Boundary Layer Solution Research Articles

A sustainable and clean method for vanadium extraction from shale ore by suspension oxidation roasting: A comparative study of leaching behavior

Vanadium (V) is an important rare metal element that is widely applied in a variety of fields. It is necessary to develop an efficient method for the utilization of V-bearing shale ore. In this work, we investigated the mechanism of enhanced vanadium extraction from shale ore by suspension oxidation roasting (SOR) by comparing the leaching behavior between traditional direct acid leaching and SOR-acid leaching. After being roasted, the V leaching rate obviously increased from 60.33 % to 83.20 % while the H2SO4 dosage, leaching temperature, and leaching time decreased 20 wt%, 30 °C, and 5 h, respectively. The non-appearance of CaF2 alleviated environment pollution from wastewater. Roasting activation improved the chemical reaction activity and transformed the dense structure into a loose and porous structure which contributed to improving the diffusion rate of acid into particles and the solutions. Kinetic studies shown that SOR contributed to decrease the activation energy of leaching system and solid layer or solution boundary layer diffusion contributed to the control of roasted product leaching rate.

The asymptotic solution of the elasticity theory problem for a radially inhomogeneous cylinder

The elasticity theory problem for a radially inhomogeneous cylinder of small thickness, whose elastic moduli are arbitrary continuous functions depending on the radius of the cylinder, is considered. It is assumed that the side surface of the cylinder is stress-free, and the boundary conditions that keep the cylinder in equilibrium are given at its seats. Asymptotic solutions are constructed by the asymptotic integration method. It is shown that the asymptotic solution consists of the sum of the penetrating solution, simple boundary effect and boundary layer solutions. The character of the stress-strain state corresponding to the penetrating solution, simple boundary effect and boundary layer solutions is determined. Asymptotic formulas are obtained for displacements and stresses, which allow to calculate the stress-strain state of a cylinder.The problem of torsion of a radially inhomogeneous cylinder is studied, with its lateral surface free from stress and the boundary conditions keeping it in equilibrium at its seats. By applying the asymptotic integration method, it is determined that the asymptotic solution for the torsion problem consists of the sum of the penetrating solution and boundary layer solutions.Numerical analysis is performed and the effect of material inhomogeneity on the stress-strain state of the cylinder is evaluated.

Mixed convection flow over a horizontal plate and the horizontal wake far downstream

The laminar mixed convection flow over a heated or cooled horizontal plate of finite length is a surprisingly intriguing problem. The hydrostatic pressure jump at the trailing edge and across the wake is to be compensated by induced circulation in the outer potential flow, similar to a plate at a non-zero angle of attack. The mixed convection flows over semi-infinite and finite horizontal plates, respectively, are thus substantially different. As the hydrostatic pressure jump remains finite along the infinitely extended wake, the induced outer potential flow around the plate would grow beyond bounds in a domain extending to infinity. Thus, boundary conditions must be imposed at the boundaries of a finite domain. However, the numerical solution of the Navier-Stokes equations remains a challenging problem due to the lack of appropriate outflow conditions. The traditional outflow condition cannot comply with the finite hydrostatic pressure jump across the wake. Thus, an asymptotic expansion for the wake far downstream from the plate is performed. Remarkably, the perturbation of the flow does not decay with increasing distance from the plate as in a classical wake. The asymptotic solution is implemented as an outflow boundary condition for the numerical solution of the Navier-Stokes equations. The results are compared with a boundary-layer solution obtained in previous work for the limiting case of vanishing Prandtl number.

ON THE PROBLEM OF THE THEORY OF ELASTICITY FOR THE RADIALLY INHOMOGENEOUS TRANSVERSAL ISOTROPIC CYLINDER WITH A FIXED LATERAL SURFACE

Using the method of asymptotic integration of the equations of the theory of elasticity, the axisymmetric problem of the theory of elasticity for a radially inhomogeneous transversally isotropic cylinder of small thickness is studied. Suppose that the elastic moduli are arbitrary continuous functions of the radius of the cylinder. It is assumed that the side part of the cylinder is fixed, and stresses are set at the ends of the cylinder, leaving the cylinder in equilibrium. Solutions have been determined having the nature of a boundary layer and are localized at the ends of the cylinder. The first terms of its asymptotic expansion coincide with the Saint-Venant edge effect in the theory of plates. The nature of the stress-strain state has been studied. It is shown that some boundary layer solutions may not be extinguished by propagating far from the ends.

