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In this work we introduce a new a posteriori error estimate of hierarchical type for the advection-diffusion-reaction equation. We prove the equivalence between the energy norm of the error and our error estimate using an auxiliary linear problem for the residual and an easy way to prove inf–sup condition.

The purpose of this paper is to mathematically investigate the formation of a plasma sheath near the surface of a ball-shaped material immersed in a bulk plasma, and to obtain qualitative information of such a plasma sheath layer. Specifically, we study existence and the quasi-neutral limit behavior of the stationary spherical symmetric solutions for the Euler–Poisson equations in a three-dimensional annular domain. We first propose a suitable condition on the velocity at the sheath edge, referred as to Bohm criterion for the annulus, and under this condition together with the constant Dirichlet boundary conditions for the potential, we show that there exists a unique stationary spherical symmetric solution. Moreover, we study the quasi-neutral limit behavior by establishing $L^2$ estimate of the difference of the solutions to the Euler–Poisson equations and its quasi-neutral limiting equations, incorporated with the correctors for the boundary layers. The quasi-neutral limit analysis employing the correctors and their pointwise estimates enables us to obtain detailed asymptotic behaviors including the convergence rates in $L^2$ and $H^1$ norms as well as the thickness of the boundary layers as a consequence of the pointwise estimates.

In this paper, we analyse the behaviour of a pile-up of vertically periodic walls of edge dislocations at an obstacle, represented by a locked dislocation wall. Starting from a continuum nonlocal energy $E_{\gamma }$ modelling the interactions — at a typical length-scale of $1/\gamma$ — of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy $E_{\gamma }$ in powers of $1/\gamma$ by $\mathrm{\Gamma }$-convergence, in the limit $\gamma \to \infty$. While the zeroth-order term in the expansion, the $\mathrm{\Gamma }$-limit of $E_{\gamma }$, captures the “bulk” profile of the density of dislocation walls in the pile-up domain, the first-order term in the expansion is a “boundary-layer” energy that captures the profile of the density in the proximity of the lock. This study is a first step towards a rigorous understanding of the behaviour of dislocations at obstacles, defects, and grain boundaries.

A mathematical model has been developed for steady-state boundary layer flow of a nanofluid past an impermeable vertical flat wall in a porous medium saturated with a water-based dilute nanofluid containing oxytactic microorganisms. The nanoparticles were distributed sufficiently to permit bioconvection. The product of chemotaxis constant and maximum cell swimming speed was assumed invariant. Using appropriate transformations, the partial differential conservation equations were non-dimensionalised to yield a quartet of coupled, non-linear ordinary differential equations for momentum, energy, nanoparticle concentration and dimensionless motile microorganism density, with appropriate boundary conditions. The dominant parameters emerging in the normalised model included the bioconvection Lewis number, bioconvection Peclet number, Lewis number, buoyancy ratio parameter, Brownian motion parameter, thermophoresis parameter, local Darcy-Rayleigh number and the local Peclet number. An implicit numerical solution to the well-posed two-point non-linear boundary value problem is developed using the well-tested and highly efficient Keller box method. Computations are validated with the Nakamura tridiagonal implicit finite difference method, demonstrating excellent agreement. Nanoparticle concentration and temperature were found to be generally enhanced through the boundary layer with increasing bioconvection Lewis number, whereas dimensionless motile microorganism density was only increased closer to the wall. Temperature, nanoparticle concentration and dimensionless motile microorganism density were all greatly increased with a rise in Peclet number. Temperature and dimensionless motile microorganism density were reduced with increasing buoyancy parameter, whereas nanoparticle concentration was increased. The present study found applications in the fluid mechanical design of microbial fuel cell and bioconvection nanotechnological devices.

