reduces the effect of this discretization on the fluid flow. The modification takes the form of the addition of a type of node that is neither solid nor fluid, called the "sticky node". Sticky nodes are used in all cells that contain both fluid and solid. They are treated like fluid nodes with modified viscosity, body force, and velocity calculation. The method is applied to the LBGK model on a D2Q9 grid, and the accuracy of the method is evaluated using several test cases. Decreased discretization artifacts and decreased sensitivity to grid alignment are demonstrated, compared to the standard link bounce-back boundary condition. The method is computationally efficient, local, and demonstrates good numerical stability.

The main objective of this paper is the derivation and validation of a free surface Neumann boundary condition for the advection-diffusion lattice Boltzmann method. Most literature boundary conditions are applied on straight walls and sometimes on curved geometries or fixed free surfaces, but dynamic free surfaces, especially with fluid motion in normal direction, are hardly addressed. A Chapman-Enskog Expansion is the basis for the derivation of the advection-diffusion equation using the advection-diffusion lattice Boltzmann method and the BGK collision operator. For this numerical scheme, a free surface Neumann boundary condition with no flux in normal direction to the free surface is derived. Finally, the boundary condition is validated in different static and dynamic test scenarios, including a detailed view on the conservation of the diffusive scalar, the normal and tangential flux components to the free surface and the accuracy. The validation scenarios reveal the superiority of the new approach to the compared literature schemes, especially for arbitrary fluid motion.

A three dimensional axisymmetric stagnation flow of hybrid nanofluid impinges on a moving plate with different slip coefficients in two orthogonal directions. On the solid boundary, no slip condition is replaced through the partial slip condition. Such anisotropic slip occurs in geometrically striated surface and super hydrophobic stripes. Using the experimental values, we have discussed the hybrid nanofluid theoretically for boundary layer flow due to stagnation point flow. Suitable similarity transformation, transform the Navier–Stokes equations to set of 12th order nonlinear ordinary differential equations. The problem is solved through asymptotic analysis for large slip and numerically integration. Effects of physical parameters are also discussed in detail.

We consider steady axisymmetric no-swirl viscous flows in a half space driven by a concentrated force. A method of finding physically meaningful similarity solutions of the Navier–Stokes equations with finite values of mass and momentum fluxes satisfying the no-slip boundary condition is developed. A new one-parameter set of analytic similarity solutions that describes impinging or emerging jets depending on the value of the parameter is found. A unique emerging-jet solution can be found for an arbitrary value of the Reynolds number. The accomplished analysis of momentum balance has resulted in an algorithm to estimate the best parameter of the similarity solution corresponding to the given physical characteristics of the jet source.

Present work concerns the pulsatile blood flow of two-fluid model through porous blood vessels under the effect of radially varying viscosity. Blood is modeled as two-phase fluid model consisting a core region by non-Newtonian (Herschel-Bulkley) fluid and a plasma region modeled as Newtonian fluid. No slip condition has been used on wall and pressure gradient is taken as periodic function of time. Up to first order approximate solutions of governing equations are obtained using perturbation approach. A comparative analysis for relative change in flow resistance between our model and previously studied single and two-fluid models without porous layer near wall has also been done. The wall of the blood vessel is composed by a thin Brinkman (porous) layer. The stress jump condition has been imposed on fluid-porous interface. Analytical expressions for the velocity profile, flow rate, wall shear stress and flow resistance have been obtained for different regions and the effect of plasma layer thickness, varying viscosity, yield stress, permeability and viscosity ratio parameter on the flow variables are pictorially discussed. It is perceived that values of flow rate for two-fluid model with porous region near wall is higher in comparison to two-fluid model without porous region near wall. Present study reveals a significant impact of glycocalyx layer on blood flow through blood vessels with a porous layer near wall.

This paper is concerned with a regularity criterion based on vorticity direction for Navier–Stokes equations in a three-dimensional bounded domain under the no-slip boundary condition. It asserts that if the vorticity direction is uniformly continuous in space uniformly in time, there is no type I blow-up. A similar result has been proved for a half space by Y. Maekawa and the first and the last authors (2014). The result of this paper is its natural but non-trivial extension based on L ∞ theory of the Stokes and the Navier–Stokes equations recently developed by K. Abe and the first author.