A novel membrane-free electrochemical separation-filtering crystallization coupling process for treating circulating cooling water

The traditional electrochemical descaling process exhibits drawbacks, including low OH− utilization efficiency, constrained cathode deposition area, and protracted homogeneous precipitation time. Consequently, this study introduces a novel membrane-free electrochemical separation-filtering crystallization (MFES-FC) coupling process to treat circulating cooling water (CCW). In the membrane-free electrochemical separation (MFES) system, OH− is rapidly extracted by pump suction from the porous cathode boundary layer solution, preventing neutralization with H+, thereby enhancing the removal of Ca2+ and Mg2+. Experimental results indicate that the pH of the pump suction water can swiftly increase from 8.13 to 11.42 within 10 min. Owing to the high supersaturation of the pump suction water, this study couples the MFES with a filtration crystallization (FC) system that employs activated carbon as the medium. This approach captures scale particles to enhance water quality and expedites the homogeneous precipitation of hardness ions, shortening the treatment time while further augmenting the removal rate. After the MFES-FC treatment, the single-pass removal rates for total hardness, Ca2+ hardness, Mg2+ hardness, and alkalinity in the effluent reached 92 %, 97 %, 64 %, and 67 %, respectively, with turbidity of 3 NTU, current efficiency of 86.6 %, and energy consumption of 7.19 kWh·kg−1 CaCO3. This coupling process facilitates an effective removal of hardness and alkalinity at a comparatively low cost, offering a new reference and inspiration for advancements in electrochemical descaling technology.

Hypersonic boundary layer over a flat plate with slip and shear nonequilibrium effects

Near-space hypersonic vehicles could encounter significant rarefied nonequilibrium effects during the flight through atmosphere, which largely influence the gas-surface momentum and heat transfer. In this paper, hypersonic boundary layer over a flat plate with velocity slip, temperature jump, and shear nonequilibrium effects is theoretically considered. The slip boundary conditions and nonlinear transport relations are embedded into the boundary-layer equations to describe the flow. Local similar solutions are derived, and key parameters for characterizing slip and shear nonequilibrium effects are determined. The velocity-slip and temperature-jump effects are determined by [(2−σu)/σu]Mae/Rex and [(2−σT)/σT]Mae/Rex respectively, and the shear nonequilibrium effect is characterized by Mae2/Rex. The obtained boundary-layer solutions are compared with the Navier–Stokes solutions for a Mach 4.5 slip flow, and the results of Direct Simulation Monte Carlo for a Mach 10 rarefied flow, good agreements are achieved. The separate and combined effects of velocity slip, temperature jump, and shear nonequilibrium on boundary-layer solutions and momentum/heat transfer are clarified. The results show that both the slip and shear nonequilibrium effects cause the boundary layer to become thinner and decrease the skin friction and Fourier heat conduction. However, with including sliding friction, the total heat flux might even increase as the slip degree increases. These results provide valuable insight into the boundary-layer characteristics of hypersonic near-continuum flows.

Analytical solutions of turbulent boundary layer beneath forward-leaning waves

Analytical solutions of turbulent boundary layer beneath forward-leaning waves

Numerical Investigation of Diffusion Flame in Transonic Flow with Large Pressure Gradient

A finite-volume method for the steady, compressible, reacting, turbulent Navier–Stokes equations is developed by using a steady-state-preserving splitting scheme for the stiff source terms in chemical reactions. Laminar and turbulent reacting flows in a mixing layer with a large streamwise pressure gradient and acceleration are studied and compared to boundary-layer solutions. It reveals that chemical reactions strongly enhance turbulent transport due to the intensive production of turbulence by the increased velocity gradients and thus produce large turbulent viscosities in the reaction region. The influence of vitiated air on the combustion process and aerodynamic performance is also investigated in the cases of turbulent mixing layers and highly loaded transonic turbine cascades. Both cases indicate the viability of the turbine-burner concept.

