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The velocity and thermal slip impacts on the magnetohydrodynamics (MHD) nanofluid flow and heat transfer through a stretched thin sheet are discussed in the paper. The no slip condition is substituted for a new slip condition consisting of higher-order slip and constitutive equation. Similarity transformation and Lie point symmetry are adopted to convert the derived governed equations to ordinary differential equations. An approximate analytical solution is gained through the homotopy analysis method. The impacts of velocity slip, temperature jump, and other physical parameters on flow and heat transfer are illustrated. Results indicate that the first-order slip and nonlinear slip parameters reduce the velocity boundary layer thickness and Nusselt number, whereas the effect on shear stress is converse. The temperature jump parameter causes a rise in the temperature, but a decline in the Nusselt number. With the increase of the order, we can get that the error reaches 10 − 6 from residual error curve. In addition, the velocity contours and the change of skin friction coefficient are computed through Ansys Fluent.

We report computations of the axisymmetric slump of viscoplastic fluid using the volume-of-fluid (VOF) method. The constitutive law is dealt with by either regularization or the augmented-Lagrangian method. The interface is tracked by the PLIC scheme, modified in order to avoid resolution issues associated with the over-ridden finger of ambient fluid that results from the no slip condition and the resulting inability to move the contact line. Numerical results are compared with asymptotic analyses for shallow gravity currents or slender vertical columns. The critical yield stress for failure is computed and bounded analytically using plasticity methods. The simulations are compared with experiments either taken from existing literature or performed using Carbopol. The comparison is satisfying for lower yield stresses; discrepancies for larger yield stresses suggest that the mechanism of release may affect the experiments.

This paper describes numerical solutions of incompressible Navier-Stokes equations with a new boundary condition. To solve this problem, we use the discretization by mixed finite element method. We use a vector extrapolation method for computing numerical solutions of the steady-state Navier-Stokes equations. In addition, two types of a posteriori error indicator are introduced and are shown to give global error estimates that are equivalent to the true error. A numerical experiment on the driven cavity flow is given to demonstrate the effectiveness of the vector extrapolation method. We compare the result with the solution from commercial code like ADINA system as well as with values from other simulations.

We present a discontinuous Galerkin (DG) algorithm with nonconformal meshes to simulate 3D elastic wave propagation in heterogeneous media with arbitrary discrete fractures. In our method, the fractures are not limited to be planar, single, and lossless, but can be curved, intersecting, and viscous. In contrast to the exact volumetric modeling for the extremely thin layer, explicitly treating an individual fracture as a geometry surface (i.e., an imperfect contact interface) requires the jump condition for displacement/velocity, but the continuity of traction vector on the fracture interface. A new upwind flux is proposed to weakly impose this jump boundary condition in the DG framework. This flux guarantees the stability and accuracy of the DG schemes to model arbitrary fractures. Unlike conventional Riemann solvers applied to continuous media, this solution involves an evolutionary update on the Godunov states. Besides this, no extra computational cost is added. In addition, we can extend the fracture interface into a perfectly matched layer to mimic an infinitely large fracture. Quantitative comparisons of the waveforms between our algorithm and an independent finite element code demonstrate the accuracy and efficiency of our algorithm.

In this article, we derive the asymptotic expansion, up to an arbitrary order in theory, for the solution of a two-dimensional elliptic equation with strongly anisotropic diffusion coefficients along different directions, subject to the Neumann boundary condition and the Dirichlet boundary condition on specific parts of the domain boundary, respectively. The ill-posedness arising from the Neumann boundary condition in the strongly anisotropic diffusion limit is handled by the decomposition of the solution into a mean part and a fluctuation part. The boundary layer analysis due to the Dirichlet boundary condition is conducted for each order in the expansion for the fluctuation part. Our results suggest that the leading order is the combination of the mean part and the composite approximation of the fluctuation part for the general Dirichlet boundary condition., 18 pages, 5 figures, 1 table