Lattice Boltzmann Method (LBM) is a statistical approach for simulating fluid flow in porous media. LBM is a method based on the kinetic theory of gases and thus is a very powerful tool for simulating such flows. Fluid flow in a nanochannel containing porous media was simulated in this study. The effect of porosity was considered as loss terms in the momentum equations. The flow domain was solved at a high Knudsen number (Kn) of 10 using a new equation for relaxation time to predict increased collisions of molecules to the wall at high Knudsen numbers. No slip condition cannot be applied at high Knudsen numbers due to wall slip. First, the results were calculated at low Knudsen numbers and compared with existing results. Then the results were calculated at high Knudsen numbers. Flow characteristics at different porosities were examined at different inlet to outlet pressure ratios and Knudsen numbers. According to the results, gas permeability and wall slip velocity increased with increasing the Knudsen number, but the maximum axial velocity at the channel center decreased. The wall slip velocity, however, increased by decreasing porosity (increasing barriers). With increasing the Knudsen number at a constant pressure ratio and a certain porosity, the volumetric flow rate decreased at Knudsen numbers below 0.1 and then began to increase at Kn>0.1. The volumetric flow rate increased with increasing porosity. Moreover, the Darcy coefficient increased with increasing the Knudsen number. With an increase in the average pressure along the channel, both permeability and Darcy coefficient decreased. The Darcy coefficient in the slip flow regime varied linearly, but showed a second-order behavior in the transient flow. The results were presented as diagrams including permeability versus Knudsen number, permeability versus porosity and volumetric flow rate as a function of Knudsen number.

Being extremely soft, brain tissue is among the most challenging materials to be mechanically quantified. Despite recent advances in mechanical testing of ultra-soft matters, there still exists a need for robust procedures to analyze their behavior at large deformation. In this paper, it is shown how failing to taking into account the precise boundary conditions can result in substantial variation from the “assumed” ideal behavior, even for the case of simple loading conditions such as the uniaxial mode. For an accurate analysis, the mathematical modeling is combined with the finite element simulation to interpret the mechanical behavior of the brain tissue based on the comprehensive experiments conducted by Budday et al. (2017). It is demonstrated herein that only an Ogden hyperelastic model with both negative and positive nonlinearity constants can predict the mechanical behavior of the brain tissue in tension and compression, and the tension-compression asymmetry might arise from the difference in compressibility behavior in tension and compression. This hypothesis is utilized for modeling the mechanical behavior of the brain tissue in uniaxial loading condition and exhibits excellent agreement with the experiments. This study also provides a comprehensive explanation for nonlinear analysis of soft matters, in general, and the brain tissue, in particular, with thoroughly describing the concept of hyperelasticity and modeling incompressible or compressible behaviors utilizing the Ogden strain energy function.

The prime aim of the present investigation is to capture the mechanism of Navier’s velocity slip and convective thermal boundary condition on the flow of MHD viscoelastic fluid over a stretching surface. Additionally, the analysis also includes the effect of natural convection and thermal radiation. The governing boundary layer equations are transformed into a set of highly non-linear ordinary differential equations using suitable similarity transforms. Galerkin Finite element method is used to solve this boundary value problem. Effects of pertinent flow parameters on the Skin friction coefficient, Nusselt number, velocity and temperature, are described graphically. Numerical results obtained in this paper are compared with earlier published results and are found to be in excellent agreement. Significant findings of the present article are the conjugate effect of partial velocity slip and viscoelasticity of the fluid on Skin friction, Nusselt number, velocity and temperature. The analysis shows that presence of partial velocity slip changes the behavior of Nusselt number and skin friction coefficients significantly in comparison to the no slip condition. The present problem has potential to serve as a model for many industrial processes such as cooling and/or drying of paper and textile, rolling sheet drawn from a die, manufacturing of polymeric sheets, sheet glass and crystalline materials, etc.