Laminar magnetohydrodynamic Marangoni-forced convection boundary layer flow of a water-based biopolymer nanofluid containing nanoparticles from a non-isothermal plate is studied. Magnetic induction effects are incorporated. A variety of nanoparticles are studied, specifically, silver, copper, aluminium oxide and titanium oxide. The Tiwari–Das model is utilized for simulating nanofluid effects. The normalized ordinary differential boundary layer equations (mass, magnetic field continuity, momentum, induced magnetic field and energy conservation) are solved subject to appropriate boundary conditions using Maple shooting quadrature. The influence of Prandtl number (Pr), magnetohydrodynamic body force parameter (β), reciprocal of magnetic Prandtl number (α) and nanofluid solid volume fraction (φ) on velocity, temperature and magnetic stream function distributions is investigated in the presence of strong Marangoni effects (ξ i.e., Marangoni parameter is set as unity). Magnetic stream function is accentuated with body force parameter. The flow is considerably decelerated as is magnetic stream function gradient, with increasing nanofluid solid volume fraction, whereas temperatures are significantly enhanced. Interesting features in the flow regime are explored. The study finds applications in the fabrication of complex biomedical nanofluids, biopolymers, etc.

Computational blood flow studies are providing an increasingly important complement to clinical experiments in 21st century biomedical engineering. Motivated by probing deeper into this topic, a theoretical and numerical study is presented of the flow induced by an impulsive acceleration to steady state hemodynamics in a curved tube which is investigated as a boundary layer developing with time from the curved entrance to a straight tube (blood vessel). The transient processes are simulated with a finite volume method solution of the Navier–Stokes equations. The rapid growth of the boundary layer with the core flow is captured in the curved entrance, along the tube to an axisymmetric flow in the downstream. Secondary flow patterns, centrifugal pressures and total head contours are correlated with longitudinal velocity distributions across various sections. It is observed that the entrance zone is controlled by uniform inlet velocity and centrifugal forces. The high pressure drop in the onset flow is associated with strong acceleration which is comparable to generating systolic pressures. The simulations further indicate that a sustained increment in volumetric flow rate is necessary to maintain the pressure wave in the aorta. Furthermore, the velocity distributions are shown to approach Hagen–Poiseuille flow in the downstream zone. The complex hemodynamic characteristics are visualized effectively with computational simulations and the study demonstrates the excellent ability of this approach in elaborating critical flow details in aortic hemodynamics.

In this article, we consider the barotropic quasigeostrophic equation of the ocean in the context of the β-plane approximation and small viscosity (see, e.g., [21, 22]). The aim is to study the behavior of the solutions when the viscosity goes to zero. To avoid the substantial complications due to the corners (see, e.g., [25]) which will be addressed elsewhere, we assume periodicity in one direction (0y). The behavior of the solution in the boundary layers at x = 0, 1 necessitate the introduction of several correctors, solving various analogues of the Prandtl equation. Convergence is obtained at all orders even in the nonlinear case. We also establish as an auxiliary result, the regularity of the solutions of the viscous and inviscid quasigeotrophic equations.

The singularly perturbed problems with a turning point were discussed in [21]. The case where the limit problem is compatible with the given data was fully resolved. However, with limited compatibility conditions on the data, the asymptotic expansions were constructed only up to the order of the level of compatibilities. In this paper, using a smooth cut-off function compactly supported around the turning point we resolve the difficulties incurred from the non-compatible data and finally provide the full asymptotic expansions up to any order.

In this paper, a coupled system of $m(\ge 2)$ second-order singularly perturbed differential equations of reaction–diffusion type with discontinuous source term subject to Dirichlet boundary conditions is studied, where the diffusive term of each equation is being multiplied by the small perturbation parameters having different magnitudes and coupled through their reactive term. A discontinuity in the source term causes the appearance of interior layers on either side of the point of discontinuity in the continuous solution in addition to the boundary layer at the end points of the domain. Unlike the case of a single equation, the considered system does not obey the maximum principle. To construct a numerical method, a classical finite difference scheme is defined in conjunction with a piecewise-uniform Shishkin mesh and a graded Bakhvalov mesh. Based on Green’s function theory, it has been proved that the proposed numerical scheme leads to an almost second-order parameter-uniform convergence for the Shishkin mesh and second-order parameter-uniform convergence for the Bakhvalov mesh. Numerical experiments are presented to illustrate the theoretical findings.