Convective flow generated in viscous liquid by isothermal and isoconcentration distribution in crown enclosure with novel aspects of inclined magnetic field

ABSTRACT Combined buoyancy shifts from temperature and concentricity gradients create double-diffusive convection. Complex Crown form enclosure with heat and mass source brings originality and practicality by enhancing use in a broad variety of technical, industrial, and geophysical operations. Current artifact is to explore thermosolutal diffusion in buoyantly driven flowing of viscid liquid contained in a crown enclosure with the installation of a uniformly heated and soluted circular cylinder. Novel physical characteristics of inclining magnetic field are introduced, and distributions affected by cylinder radius change are compared. Dimensionless PDEs are solved numerically using finite elements. Grid independence test ensures simulation mesh level. Hybrid meshing with triangular and rectangular components discretizes domain. Linear interpolating polynomials approximate pressure, whereas quadratic polynomials approach velocity, temperature, and concentricity. PARDISO solves a non-linear system of equations efficiently. Average Nusselt and Sherwood numbers are estimated by comparing relevant parameters to cylinder radius change. Heat transference rate rises to 40% and mass transport upsurges to 60% when radius of cylinder is increased from 0.1 to 0.2. Hartmann number tends to affect a decline in Nusselt and Sherwood quantities. Reduction in thermal and solutal boundary layers near surface of the cylinder is observed against the improve in radius of cylinder.

Brownian motion and thermophoresis influence in magnetized Maxwell upper-convected stagnation point fluid flow via a stretching porous surface

This article focuses on the effects of Brownian motion and thermophoresis convection in the stagnation point flow of a Maxwell upper-convected fluid over a non-Darcian porous surface with slip conditions and a magnetic inclination effect. The Maxwell dissipative fluid accounts for Joule heating due to an imposed magnetic field and porous medium resistance. At the same time, the Cattaneo-Christov heat flux model represents thermal relaxation in contrast to the conventional Fourier law. The resulting nonlinear partial differential equations are transformed into ordinary differential equations (ODEs) using similarity variables. The analysis revealed the influence of dimensionless numbers, including Deborah, Eckert, Prandtl, chemical reaction, thermal relaxation, inclination and slip parameters. The findings were presented using graphs and tables. An increase in the thermophoresis parameter notably led to higher concentration and temperature profiles. In contrast, increasing Brownian motion (Nb) decreased the solutal boundary layer thickness but enhanced the thermal boundary layer.

Local correlation, compressibility, and crossflow corrections of γ-Reθ transition model for high-speed flows

Based on the original γ-Reθ transition model framework, in this work, an improved local correlation-based transition closure model is developed for high-speed flows. The local correlation between the vorticity Reynolds number and the momentum thickness Reynolds number obtained by compressible boundary-layer self-similar solutions, local compressibility correction including the pressure gradient parameter and momentum thickness Reynolds number, and local crossflow correlation are applied to improve the original γ-Reθ model for hypersonic transition predictions. The function Fonset1 used to control the transition onset and several relevant model parameters are also modified to make the improved model suitable for high-speed flow. The improved transition model is validated through several basic test cases under a wide range of flow conditions, including high-speed flat plates, sharp cones, double ramp, Hypersonic International Flight Research Experimentation, and complex hypersonic configuration X-33 vehicles. The numerical results show that the transition onset locations and the changes of heat transfer rate predicted by the present improved transition model are reasonably consistent with experimental results. The proposed model predicts the high-speed boundary layer transition behaviors induced by streamwise and crossflow instabilities with reasonable precision.