Immersed boundary methods are frequently employed for simulating flows at low Reynolds numbers or for applications where viscous boundary layer effects can be neglected. The primary shortcoming of Cartesian mesh immersed boundary methods is the inability of efficiently resolving thin turbulent boundary layers in high-Reynolds number flow application. The inefficiency of resolving the thin boundary is associated with the use of constant aspect ratio Cartesian grid cells. Conventional CFD approaches can efficiently resolve the large wall normal gradients by utilizing large aspect ratio cells near the wall. This paper presents different approaches for immersed boundary methods to account for the viscous boundary layer interaction with the flow-field away from the walls. Different wall modeling approaches proposed in previous research studies are addressed and compared to a new integral boundary layer based approach. In contrast to common wall-modeling approaches that usually only utilize local flow information, the integral boundary layer based approach keeps the streamwise history of the boundary layer. This allows the method to remain effective at much larger y+ values than local wall modeling approaches. After a theoretical discussion of the different approaches, the method is applied to increasingly more challenging flow fields including fully attached, separated, and shock-induced separated (laminar and turbulent) flows.

The lid driven cavity problem has been studied numerically. The present work is focused on reviewing the fluid flow patterns for the lid driven cavity problem. The left, right and bottom walls are maintained at no slip boundary condition and the top wall is moved in the right direction with uniform velocity (u = 1.0). The governing equations are processed by the mass and momentum equations and solved by using the radial basis function (RBF) method. The governing equations are expressed using a primitive variable formulation. The method was based on implicit Euler scheme for temporal discretization and RBF method for spatial discretization. The effects of the Reynolds number on the fluid flow in the square cavity are investigated. Stream functions and pressure contours are presented for various Reynolds numbers, such as 102, 4x102 and 103. Comparing the numerical results obtained using the present method with other numerical methods from the literatures shows very good agreement.

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.

We present homogenization of the viscous incompressible porous media flows under stress boundary conditions at the outer boundary. In addition to Darcy’s law describing filtration in the interior of the porous medium, we derive rigorously the effective pressure boundary condition at the outer boundary. It is a linear combination of the outside pressure and the applied shear stress. We use the two-scale convergence in the sense of boundary layers, introduced by Allaire and Conca (1997) to obtain the boundary layer structure next to the outer boundary. The approach allows establishing the strong L 2 -convergence of the velocity corrector and identification of the effective boundary velocity slip jump. The theoretical results are confirmed through numerical experiments.

We have found that only a few attempts are available in the literature relatively to the tangent hyperbolic fluid flow induced by stretching cylindrical surfaces. In particular, temperature and concentration stratification effects have not been investigated until now with respect to the tangent hyperbolic fluid model. Therefore, we have considered the tangent hyperbolic fluid flow induced by an acutely inclined cylindrical surface in the presence of both temperature and concentration stratification effects. To be more specific, the fluid flow is attained with the no slip condition, which implies that the bulk motion of the fluid particles is the same as the stretching velocity of a cylindrical surface. Additionally, the flow field situation is manifested with heat generation, mixed convection and chemical reaction effects. The flow partial differential equations give a complete description of the present problem. Therefore, to trace out the solution, a set of suitable transformations is introduced to convert these equations into ordinary differential equations. In addition, a self-coded computational algorithm is executed to inspect the numerical solution of these reduced equations. The effect logs of the involved parameters are provided graphically. Furthermore, the variations of the physical quantities are examined and given with the aid of tables. It is observed that the fluid temperature is a decreasing function of the thermal stratification parameter and a similar trend is noticed for the concentration via the solutal stratification parameter.

The development of secondary boundary layer for a steady entry of fluid flow in a horizontal curved pipe is analysed using the turbulent boundary layer approach. Outside the boundary layer (within the central core), the inviscid fluid motion is treated by using the Euler and the continuity equations, whereas inside the boundary layer, the Pohlhausen method is applied, assuming a one-seventh power law of velocity for solving the integrals of the Navier–Stokes equations. The growth of the secondary boundary layer is slow with an increase in azimuthal angle, while it is gradual along the outer pipe-wall with an increase in radial angle and becomes abruptly faster near the separation point. The wall shear stress decreases with azimuthal angle, whereas it increases with radial angle attaining a peak and then decreases to zero at the separation point along the outer pipe-wall.