In this study, the smoothed profile-lattice Boltzmann method (SP-LBM) is proposed to determine the contact line dynamics on a hydrophobic or a hydrophilic curved wall. Two types of smoothed indicator functions are introduced, namely a function that identifies the solid domain for non-slip and non-penetration conditions and a function that denotes the boundary layer for no mass-flux and the wetting boundary conditions. In order to prevent fluid penetration into the solid boundary, the fluid-solid interaction force is computed based on the definition of the fluid velocity as proposed by Guo et al. [1]. In order to implement the Neumann boundary conditions for the order parameter and the chemical potential, the fluxes from the solid surfaces are distributed to relevant physical valuables through a smoothed profile. Several two-dimensional and three-dimensional numerical investigations including those determining the Couette flows, flow around a circular cylinder, transition layer on a wetting boundary, and dynamic behavior of a droplet on a flat or curved plate demonstrate the efficiency of the present method in calculating the contact angle of a droplet on curved surfaces with wall impermeability. The present model provides a simple algorithm to compute the surface normal vector and contact line dynamics on an arbitrarily shaped boundary by using a smoothed-profile.

In this paper, we show the existence of weak solutions for steady flows of electrorheological fluids with nonhomogeneous Dirichlet boundary condition under the condition p ( x ) > 2 n n + 2 , n = 2 , 3 . In particular, we improve a priori estimate for weak solutions to the problem with min p ( x ) 2 , which is an improvement even for constant p .

The numerical modeling of the aerodynamic interactions at high-Altitudes and high-Mach numbers is considered in view of its importance when studying problems where the continuum hypothesis at the foundation of the Navier- Stokes equations becomes invalid. One of the difficulties associated with these flight conditions is that both the velocity and the temperature of the fluid do not abide by the no-slip conditions at the wall. A weak Galerkin finite element formulation of the Maxwell-Smoluchowki model is introduced to discretize the velocity slip and temperature jump conditions with better accuracy than the standard finite element approximation. The methodology is assessed on configurations such as cylinders and spheres for flow conditions ranging from quasi-equilibrium to nonequilibrium. Improvements are observed in the slip regime compared with available data. Nonetheless, the results in the transition regime highlight the need for more sophisticated physical modeling to address nonequilibrium at the wall.

In this paper, we study strong solutions to the steady compressible heat-conductive fluid near a non-zero constant flow with the Dirichlet boundary condition for the velocity on the inflow and outflow part of the boundary. We also consider the Dirichlet boundary condition for the temperature, and we do not need the thermal conductivity coefficient κ to be large. The existence of strong solutions is established for any Reynolds number and Mach number in the framework of perturbation.

The purpose of present analysis is to explore the peristaltic flow of Williamson nanofluid in an endoscope. Inner tube is rigid while the outer is subject to sinusoidal peristaltic wave. The outer tube satisfied partial slip condition while inner tube has no slip condition. Complaint wall properties are considered. Such preferences are practical in blood capillaries where inner radius of tube is very small. Long wave length and small Reynolds number are taken into account. Numerical study is performed for the outcoming problem. Impact of physical quantities are discussed through graphs for velocity, temperature and concentration respectively. The attributes of this study are presented in last section and a good agreement is found with existing literature. It is concluded that slip causes decrease in velocity and temperature. Whereas temperature and concentration give opposite response wih rise in thermophoresis and Brownian diffusion.

We consider the system of MHD equations in Ω × ( 0 , T ) , where Ω is a domain in R 3 and T > 0 , with the no slip boundary condition for the velocity u and the Navier-type boundary condition for the magnetic induction b . We show that an associated pressure p , as a distribution with a certain structure, can be always assigned to a weak solution ( u , b ) . The pressure is a function with some rate of integrability if the domain Ω is “smooth”, see section 3. In section 4, we study the regularity of p in a sub-domain Ω 1 × ( t 1 , t 2 ) of Ω × ( 0 , T ) , where u (or, alternatively, both u and b ) satisfies Serrin’s integrability conditions. Regularity criteria for weak solutions to the MHD equations in terms of π ≔ p + 1 2 | b | 2 are studied in section 5. Finally, section 6 contains remarks on analogous results in the case of Navier’s or Navier-type boundary conditions for the velocity u .