Analytical and Numerical Solutions for Heat, Mass and Entropy Generation of Fourth Grade Nanofluid Flow Over A Vertical Plate

The focus of this research is to investigate magnetohydrodynamic flow, entropy generation, heat and mass transfer analysis for fourth grade nanofluid over a vertical plate in a porous medium subject to slip and convective boundary conditions. Lie group invariant similarity variable is used to transform the partial differential equations (PDEs) governing the problem to system of coupled nonlinear ordinary differential equations (ODEs). The resulting dimensionless ODEs are solved using Homotopy Perturbation Method (HPM). HPM results are validated with the numerical solutions via shooting method alongside the six order Runge-Kutta integration technique. The results revealed effects of new embedded governing flow parameters on velocity, temperature, entropy generation and concentration profiles.Furthermore, the impact of new embedded flow parameters on skin friction coefficient, Nusselt number and Sherwood number at the plate surface are investigated and the results are shown in tabular forms. The results indicate that suction on the plate can be used to control momentum, thermal and solutal boundary layer thickness. This research discovered that magnetic parameter, Eckert number, Biotsolutal number and Hartmann number can be respectively used to adjust fluid flow’s velocity, temperature, concentration and entropy generation. This research also detected that the force driving the fourth grade nanofluid flow called buoyancy driven force can be accelerated or decelerated by adjusting angle of magnetic field inclination. The present investigation has applications in industries and engineering, such as petroleum industries, chemical industries and metallurgy sciences

Hiemenz stagnation point flow of a second-order micropolar slip flow with heat transfer

Purpose This paper aims to study a non-Newtonian micropolar fluid flow over a bidirectional flexible surface for multiple exact solutions of momentum boundary layer and thermal transport phenomenon subject to wall mass flux, second-order slip and thermal jump conditions. Design/methodology/approach The coupled equations are transformed into ordinary differential equations using similarity variables. Analytical and numerical techniques are used to solve the coupled equations for single, dual or multiple solutions. Findings The results show that the stretching flow, shrinking flow, the wall drag, thermal profile and temperature gradient manifest large changes when treated for special effects of the standard parameters. The role of critical numbers is definitive in locating the domains for the existence of exact solutions. The nondimensional parameters, such as mass transfer parameter, bidirectional moving parameter, plate deformation strength parameter, velocity slips, material parameter, thermal jump and Prandtl number, are considered, and their physical effects are presented graphically. The presence of governing parameters exhibits special effects on the flow, microrotation and temperature distributions, and various exact solutions are obtained for the special parametric cases. Originality/value The originality and value of this work lie in its exploration of non-Newtonian micropolar fluid flow over a bidirectional flexible surface, highlighting the multiple exact solutions for momentum boundary layers and thermal transport under various physical conditions. The study provides insights into the effects of key parameters on flow and thermal behavior, contributing to the understanding of complex fluid dynamics.

Thermal boundary-layer solutions for forced convection in a porous domain above a flat plate

Thermal boundary-layer solutions for forced convection in a porous domain above a flat plate

MHD Casson fluid flow with chemical reaction and suction in a porous medium over an exponentially stretching sheet

MHD Casson fluid flow with chemical reaction and suction in a porous medium over an exponentially stretching sheet was studied. The mathematical problem was solved numerically using shooting technique with fourth order Runge-Kutta method after converting the resulting partial differential equations to a system of ordinary differential equations and applying similarity transformation on the latter. The effects of some fluid parameters on the momentum and concentration profiles are examined through graphs using MATLAB. The results from the graphs show that increase in the values of Casson parameter decreases the velocity profiles and causes increase in the concentration profiles. The momentum boundary layers decrease while the solute boundary layers increase with increasing values of magnetic parameter and the permeability parameter. The concentration profile reduces with increasing values of reaction parameters.

A multiscale asymptotic expansion for combustion system with composite materials

This paper discusses the multiscale asymptotic expansion for combustion system with composite materials or porous media in thermal protection system. The combustion system consists of two nonlinear equations with rapidly oscillating coefficients, which are posed on a time-varying domain and computationally expensive to solve with the finite element method. To efficiently solve this system, we propose the standard/novel multiscale asymptotic expansions by combining them with boundary layer solutions. Numerical simulation is performed to validate the accuracy and efficiency of the newly proposed methods, which show that the novel multiscale asymptotic expansion is more accurate than the standard one.