The asymptotic analysis of boundary layers of random thinness and of higher Reynolds number separating two interacting incompressible viscous fluid flows is described using Γ-convergence methods. An asymptotic interfacial contact law is derived, which involves the jumps of the velocity and of the pressures of the fluids through an ergodic coefficient.

A theoretical algorithm by united Lagrangian-Eulerian method for the problem in dealing with viscous fluid and a circular cylindrical shell is presented. In this approach, each material is described in its preferred reference frame. Fluid flows are given in Eulerian coordinates whereas the elastic circular cylindrical shell is treated in a Lagrangian framework. The fluid velocity in a two-dimensional uniform elastic circular cylindrical shell filled with viscous fluid is studied under the assumption of low Reynolds number. The coupling between the viscous fluid and the elastic circular cylindrical shell shows kinematic conditions at the shell surface. Also, the radial velocity and axial velocity of the fluid are discussed with the help of graphs.

In this paper, a method that combines the characteristic-based split finite element method (CBS-FEM) and the direct forcing immersed boundary (IB) method is proposed for the simulation of incompressible viscous flows. The structured triangular meshes without regarding the location of the physical boundary of the body is adopted to solve the flow, and the no-slip boundary condition is imposed on the interface. In order to improve the computational efficiency, a grid stretching strategy for the background structured triangular meshes is adopted. The obtained results agree very well with the previous numerical and experimental data. The order of the numerical accuracy is shown to be between 1 and 2. Moreover, the accuracy control by adjusting the number density of the mark points purely at certain stages is explored, and a second power law is obtained. The numerical experiments for the flow around a cylinder behind a backward-facing step show that the location of the cylinder can affect the sizes and the shapes of the corner eddy and the main recirculation region. The proposed method can be applied further to the fluid dynamics with complex geometries, moving boundaries, fluid-structure interactions, etc..

Analytical solutions for axial annular flow of viscoelastic fluid obeying the Giesekus model are obtained including effect of slip condition at both walls. A power-law relation between wall shear stress and wall slip velocity called nonlinear Navier model was employed for the slip equation. The wall slip condition is assumed to be the same for both walls because both walls have similar properties. For the slip model it is considered that wall slip occurs only when a critical value is reached which is known as slip critical shear stress. Results indicate that slip initially occurs at the inner wall and then at the outer wall. Thus there are three flow regimes: no slip condition, slip only at the inner wall and slip at both walls. The flow regime is characterized by the value of slip critical shear stress. Results show that increasing slip effect causes pressure gradient, shear and normal stresses to decrease but, they increase by increasing slip critical shear stress. Also increasing elasticity decreases slip velocity at walls.

In this paper we are concerned with the regularity of solutions to the Navier-Stokes equations with the condition on the pressure on parts of the boundary where there is flow. For the steady Stokes problem a result similar to L-q-theory for the one with Dirichlet boundary condition is obtained. Using the result, for the steady Navier-Stokes equations we obtain regularity as the case of Dirichlet boundary conditions. Furthermore, for the time-dependent 2-D Navier-Stokes equations we prove uniqueness and existence of regular solutions, which is similar to J.M.Bernard's results([6]) for the time-dependent 2-D Stokes equations.

Under study is some boundary value problem for stationary equations of magnetohydrodynamics of a viscous heat-conducting liquid considered together with the Dirichlet condition for the velocity and mixed boundary conditions for the electromagnetic field and temperature. Some sufficient conditions are established on the initial data providing the global solvability of this problem and the local uniqueness of the solution.

Summary This work proposes an innovative numerical method for simulating the interaction of fluid with irregularly shaped stationary structures based on Cartesian grids. Instead of prescribing an artificial force to enforce the no-slip boundary condition at the solid–fluid interface, this work imposes two boundary velocities, referred to as the solid and mass-conserving boundary velocities, to satisfy the no-slip boundary condition and mass conservation in the ghost cells around the immersed solid boundary. Both the traditional level set method [41] and the hybrid particle level set method [45] were used to represent the solid boundary and the complex free-surface evolution, respectively. Consequently, the boundary velocities close to the immersed solid boundary can be determined in terms of the level set function and the neighboring fluid velocity. The projection method is further modified to incorporate the solid and mass-conserving boundary velocities into the solution algorithm. A series of numerical experiments were conducted to demonstrate the feasibility of the proposed method. They involved uniform flow past a stationary circular cylinder and the propagation of water waves over a submerged trapezoidal breakwater. Comparisons between the numerical results and experimental data showed very good agreement in all cases of interest. Copyright © 2015 John Wiley & Sons, Ltd.

