Boundary Layer Solution Research Articles

Construction of homogeneous solutions in the torsion problem for a transversally isotropic sphere with variable elastic moduli

The object of research is the problem of torsion for a radially inhomogeneous transversally isotropic sphere and the study based on this three-dimensional stress-strain state. To establish the scope of applicability of existing applied theories and to create more refined applied theories of inhomogeneous shells, it is important to study the stress-strain state of inhomogeneous bodies based on three-dimensional equations of elasticity theory. The problem of torsion of a radially inhomogeneous transversally isotropic non-closed sphere containing none of the poles 0 and π is considered. It is believed that the elastic moduli are linear functions of the radius of the sphere. It is assumed that the lateral surface of the sphere is free from stresses, and arbitrary stresses are given on the conic sections, leaving the sphere in equilibrium. The formulated boundary value problem is reduced to a spectral problem. After fulfilling the homogeneous boundary conditions specified on the side surfaces of the sphere, a characteristic equation is obtained with respect to the spectral parameter. The corresponding solutions are constructed depending on the roots of the characteristic equation. It is shown that the solution corresponding to the first group of roots is penetrating, and the stress state determined by this solution is equivalent to the torques of the stresses acting in an arbitrary section θ=const. The solutions corresponding to the countable set of the second group of roots have the character of a boundary layer localized in conic slices. In the case of significant anisotropy, some boundary layer solutions decay weakly and can cover the entire region occupied by the sphere. On the basis of the performed three-dimensional analysis, new classes of solutions (solutions having the character of a boundary layer) are obtained, which are absent in applied theories. In contrast to an isotropic radially inhomogeneous sphere, for a transversely isotropic radially inhomogeneous sphere, a weakly damped boundary layer solution appears, which can penetrate deep far from the conical sections and change the picture of the stress-strain state.

Dynamics of eddying abyssal mixing layers over sloping rough topography

Abstract ABSTRACT: The abyssal overturning circulation is thought to be primarily driven by small-scale turbulent mixing. Diagnosed watermass transformations are dominated by rough topography “hotspots”, where the bottom-enhancement of mixing causes the diffusive buoyancy flux to diverge, driving widespread downwelling in the interior—only to be overwhelmed by an even stronger up-welling in a thin Bottom Boundary Layer (BBL). These watermass transformations are significantly underestimated by one-dimensional (1D) sloping boundary layer solutions, suggesting the importance of three-dimensional physics. Here, we use a hierarchy of models to generalize this 1D boundary layer approach to three-dimensional eddying flows over realistically rough topography. When applied to the Mid-Atlantic Ridge in the Brazil Basin, the idealized simulation results are roughly consistent with available observations. Integral buoyancy budgets isolate the physical processes that contribute to realistically strong BBL upwelling. The downwards diffusion of buoyancy is primarily balanced by upwelling along the sloping canyon sidewalls and the surrounding abyssal hills. These flows are strengthened by the restratifying effects of submesoscale baroclinic eddies and by the blocking of along-ridge thermal wind within the canyon. Major topographic sills block along-thalweg flows from restratifying the canyon trough, resulting in the continual erosion of the trough’s stratification. We propose simple modifications to the 1D boundary layer model which approximate each of these three-dimensional effects. These results provide local dynamical insights into mixing-driven abyssal overturning, but a complete theory will also require the non-local coupling to the basin-scale circulation.

Magnetized Flow of Electrically Induced Maxwell Nanofluid over Reactive Stretching Plate with Thermal Stratification

Effects of thermal stratification on magnetized flow of electrically induced Maxwell nanofluid over reactive stretching plate have been analyzed. The nonlinear ordinary differential equations governing the flow problem were obtained by applying Similarity transformation. The resulting model was then solved with the aid of the fourth order Runge-Kutta algorithm along with the shooting technique. Results for pertinent flow parameters were tabulated and analyzed graphically. The Richardson number was noted to appreciate the momentum boundary layer thickness but it decayed both the thermal and solutal boundary layer thicknesses.