This paper presents topology optimization of capillary, the typical two-phase flow with immiscible fluids, where the level set method and diffuse-interface model are combined to implement the proposed method. The two-phase flow is described by the diffuse-interface model with essential no slip condition imposed on the wall, where the singularity at the contact line is regularized by the molecular diffusion at the interface between two immiscible fluids. The level set method is utilized to express the fluid and solid phases in the flows and the wall energy at the implicit fluid-solid interface. Based on the variational procedure for the total free energy of two-phase flow, the Cahn-Hilliard equations for the diffuse-interface model are modified for the two-phase flow with implicit boundary expressed by the level set method. Then the topology optimization problem for the two-phase flow is constructed for the cost functional with general formulation. The sensitivity analysis is implemented by using the continuous adjoint method. The level set function is evolved by solving the Hamilton-Jacobian equation, and numerical test is carried out for capillary to demonstrate the robustness of the proposed topology optimization method. It is straightforward to extend this proposed method into the other two-phase flows with two immiscible fluids.

We study the homogenization of a slow viscous two-phase incompressible flow in a domain consisting of a free fluid domain, a porous medium, and the interface between them. We take into account the capillary forces on the fluid-fluid interfaces. We construct boundary layers describing the flow at the interface between the free fluid and the porous medium. We derive a macroscopic model with a viscous two-phase fluid in the free domain, a coupled Darcy law connecting two-phase velocities in the porous medium, and boundary conditions at the permeable interface between the free fluid domain and the porous medium.

The consideration of viscoelasticity within fluid dynamical boundary element methods has traditionally required meshing over the whole flow domain. In turn, a major advantage of the boundary element method is lost, namely the need to consider only surface boundary integrals. Here, using a generalised reciprocal relation and viscoelastic force singularities, a boundary element method is developed for linear viscoelastic flows. We proceed to explore finite-deformation microswimming in a linear Maxwell fluid. We firstly deduce a finite-amplitude generalisation of a previously reported result that the flow field is unchanged between a Newtonian and linear Maxwell fluid for prescribed small-amplitude deformations. Hence Purcell’s theorem holds for a linear Maxwell fluid. We proceed to consider deformation swimming in a linear Maxwell fluid given an external forcing. Boundary scattering trajectories for an exemplar squirmer approaching a surface are observed to exhibit a weak dependence on the Deborah number, while the trajectories of a sperm and monotrichous bacterium near a surface are predicted to be essentially unaffected at moderate Deborah number. In turn, the latter supports the common simplification of using Newtonian Stokes flows for studying flagellate swimming in linear Maxwell media. In addition, the motion of a magnetic helix under the influence of an external magnetic field is considered, and highlights that linear viscoelasticity can significantly impact the propagation of the helix, in turn demonstrating that even linear rheology is important to consider for forced swimmers. Finally, the presented framework requires minimalistic adjustments to Newtonian boundary element codes, enabling rapid implementation, and is more generally applicable, for instance to studies of particle interactions in active linear rheology on the microscale.

The purpose of this paper is to study the boundary layer problem and zero viscosity-diffusion limit of the initial boundary value problem for the incompressible viscous and diffusive magnetohydrodynamic (MHD) system with (no-slip characteristic) Dirichlet boundary conditions and to prove that the corresponding Prandtl's type boundary layer are stable with respect to small viscosity-diffusion coefficients. The main difficulty here comes from Dirichlet boundary conditions for the velocity and magnetic field. Firstly, we identify a non-trivial class of initial data for which we can establish the uniform stability of the Prandtl's type boundary layers and prove rigorously the solution of incompressible viscous-diffusion MHD system converges strongly to the sum of the solution to the ideal MHD system and the approximating solution to Prandtl's type boundary layer equation by using the elaborate energy methods and the special structure of the solution to ideal MHD system with the initial data we identify here, which yields that there exists the cancellation between the boundary layer of the velocity and that of the magnetic field. Secondly, for general initial data, we obtain zero viscosity-diffusion limit of the incompressible viscous and diffusive MHD system with the different horizontal and vertical viscosities and magnetic diffusions, when they go to zero with the different speeds, and, we prove rigorously the convergence of the incompressible viscous and diffusion MHD system to the ideal MHD system or the anisotropic MHD system by constructing the exact boundary layers and using the elaborate energy methods. We also mention that these results obtained here should be the first rigorous ones on the stability of Prandtl's type boundary layer for the incompressible viscous and diffusion MHD system with no-slip characteristic boundary condition.