International audience; The aim of this study is to discus different numerically models for the simulation of moving contact lines in the context of a Volume of Fluid–Continuum Surface Force (VoF–CSF) method. We focus on the particular situation of spreading drops. We first present the numerical methods used for the simulation of moving contact line i.e. static contact angle versus dynamic contact angle, no slip condition versus slip condition. A grid and time convergence is performed for the different models. We show that the integration of the Continuum Surface Force using the finite volume method results in a grid dependence at the onset of the spreading. The static and dynamic models are compared to experiments. It is shown that the dynamic models based on the Cox’s relation for the dynamic contact angle are able to reproduce experiments while static models overestimate the spreading time and are not able to reproduce the Tanner regime. The difference between static and dynamic models is shown to increase with the Ohnesorge number.

In this paper we prove a local result of existence and uniqueness for a free boundary problem for snow avalanche arising from a new model proposed in [5]. The mathematical problem consists of a parabolic free boundary problem with non-standard free boundary conditions (erosion dynamics). The proof is essentially based on a fixed point argument.

In this paper, we study the interaction of an incompressible viscous fluid occupying two-dimensional channel with an elastic plate located on one part of fluid boundary. We assume that the deformation of the boundary is small enough and consider the fluid flow equations in the initial configuration. Non-steady Stokes equations are used to model the flow, and the pressure boundary conditions are prescribed on the channel’s ends. The results on existence, uniqueness, and regularity of solution are proved.

The flow past an impermeable sphere embedded in a fluid-saturated porous medium was investigated numerically. The flow is considered viscous, laminar, axisymmetric, steady and incompressible. The generalized model for fluid flow through an isotropic, rigid and homogeneous porous medium was used. The stream function–vorticity equations were solved numerically in spherical coordinates system by a defect correction multigrid method. The computations were focused on the influence of the Darcy number (Da) and Forchheimer constant on the flow field for two boundary conditions on the surface of the sphere: slip and no-slip. The conditions when the macroscopic inertial terms of the generalized model can be neglected are presented. For high-porosity porous media, the macroscopic inertial terms of the generalized model can be neglected if Da $$Re< 1$$ (Re is the sphere Reynolds number).

Phase velocities and attenuation of viscous plane and cylindrical waves in free space and in gaps are investigated analytically and numerically under various boundary conditions. It is shown that boundary conditions in a gap determine the nature of spatial dispersion of a viscous wave. The influence of finite dimensions of the source and the size of the gap on the phase velocity gradient and on viscous wave attenuation is analyzed. The viscous waves being investigated are generated during operation of acoustic oscillating transducers in small volumes of viscous incompressible liquids used for technological purpose.

A novel scheme for implementation of the no-slip boundary conditions in the lattice Boltzmann method is presented. In detail, we have substituted the classical bounce-back idea by the direct velocity boundary condition specification employing geometric-based manipulation of the equilibrium distribution functions. In this way we have constructed the equilibrium density function in such a way that it imposes the desired Dirichlet boundary conditions at numerical boundary points. Therefore, in fact a kind of equilibrium boundary condition is made. This specification for general curved solid surfaces is made by means of immersed boundary concepts, but without any need to interpolating density distribution values. On the other hand, the results show that the method presents a faster solution procedure in comparison to the bounce-back scheme.