Dual Stratification Effects on Mixed Convective Electro-magnetohydrodynamic Flow over a Stretching Plate with Multiple Slips and Cross Diffusion

This paper analyzed the effects of dual stratification on mixed convective electro-magnetohydrodynamic flow over stretching plates with multiple slips. With the aid of the similarity transformation technique were, the governing boundary equations, that were partial differential equations, were changed to a couple of ordinary differential equations and then solved with fourth order Runge Kutta method and Newton’s Raphson shooting techniques. It was observed that the magnetic field, Buoyancy ratio, permeability, momentum slip parameters, Dufour, Soret and Brinkmann numbers made the thermal boundary layer thickness to increase but the solutal stratification, electric field, chemical reaction, solutal slip, suction, thermal slip and thermal stratification parameters, Prandtl, Richardson and Lewis number decreased the thickness of the thermal boundary layer. The Buoyancy ratio, permeability, momentum slip, thermal slip and thermal stratification parameters and Soret number enhanced the solutal boundary layer thickness.

Soret Effect with Chemical Reaction on Unsteady MHD Flow of Nanofluid Past an Impulsively Started Infinite Vertical Plate Embedded in a Porous Medium

This article deals with the analysis of the thermal-diffusion effect, chemical reaction and heat generation on the convective hydromagnetic flow of water-based nanofluid past an instantaneously accelerated infinite vertical plate nested in a porous medium. Simultaneous application of ramped temperature, ramped velocity, and ramped concentration has been considered. With the help of Laplace transformation, the set of transformed domain equations has been resolved. The consequences of various flow parameters involved in the study are analysed graphically. The results exhibit that the hydrodynamic and solutal boundary layer elevates for the higher value of the Soret effect Sr. Moreover, the rate of heat transfer hikes and on the other hand, the rate of mass transfer drops on account of the volume concentration of nanoparticles φ. Again, it is observed that the temperature, concentration and velocity field are dominated in the ramped condition by that of the isothermal condition.

Efficient Two-Way Coupled Analysis of Steady-State Particle-Laden Hypersonic Flows

A direct solution approach for surface erosion in particle-laden hypersonic flows is extended for use in low-cost two-way coupled solutions of dilute gas-particle flows. The trajectory control volume method, which uses a sparse set of probe particles to predict surface erosion distributions on general vehicles, is reformulated for the solution of source terms by mean trajectory subdivision and computing a flux differencing. The approach is verified successfully against a boundary-layer solution and shown to agree well with experimental measurements. A representative Mars entry case, with conditions and geometry based on the ExoMars Schiaparelli capsule, is solved with the approach to study the impact of two-way coupling on surface heating and erosion. Results indicate that, for realistic loading conditions, heating is largely unmodified compared to one-way coupled results at peak heating trajectory conditions, and no measureable difference is observed in the surface erosion rate. At exaggerated loading conditions high enough to observe coupling effects, the worst-case collisional heating can increase heating by up to 60%.

Postcritical Imperfection Sensitivity of Functionally Graded Piezoelectric Cylindrical Nanoshells Using Boundary Layer Solution

The nonlocal (NL) and the strain gradient (SG) based equivalent continuum theories were proposed for nanomechanics by accommodating the long-range molecular interactions and to bypass the computationally expensive atomistic simulations. Nanostructures are often made of smart materials (e.g., piezoelectric, piezoceramic, and flexoelectric) for multifunctional properties. Applications of nanostructures in critical systems deserve accurate analysis. The buckling and postcritical behavior of an axially loaded, piezoelectric, nonlocal strain gradient (NLSG) thin cylindrical shells with functionally graded elastic properties were presented in earlier studies following Donnell’s approach, leading to a set of stiff, nonlinear, partial differential equations, accommodating the prebuckling nonlinearity and large postcritical deflection. This study extends the previous work by accommodating geometric imperfections in the formulation. Furthermore, a consistent thickness-wise distribution for the electric potential is also adopted following the Maxwell equation. The boundary layer (BL) concept is employed to solve the resulting nonlinear equations via asymptotic expansions of the regular and the BL fields. The solution is numerically illustrated on moderately short shells, illustrating the influence of the geometric imperfections. The critical load remains imperfection sensitive, yet the imperfections change an unstable postcritical path into a stable one. The influence of the functional gradation and the external electric fields on the imperfection sensitivity behavior are illustrated.