Casson nanofluid performance on MHD convective flow past a semi‐infinite inclined permeable moving plate in presence of heat and mass transfer

AbstractThis article investigates the radiation and chemical reaction effects on the free convection heat and mass transfer flow of Casson nanofluids (Cu and TiO2) over an inclined porous vertical plate embedded in a porous medium in the presence of radiation absorption and constant heat source under fluctuating boundary conditions. The plate is moved with a constant velocity of U0, and temperature and the concentration are assumed to be fluctuating with time harmonically from a constant mean at the plate. The perturbation technique is applied to solve the governing equations of the flow and pointed out the variations in velocity, temperature, and concentration with the use of graphical presentations. The impact of several parameters on local skin friction, Nusselt number, and Sherwood number is also noticed and discussed. It is concluded that the resultant velocity reduces with increasing Casson parameter and suction parameter, velocity, and temperature enhanced with increasing radiation absorption and diffusion thermo parameters. Also, it is noticed that the solutal boundary layer thickness decreases with an increase in chemical reaction parameters. It is because chemical molecular diffusivity reduces for higher values of Kr.

Eyring-Powell nano liquid flow through permeable elongated sheet conveying inclined magnetic field subject to constructive chemical reaction and multiple slip effects

ABSTRACT The present study examines the multiple slip effects and constructive chemical reaction on Eyring-Powell nanoliquid through a permeable elongated sheet on an inclined magnetic field. The energy balance equation includes Heat and Joule dissipation terms. The renowned Buongiorno nanofluid model is employed extensively to explore the thermophoresis and Brownian motion phenomena. The primary partial differential equations (PDEs) of governing flow phenomena are remodeled to non-linear ordinary differential equations (ODEs) by adopting proper similarity invariants. Runge-Kutta 4th order quadrature employing shooting technique is utilized to transform boundary value problem to initial value problem to seek numerical results via graphs and tables on physically interesting parameters. Current outcomes are validated through the comparison with previously published studies. An applied magnetic field slows down the fluid and raises both thermal and solutal boundary layers. It is significant to notice that the thermal boundary layer rises, as the Brownian and Thermophoresis are amplified.

Convective transport from geothermal borehole heat exchangers embedded in a fluid-saturated porous medium

Multi-dimensional natural convection and mixed convection have been studied numerically for the vertical borehole, a heated cylinder characterized by length Lb and diameter D, of a ground source heat pump in a fluid-saturated porous medium. The effect of natural convection over a range of ground properties characterized by Rayleigh number (RaL) is found to be significant and deviates strongly from boundary layer solutions for Rayleigh number less than 106, due to the flow turning required by the ground surface. For instance, the natural convection heat flux deviates from boundary layer prediction by over 20% for RaL = 104. Only at Rayleigh numbers of less than 2 × 103 can the heat transfer be treated as simple conduction, with the deviation less than approximately 5%, for fluid-saturated soil properties. The presence of groundwater flow, due to a hydraulic gradient, produces mixed convection leading to a convective heat transfer depending also upon Peclet number (PeD). A sufficient range of properties are examined such that the multi-dimensional mixed convection is shown to be captured between asymptotes to natural and forced convection. The dimensionless group (Lb/D)PeD/RaL is identified and shown to parameterize the variation between these two limits, with values less than 1.4 asymptoting to natural convection and values greater than 2.9 asymptoting to forced convection. Finally, a correlation encompassing all three regimes, natural, forced and mixed, is proposed.

INFLUENCE OF SORET AND DUFOUR, ACTIVATION ENERGY ON MHD CHEMICALLY REACTING NANOFLUID FLOW PAST A VERTICAL SURFACE WITH RADIATION

The current article aims to investigate Soret and Dufour's effects on MHD flow of heat and mass transfer nanofluid over a vertical stretching sheet with activation energy and radiation. Using Similarity transformations, the non-linear momentum, energy, solute, and nanoparticle concentration boundary layer equations are simplified. The resulting boundary layer equations are solved numerically with bvp4c and along with the shooting technique. The impact of various parameters on velocity, temperature, concentration, and nanoparticle volume fraction, Skin friction, local Sherwood and local Nusselt numbers are presented through graphs. It is found that the temperature, fluid velocity, and nanoparticle volume fraction enhanced with both Dufour and Soret effects.