Dynamic fluid-solid interactions are widely found in chemical engineering, such as in particle-laden flows, which usually contain complex moving boundaries. The immersed boundary method (IBM) is a convenient approach to handle fluid-solid interactions with complex geometries. In this work, Uhlmann's direct-forcing IBM is improved and implemented on a supercomputer with CPU-GPU hybrid architecture. The direct-forcing IBM is modified as follows: the Poisson's equation for pressure is solved before evaluation of the body force, and the force is only distributed to the Cartesian grids inside the immersed boundary. A multidirect forcing scheme is used to evaluate the body force. These modifications result in a divergence-free flow field in the fluid domain and the no-slip boundary condition at the immersed boundary simultaneously. This method is implemented in an explicit finite-difference fractional-step scheme, and validated by 2D simulations of lid-driven cavity flow, Couette flow between two concentric cylinders and flow over a circular cylinder. Finally, the method is used to simulate the sedimentation of two circular particles in a channel. The results agree very well with previous experimental and numerical data, and are more accurate than the conventional direct-forcing method, especially in the vicinity of a moving boundary. (C) 2014 Published by Elsevier B.V. on behalf of Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences.

Boundary conditions have a vital role in the numerical solution of the partial differential equations governing fluid flows, and they have a great influence over the numerical stability and accuracy of the final solutions. One of the most important physical boundary conditions in flow field analysis, especially for inviscid flows, is the wall boundary condition imposed on body surfaces. To solve a three-dimensional compressible Euler equation (with five PDE's), a total of five boundary conditions on the body surface should be prescribed. The wall's velocity magnitude is one of the parameters to be determined, and the way this velocity magnitude is calculated affects the accuracy and stability of the numerical approach. In this paper, four different methods for calculation of the wall velocity magnitude are introduced, tested and compared against several test cases of subsonic and supersonic flows. Since there are many problems where both subsonic and supersonic flows coexist, a mixed boundary condition ta...

The steady laminar two-dimensional flow of nanofluid due to nonlinearly stretching sheet is discussed. Convective surface boundary condition is employed for a thermal boundary layer problem. The newly proposed boundary condition is considered that requires nanoparticle volume fraction at the wall to be passively rather than actively controlled. Suitable similarity transformations are introduced to non-dimensionalize the governing boundary layer equations. The velocity, temperature and nanoparticle volume fraction distributions are determined by two methods namely 1) optimal homotopy analysis method and 2) fourth–fifth-order Runge–Kutta method based shooting technique. The results obtained by two solutions are in excellent agreement. Behavior of interesting parameters on the flow fields is thoroughly presented and discussed.

The appropriate boundary condition between an unconfined incompressible viscous fluid and a porous medium is given by the law of Beavers and Joseph. The latter has been justified both experimentally and mathematically, using the method of periodic homogenization. However, all results so far deal only with the case of a planar boundary. In this work, we consider the case of a curved, macroscopically periodic boundary. By using a coordinate transformation, we obtain a description of the flow in a domain with a planar boundary, for which we derive the effective behavior: The effective velocity is continuous in normal direction. Tangential to the interface, a slip occurs. Additionally, a pressure jump occurs. The magnitude of the slip velocity as well as the jump in pressure can be determined with the help of a generalized boundary layer function. The results indicate the validity of a generalized Beavers-and-Joseph-type law, where the geometry of the interface has an influence on the slip and jump constants.

The MHD unsteady flow heat and mass transfer of a viscous incompressible fluid induced by a stretching surface in a rotating fluid taking the effects of thermal radiation, heat absorption and first order chemical reaction into account with convective boundary condition. The rotation parameter is found to reduce the velocities and the magnetic parameter prevents the flow reversal in the x – direction. The increasing values of Biot number generate thicker thermal boundary layers resulting in the rise of temperature. The Schmidt number and chemical reaction parameter are found to have a strong influence on the species concentration resulting in small values. The rate of heat transfer is enhanced by magnetic field, thermal radiation parameter and rotation parameter. The rotation parameter and time are observed to enhance the rate of mass transfer.

We consider flows of a viscous fluid in a domain on the boundary of which ether velocity or stress is specified. It is shown that the solutions of boundary-value problems for the Navier-Stokes and linear Stokes equations with the same stresses on the boundary satisfy an inequality, which makes it possible to obtain an estimate from above for the functionals of the velocity fields generated by the stresses applied on the boundary. For a flow in a closed domain, generated by the shear stress on the boundary, for which at high Reynolds numbers Batchelor’s theorem is valid and the condition of monotonic dependence of the velocity functional on the Reynolds number holds, two-sided estimates for the average velocity value on the boundary are obtained for all Reynolds numbers.