The present study examines the steady, axisymmetric Stokes flow past a viscous fluid spheroid whose shape deviates slightly from that of a sphere in a micropolar fluid spheroidal cavity. Inertial effects are neglected for both the inner and outer fluids. The boundary value problem is solved analytically using the cell model technique. The boundary conditions used are the vanishing of the normal velocities, the continuity of the tangential velocities, the continuity of shear stresses and the spin-vorticity relation at the surface of the inner viscous fluid spheroid. On the outer spheroidal surface (filled with micropolar fluid), four known boundary conditions, namely Happel’s, Kvashnin’s, Kuwabara’s and Cunningham’s (Mehta–Morse) are considered. The dependence of the wall correction factor on the spin parameter, viscosity ratio, volume fraction, deformation parameter and the micropolarity parameter is studied numerically and its variation is presented graphically. Both types of spheroids, prolate and oblate are considered. As a limiting case, the drag force acting on viscous fluid spheroid by the micropolar fluid in an unbounded medium and the wall correction factor for slow motion of a viscous fluid sphere located at the center of a spherical cell are obtained from the present analysis.

The planar flow of a Newtonian incompressible fluid in a T-shaped channel is investigated. Three fluid interaction models with solid walls are considered: no slip boundary condition, Navier slip boundary condition and slip boundary condition with slip yield stress. The fluid flow is provided by uniform pressure profiles at the boundary sections of the channel. The problem is numerically solved using a finite difference method based on the SIMPLE procedure. Characteristic flow regimes have been found for the described models of liquid interaction with solid walls. The estimation of the influence of the Reynolds number, pressure applied to the boundary sections and the parameters of these models on the flow pattern was performed. The criterial dependences describing main characteristics of the flow under conditions of the present work have been demonstrated.

Receptivity of high-speed boundary layers in calorically perfect gas is considered within the framework of fluctuating hydrodynamics introduced by Landau and Lifshitz (“Hydrodynamic Fluctuations,” ...

Summary A novel implicit immersed boundary method of high accuracy and efficiency is presented for the simulation of incompressible viscous flow over complex stationary or moving solid boundaries. A boundary force is often introduced in many immersed boundary methods to mimic the presence of solid boundary, such that the overall simulation can be performed on a simple Cartesian grid. The current method inherits this idea and considers the boundary force as a Lagrange multiplier to enforce the no-slip constraint at the solid boundary, instead of applying constitutional relations for rigid bodies. Hence excessive constraint on the time step is circumvented, and the time step only depends on the discretization of fluid Navier-Stokes equations, like the CFL condition in present work. To determine the boundary force, an additional moving force equation is derived. The dimension of this derived system is proportional to the number of Lagrangian points describing the solid boundaries, which makes the method very suitable for moving boundary problems since the time for matrix update and system solving is not significant. The force coefficient matrix is made symmetric and positive definite so that the conjugate gradient method can solve the system quickly. The proposed immersed boundary method is incorporated into the fluid solver with a second-order accurate projection method as a plug-in. The overall scheme is handled under an efficient fractional step framework, namely, prediction, forcing, and projection. Various simulations are performed to validate current method, and the results compare well with previous experimental and numerical studies.

Sedimentation processes of solid particles in a fluid with heat transfer are simulated using a coupled Lattice Boltzmann Method, Immersed Boundary Method and Discrete Element Method (LBM-IBM-DEM) scheme. In the numerical simulations, solid particles are specified either by a given temperature which is termed Dirichlet boundary condition or by a temperature gradient which is termed Neumann boundary condition. Several cases are examined containing one, two and 504 solid particles settling in a fluid, respectively. All the considered cases could be divided into two groups: Group Dirichlet and Group Neumann according to different styles of boundary conditions employed but under exactly the same initial states. The effects of these two boundary conditions on the particle behavior are quantized.