This paper develops efficient high-order convergent discretization methods for coupled systems of singularly perturbed parabolic convection-diffusion problems exhibiting boundary layers. The discretization of the problem consists of two splitting schemes in time on a uniform mesh and a high-order hybrid finite difference scheme in space on appropriate layer-adapted Shishkin mesh. The splitting schemes decoupled the vector approximate solution at each time level, resulting in a reduced computational time. The convergence of the discretization methods is discussed in two steps using the barrier function approach. The proposed methods are proved to be uniformly convergent, having first-order accuracy in time and almost second-order accuracy in space. The splitting schemes are very efficient in comparison with the standard discretization. Numerical results confirm the theoretical findings and illustrate the efficiency of the proposed discretization methods.

In this paper, a class of linear parabolic systems of singularly perturbed second-order differential equations of reaction–diffusion type with initial and Robin boundary conditions is considered. The components of the solution $\overrightarrow{u}$ of this system are smooth, whereas the components of $\frac{\partial \overrightarrow{u}}{\partial x}$ exhibit parabolic boundary layers. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first-order convergent in time and essentially first-order convergent in the space variable in the maximum norm uniformly in the perturbation parameters.

Bioconvection has shown significant promise for environmentally friendly, sustainable “green” fuel cell technologies. The improved design of such systems requires continuous refinements in biomathematical modeling in conjunction with laboratory and field testing. Motivated by exploring deeper the near-wall transport phenomena involved in bio-inspired fuel cells, in the present paper, we examine analytically and numerically the combined free-forced convective steady boundary layer flow from a solid vertical flat plate embedded in a Darcian porous medium containing gyrotactic microorganisms. Gyrotaxis is one of the many taxes exhibited in biological microscale transport, and other examples include magneto-taxis, photo-taxis, chemotaxis and geo-taxis (reflecting the response of microorganisms to magnetic field, light, chemical concentration or gravity, respectively). The bioconvection fuel cell also contains diffusing oxygen species which mimics the cathodic behavior in a proton exchange membrane (PEM) system. The vertical wall is maintained at iso-solutal (constant oxygen volume fraction and motile microorganism density) and iso-thermal conditions. Wall values of these quantities are sustained at higher values than the ambient temperature and concentration of oxygen and biological microorganism species. Similarity transformations are applied to render the governing partial differential equations for mass, momentum, energy, oxygen species and microorganism species density into a system of ordinary differential equations. The emerging eight order nonlinear coupled, ordinary differential boundary value problem features several important dimensionless control parameters, namely Lewis number (Le), buoyancy ratio parameter i.e. ratio of oxygen species buoyancy force to thermal buoyancy force (Nr), bioconvection Rayleigh number (Rb), bioconvection Lewis number (Lb), bioconvection Péclet number (Pe) and the mixed convection parameter ($𝜀)$ spanning the entire range of free and forced convection. The transformed nonlinear system of equations with boundary conditions is solved numerically by a finite difference method with central differencing, tridiagonal matrix manipulation and an iterative procedure. Computations are validated with the symbolic Maple 14.0 software. The influence of buoyancy and bioconvection parameters on the dimensionless temperature, velocity, oxygen concentration and motile microorganism density distribution, Nusselt, Sherwood and gradient of motile microorganism density are studied. The work clearly shows the benefit of utilizing biological organisms in fuel cell design and presents a logical biomathematical modeling framework for simulating such systems. In particular, the deployment of gyrotactic microorganisms is shown to stimulate improved transport characteristics in heat and momentum at the fuel cell wall.