An improved momentum-exchanged immersed boundary-based lattice Boltzmann method (MEIB-LBM) is proposed for incompressible viscous flows in this paper. In present work, we come back to the intrinsic feature of LBM, which uses the density distribution function as a dependent variable to evolve the flow field, and use the non-equilibrium density distribution function correction at the neighbour Euler mesh points to satisfy the non-slip boundary condition on the immersed boundary. The improvements for the original MEIB-LBM are that the intrinsic feature of LBM is kept and the flow penetration is removed. Examples, including force convection over a stationary heated circular cylinder for heat flux condition, and natural convection with a buoyant circle particle in viscous fluid, are provided to validate the present method.

When molten plastic is extruded, the upper limiting throughput is often dictated by fine irregular distortions of the extrudate surface. Called sharkskin melt fracture, plastics engineers spike plastics formulations with processing aids to suppress these distortions. Sharkskin melt fracture is not to be confused with gross melt fracture, a larger scale distortion arising at throughputs higher than the critical throughput for sharkskin melt fracture. Sharkskin melt fracture has been attributed to a breakdown of the no slip boundary condition in the extrusion die, that is, adhesive failure at the die walls, where the fluid moves with respect to the wall. In this article, we account for the frictional heating at the wall, which we call slip heating. We focus on slit flow, which is used in film casting, sheet extrusion, curtain coating, and when curvature can be neglected, slit flow is easily extended to pipe extrusion and film blowing. In slit flow, the magnitude of the heat flux from the slipping interface is the product of the shear stress and the slip speed. We present the solutions for the temperature rise in pressure-driven slit flow and simple shearing flow, each subject to constant heat generation at the adhesive slip interface, with and without viscous dissipation in the bulk fluid. We solve the energy equation in Cartesian coordinates for the temperature rise, for steady temperature profiles. For this simplest relevant nonisothermal model, we neglect convective heat transfer in the melt and use a constant viscosity. We arrive at a necessary dimensionless condition for the accurate use of our results: Pe≪1. We find that slip heating can raise the melt temperature significantly, as can viscous dissipation in the bulk. We conclude with two worked examples showing the relevance of slip heating in determining wall temperature rise, and we show how to correct wall slip data for this temperature rise. POLYM. ENG. SCI., 55:2042–2049, 2015. © 2014 Society of Plastics Engineers

The paper presents the application of an indirect variant of the boundary element method (BEM) to solve the two-dimensional steady flow of a Stokes liquid. In the BEM, a system of differential equations is transformed into integral equations. This makes it possible to limit discretization to the border of the solution. Numerical discretization of the computational domain was performed with linear boundary elements, for which a constant value of unknown functions was assumed. The verification was carried out for the case of flow in a square cavity with one moving wall. The results obtained show that the use of approximations by simple linear functions is relatively easy for different shapes of the area, but the result may be affected by significant errors.

The acoustics of a mean flow boundary layer over an impedance surface or acoustic lining is considered. By considering a thick mean flow boundary layer (possibly due to turbulence), the boundary-layer structure is separated asymptotically into two decks, with a thin weakly viscous mean flow boundary layer and an even thinner strongly viscous acoustic sublayer, without requiring a high frequency. Using this, analytic solutions are found for the acoustic modes in a cylindrical lined duct. The mode shapes in each region compare well with numerical solutions of the linearized compressible Navier–Stokes equations, as does a uniform composite asymptotic solution. A closed-form effective impedance boundary condition is derived, which can be applied to acoustics in inviscid slipping flow to account for both shear and viscosity in the boundary layer. The importance of the boundary layer is demonstrated in the frequency domain, and the new boundary condition is found to correctly predict the attenuation of upstream-propagating cut-on modes, which are poorly predicted by existing inviscid boundary conditions. Stability is also investigated, and the new boundary condition is found to yield good results away from the critical layer. A time-domain formulation of a simplified version of the new impedance boundary condition is proposed.