A typical process in many industrial applications is rising bubble dynamic in viscous liquids like two-phase reactors. Examining the physical behavior of bubbles may improve the understanding of systems regarding design and operation. This study focused on the splitting of bubbles resulting from their impact on solid obstacles. Fragmentation of the bubbles appears in many applications such as lab on a chip in small scale or slug bubbly flow moving upward in a tube in large scales. Using a new index-function model in Lattice Boltzmann technique proposed by “He”, we simulated the deformation and motion of a bubble in different regimes, through which, we accurately captured a sharp interface between the two phases. We extended the aforementioned technique from 2D to 3D modelling of buoyancy-driven motion of a single bubble in quiescent viscous liquid. It was demonstrated that there is a reasonable agreement in terms of terminal rising velocity as well as bubble shape. This was found by comparing it with other available experimental and numerical results in different regimes through varying the two non-dimensional numbers (Eotvos and Morton) to characterize the fluid regime behind the rising bubble. In addition, by applying Bounce-Back no slip boundary condition to the surface area of the tubes with circular cross sections, we simulated the impact of the bubble during its upward motion. Changing the distance between the tubes and their corresponding diameters causes different shapes in the bubbles. Our simulation demonstrated that the 3D model based on index-function model of LBM is a suitable tool for 3D numerical simulation of rising bubbles in the presence of obstacles.

In this paper, artificial boundary conditions are designed for out-of-plane waves in penta-graphene, a newly proposed allotrope of carbon. By matching the dispersion relation for acoustic branch phonons in the long wave limit, we determine parameters in proposed linear constraints among displacements and velocities at the boundary and nearby atoms. Reflection analysis for normal incidences and a numerical test demonstrate the effectiveness of the artificial boundary conditions. These conditions may be used for studying mechanical behaviours of the novel complex lattice of penta-graphene.

We shall consider the problem of the motion of a rigid body in an incompressible viscous fluid filling a bounded domain. This problem was studied by several authors. They mostly considered classical non-slip boundary conditions, which gave them very paradoxical result of no collisions of the body with the boundary of the domain. Only recently there are results when the Navier type of boundary is considered. In our paper we shall consider the Navier condition on the boundary of the body and the non-slip condition on the boundary of the domain. This case admits collisions of the body with the boundary of the domain. We shall prove the global existence of weak solution of the problem.

In this paper, using mixture theory we study the flow of a dense suspension, composed of solid particles and a fluid; the emphasis is on the influence of the slip boundary condition and the effect of normal stress differences. Very little work has been done considering both the slip at the walls and the normal stress effects in the frame of a two-component flow. In this paper, the stress tensor for the solid component is modeled as a nonlinear fluid which not only includes the viscous effects but also the normal stress effects; the fluid constituent is modeled as a viscous fluid. We look at the flow between two flat plates.

We study a system of nonlinear equations describing unsteady flows of a viscoelastic fluid of the Oldroyd type in a bounded three-dimensional domain under the mixed boundary conditions: the Navier slip condition is set on one part of the boundary, while on the other part, the no-slip condition is set. We prove the theorem concerning the existence, uniqueness, and energy estimates for weak solutions.

In this paper, we study boundary layer problem for the incompressible magneto-hydrodynamical (MHD) system in the presence of physical boundaries with the standard Dirichlet boundary conditions with small generic viscosity and diffusion coefficients. We identify a non-trivial class of initial data for which we can establish the uniform stability of the Prandtls type boundary layers and prove rigorously that the solutions to the viscous and diffusive incompressible MHD system converges strongly to the superposition of the solution to the ideal MHD system with a Prandtls type boundary layer corrector. One of the main difficulties is to deal with the effect of the difference between viscosity and diffusion coefficients and to control the singular boundary layers resulting from the Dirichlet boundary conditions for both the viscosity and the magnetic fields. One key derivation here is that for the class of initial data we identify here, there exist cancelations between the boundary layers of the velocity field and that of the magnetic fields so that one can use an elaborate energy method to take advantage of this special structure. In addition, in the case of fixed positive viscosity, we also establish the stability of diffusive boundary layer for the magnetic field and convergence of solutions in the limit of zero magnetic diffusion for general initial data.

Present analysis is made for the laminar flow of Oldroyd-B fluid induced by a deforming sheet in the existence of transverse magnetic field. Flow model is constructed in the presence of slip boundary condition. The governing problem even after utilizing boundary layer approximations comprises of non-linear differential equation with the non-linear boundary condition. Such boundary condition for the no slip case is linear. Highly accurate analytic solutions for velocity distribution are derived by powerful homotopy analysis approach. Permissible values of the convergence control parameter are obtained by the so-called h-curves. The behavior of parameters on the solutions is shown graphically. In the light of numerical results, we predict that slip effect gives opposition to the momentum transport phenomenon. Moreover, the behaviors of relaxation and retardation time are qualitatively opposite. A comparative study of slip and no-slip cases is also discussed.