ACCURATE BENCHMARK RESULTS OF BLASIUS BOUNDARY LAYER PROBLEM USING A LEAPING TAYLOR’S SERIES THAT CONVERGES FOR ALL REAL VALUES

Blasius boundary layer solution is a Maclaurin series expansion of the function \(f(\eta)\), which has convergence problems when evaluating for higher values of \(\eta\) due to a singularity present at \(\eta\approx-5.69\). In this paper we are introducing an accurate solution to \(f(\eta)\) using Taylor's series expansions with progressively shifted centers of expansion(Leaping centers). Each series is solved as an IVP with the three initial values computed from solution of the previous series, so the gap between the centers of two consecutive expansions is selected from within the convergence disc of the first series. The last series is formed such that it is convergent for a reasonable high \(\eta\) value needed for implementing the boundary condition at infinity. The present methodology uses Newton-Raphson method to compute the value of the unknown initial condition viz. \(f''(0)\) in an iterative manner. Benchmark accurate results of different parameters of flat plate boundary layer with no slip and slip boundary conditions have been reported in this paper.

Entropy optimization of MHD Casson-Williamson fluid flow over a convectively heated stretchy sheet with Cattaneo-Christov dual flux

This work confers the comparative study of Casson-Williamson fluid flow over a heated porous stretchy sheet. The energy and mass transfer equations are modeled by Cattaneo-Christov theory. The governing flow models were altered into an ODE model with the use of suitable transformations. The HAM scheme is applied to find the series solutions. The response of diverse flow variables on fluid speed, fluid warmness, liquid concentration, skin friction coefficient, local Nusselt number, local Sherwood number, local entropy generation number and Bejan number are analyzed through graphs and charts. It is found that the fluid speed subsides when surging values of the magnetic field, porosity, Casson fluid, Williamson fluid and injection/suction parameters. The fluid warmness escalates for a high amount of radiation, convective heating and heat generation/absorption parameters and its suppresses when enriching the convective cooling parameter. The chemical reaction parameter leads to rise in the thickness of the solutal boundary layer. The higher quantity of skin friction coefficient occurs in Casson fluid compared to Williamson fluid. The local entropy generation decimates when growing the Casson and Williamson parameters and it aggravates when raising the Biot number. The Bejan number exalts when upgrading the Reynolds, Brinkman and Biot numbers.

Boundary Layer Solution of the Boltzmann Equation for Diffusive Reflection Boundary Conditions in Half-Space

We study steady Boltzmann equation in half-space, which arises in the Knudsen boundary layer problem, with diffusive reflection boundary conditions. Under certain admissible conditions and the source term decaying exponentially, we establish the existence of boundary layer solution for both linear and nonlinear Boltzmann equation in half-space with diffusive reflection boundary condition in $L^\infty_{x,v}$ when the far-field Mach number of the Maxwellian is zero. The continuity and the spacial decay of the solution are obtained. The uniqueness is established under some constraint conditions.

Wave-averaged properties for non-breaking waves in a canopy: Viscous boundary layer and vertical shear stress distribution

The wave-averaged contributions to the vertical shear stress distribution in both submerged and emerging canopies has been derived. Additionally, the contribution from a viscous boundary layer on top of a submerged canopy on both the vertical shear stress distribution and the Lagrangian Stokes drift velocity are quantified. The analytical model is in principal restricted to a description of the local behaviour, but it is shown, how the model can be used to describe the wave damping and the velocity field through an entire canopy. The theoretical findings are validated with laboratory experimental data on the wave attenuation across a canopy and the in-canopy wave-induced velocities. The combined analytical and experimental findings are synthesised to provide recommendations of how to incorporate effect of waves in large-scale practical engineering models for Eulerian-mean hydrodynamics. • An analytical shear stress distribution due to wave dissipation in canopies. • A constant-viscosity boundary layer solution in a submerged canopy is proposed. • Viscous effects on Lagrangian Stokes drift is quantified. • Wave attenuation is predictable with reduced velocities and fixed force coefficients. • Procedure to link spectral wave models and hydrodynamic models is outlined.