A robust Incompressible Smoothed Particle Hydrodynamics (ISPH) framework is developed to simulate specified inflow and outflow boundary conditions for open-channel flow. Being purely divergence-free, the framework offers smoothed and structured pressure distribution. An implicit treatment of Pressure Poison Equation and Dirichlet boundary condition is applied on free-surface to minimize error in velocity-divergence. Beyond inflow and outflow threshold, multiple layers of dummy particles are created according to specified boundary condition. Inflow boundary acts as a soluble wave-maker. Fluid particles beyond outflow threshold are removed and replaced with dummy particles with specified boundary velocity. The framework is validated against different cases of open channel flow with different boundary conditions. The model can efficiently capture flow evolution and vortex generation for random geometry and variable boundary conditions.

In this article, mathematical modeling for peristaltic flow of Rabinowitsch fluid model is considered in a non-uniform tube with combined effects of viscous dissipation and convective boundary conditions. Wall properties analysis is also taken into account. Non-dimensional differential equations are simplified by using the well-known assumptions of low Reynolds number and long wavelength. The influence of various parameters connected with this flow problem such as rigidity parameter E1, stiffness parameter E2, viscous damping force parameter E3, Brickman number and Biot number are plotted for velocity distribution, temperature profile and for stream function. Results are plotted and discussed in detail for shear thinning, shear thickening and for viscous fluid. It is found that velocity profile is an increasing function of rigidity parameter, stiffness parameter, and viscous damping force parameter for shear thinning and for viscous fluid, due to the less resistance offered by the walls but, quite opposite behavior is depicted for shear thickening fluids. It is seen that Brickman number relates to the viscous dissipation effects, so it contributes in enhancing fluid temperature for all cases.

The fluid flow and heat transfer over a permeable shrinking sheet embedded in a porous medium with a second-order slip is studied. Using appropriate similarity variables, the governing partial differential equations are reduced to ordinary differential equations before being solved numerically by a shooting method for different values of the selected governing parameters. The numerical results reveal that two (dual) solutions are possible for a certain range of these parameters. A stability analysis is performed to determine which solution is stable and which one is not. The effects of suction and the second order slip parameters on the skin friction coefficient and the local Nusselt number as well as on the velocity and temperature profiles are plotted and discussed. It is found that applying the slip condition increases the range of solutions. Also, the skin friction coefficient is higher for the no slip condition compared to that of with slip condition.

Objectives: Effects of slip velocity and aligned Magneto hydro-dynamic on Jeffrey fluid which passing across a stretching sheet with Newtonian heating boundary condition are investigated. Methods/Statistical Analysis: Governing partial differential equations are first transformed into ordinary differential equation by applying the similarity transformations before undergo computation process using y bvp4c in MATLAB. Findings: For validation purposes, the present results are comparing with the outcome from previous publications for the case Constant Wall Temperature (CWT) and it shows a very strong agreement on the values of −θ′(0) . From the study, the increasing on the values of magnetic parameter, M drop the fluid velocity for the range value of boundary layer thickness 0 3.5 , the value of fluid velocity is increasing on the larger magnetic parameter. However, the distribution on temperature of fluid shows an opposite trend as compare with fluid velocity. On the range of 0 2.5 it evidently reduce the velocity of fluid. The fluid temperature is high at smaller values of Deborah number. The distribution on velocity and temperature of fluid presented in this article are strictly asymptotically fulfilling the boundary conditions which then contribute to the adequate of the present results. Application/Improvements: The results from this project will enhance the knowledge in non-Newtonian fluid problem via mathematical approach.

We investigate the dynamics of the gas bounded by an infinite flat plate which is initially in equilibrium and set at some instant impulsively into uniform motion in its own plane. We use the Boltzmann equation to describe intermolecular collisions and assume the diffuse reflection to describe the interaction of the gas with the boundary. The Mach number of the plate is assumed to be small so that we can linearize the Boltzmann equation as well as the boundary condition. We show that the asymptotic behavior of the gas represents a perturbation to the free molecular gas when the time is much less than the mean free time. On the other hand, if the time is much greater than the mean free time, we show that the gas dynamics is governed by the linearized Navier–Stokes equation with a slip flow on the boundary and establish a boundary layer correction with thickness of the order of the mean free path. We also establish the singularity of velocity distribution function along the particle trajectory near the boundary.

In this paper we consider a one-dimensional free boundary problem describing adsorption phenomena in a porous media. This problem was already proposed and we established a local existence in time and a uniqueness result in [1] , [4] . Here, through the discussion for the dynamics of the free boundary in wetting and drying processes we suppose a specific form of the growth rate condition for the free boundary. By using this form in our problem we can obtain the global existence of a solution in time and convergence of the free boundary as t → ∞ .

The theme of present work is to report the numerical solution of mixed convection tangent hyperbolic fluid flow towards stretching cylindrical surface immersed in a double stratified media. The fluid flow is attained through no slip condition. The flow regime characteristics are modelled in terms of partial differential equations. A similarity transformation is used to transform partial differential equations into coupled non-linear ordinary differential equations. A computational algorithm is executed to predict numerical results. The effects of flow controlling parameters namely, mixed convection parameter, thermal stratification and solutal stratification parameters on velocity, temperature and concentration are examined and offered by means of graphical outcomes. It is noticed that in the presence of mixed convection effect both the fluid temperature and concentration are decreasing function of thermal stratification and solutal stratification parameters respectively. The obtained values are certified by developing comparison with existing values and an excellent agreement is observed which confirms the execution of computational algorithm. Keywords: Tangent hyperbolic fluid model, Mixed convection, Temperature stratification, Concentration stratification, An inclined cylinder, Shooting method

If the non-Newtonian fluid equation with a diffusion coefficient is degenerate on the boundary, the weak solution lacks the regularity to define the trace on the boundary. By introducing a new kind of weak solutions, the stability of the solutions is established without any boundary condition.

In this paper, an immersed boundary-gas kinetic flux solver (IB-GKFS) is presented for simulation of incompressible viscous flows. In the present scheme, a simple Cartesian mesh is applied for the flow field and a set of Lagrangian points are used to represent the solid boundary. The solution process of the IB-GKFS can be separated into two steps, the predictor step and the velocity correction step. In the predictor step, the intermediate flow field is obtained by applying the gas kinetic flux solver. As the solid boundary is not considered in this step, there is no external force added in gas distribution function during the evaluation of numerical flux at each cell interface. In the velocity correction step, no-slip boundary condition is imposed at all boundary points to make velocity correction on the surrounding Eulerian points. The implicit boundary condition-enforced immersed boundary method is applied so that no-slip boundary condition can be accurately fulfilled and flow penetration is entirely avoided. The decoupled feature of the predictor step and velocity correction step makes the current scheme simple and efficient because the flux on each cell interface only needs to be calculated once in every time step. With a simple Cartesian mesh and flexible boundary condition treatment, the IB-GKFS can be conveniently applied to solve complex and moving boundary problems. Several numerical experiments are conducted to validate the present scheme, including the flow past a stationary circular cylinder and the NACA0012 airfoil, flow past an in-line oscillating cylinder with prescribed motions. After that, the typical fluid-structure interaction problem of one particle sedimentation in a rectangular domain is further considered. The numerical results of those test cases demonstrate the good capability of the present scheme.

Phase field (diffuse interface) models accommodate diffusive triple line motion with variable contact angle, thus allowing for the no-slip boundary condition without the stress singularities. We consider two commonly used classes of phase field models: the compositionally compressible (CC) model with compressibility limited to the fluid mix within the diffuse interface, and the incompressible (IC) model. First, we show that the CC model applied to fluids with dissimilar mass densities exhibits the computational instability leading to the breakup of the triple line. We provide a qualitative physical explanation of this instability and argue that the compositional compressibility within the diffuse interface is inconsistent with the global incompressible flow. Second, we derive the IC model as a systematic approximation to the CC model, based on a suitable choice of continuum velocity field. Third, we benchmark the IC model against sharp interface theory and experimental kinetics. The triple line kinetics is well represented by the triple line mobility parameter. Finally, we investigate the effects of the bulk phase field diffusional mobility parameter on the kinetics of the wetting process and find that within a wide range of magnitudes the bulk mobility does not affect the flow.