Thermal and diffusion Cattaneo-Christov heat flux models on Walter’s-B Buongiorno nanofluid over an electromagnetic surface with second order velocity slip

The intension of the current study is to explore optimization in heat transfer using Cattaneo-Christovs thermal and the solutal diffusions along with heat source/sink which are non-uniform on a stagnation points flow of nanoWalter’s B fluid over an electromagnetic sheet subjected to the multiple slip mechanisms. This study also scrutinizes the role of electromagnetic fields. The flow equations are modified via incorporating suitable transformations into a self-similarity equalities. Further numerically solved by employing Runge–Kutta method of shooting technique. The acquired results shows good agreement with the previous published works. The noteworthy findings are- Walter’s B nanofluid which flow parallel to the electromagnetic sheet is assisted by the Lorentz force. Electromagnetic sheet can be used for better cooling since improvement in Hartmann declines thermal boundary layer. Augmentation in thermal, velocity, and the solutal slip parameter shrink the hydrodynamic, solutal and thermal boundary layer.

Numerical study of heat and mass transfer enhancement in Prandtl fluid MHD flow using Cattaneo-Christov heat flux theory

Heat and mass transfer have numerous industrial applications. The classical heat and mass transfer laws (Fourier and Fick laws) do not predict thermal and solute relaxation time phenomena. However, in this article, generalized modeling related to simultaneous heat and mass transfer in non-Newtonian fluid in the presence of chemical reaction and heat generation is presented and models are numerically solved by the finite element method (FEM). Hybrid nanoparticles A g and F e 3 O 4 are considered and novel correlations are inserted during numerical simulations. The present results have a good agreement with already published benchmark. Thermal relaxation time is the characteristics of the fluid due to which it avoids or tries to avoid the thermal changes. The fluid with thermal relaxation characteristic tries to restore the thermal equilibrium and hence the temperature of the fluid is decreased. The solute relation is incorporated in the concentration equation from generalized Fick's law and solute relaxation time has shown a decreasing tendency in the concentration field. The solute boundary layer region can be controlled via an increase in the solute relaxation parameter. Ohmic dissipation in hybrid nanofluid A g − F e 3 O 4 − Prandtl fluid) is stronger than that in mono nanofluid ( A g − Prandtl fluid). Hybrid nanofluid ( A g − F e 3 O 4 − Prandtl fluid) produces more heat due to Joule heating than that produced by mono nanofluid ( A g − Prandtl fluid).

Impact of the Cattaneo–Christov heat flux on heat and mass transfer analysis of a hybrid nanofluid flow over a vertical cone

This article presents the sway of thermal radiation and chemical reaction on the MHD flow of a Ag/SWCNT–Water–based Maxwell hybrid nanoliquid over a vertical cone by considering the Cattaneo – Christov heat flux, convective boundary condition and suction parameter. The governing dimensionless equations are solved by using the finite-element method. The sway of dissimilar parameters on a hydrodynamic, thermal and solutal boundary layer is perceived and the results are laid out graphically. Additionally, the values of the skin friction coefficient, Nusselt number and Sherwood number are authorised for different values of pertinent parameters and the results are demonstrated in tabular form. The key observations of the present study are that the nanoparticle volume fraction increases the velocity of the hybrid nanofluid. The temperature of the hybrid nanofluid increases with augmenting Biot number values. The values of the skin friction coefficient and Sherwood number decline; nevertheless, the value of the Nusselt number escalates as the values of the Deborah number improve.

Compressible high-pressure lubrication flows in thrust bearings

We present a detailed derivation of the Reynolds equation and its corresponding energy equation for three-dimensional, steady, laminar, compressible flows of single-phase Navier–Stokes fluids in thrust bearings. These equations are shown to be valid over most of the dense and supercritical gas regime except for the vicinity of the thermodynamic critical point. It is shown that the primary thermodynamic function governing the lubrication flow of high-pressure gases is the effective bulk modulus defined as the ratio of the bulk modulus to the shear viscosity. Numerical solutions to our Reynolds equation are obtained using a finite difference scheme for both moderate and high-speed flows. Approximate solutions to our Reynolds equation for high-speed flows are also derived through a perturbation analysis. It is found that boundary layers form on three out of four edges of the thrust pad. At the inner and outer radii of the pad, the flow is governed by a nonlinear heat equation. As the main flow leaves the pad, the flow is governed by a nonlinear relaxation equation. These three boundary layer solutions are rendered consistent by the construction of boundary layer solutions in the corner regions. A composite solution is developed which provides a single approximation and has the same accuracy as the individual approximations in their respective regions of validity.

Non‐Newtonian nanofluid characteristics over a melting wedge: A numerical study

AbstractA study is conducted to determine the impact of joule heating, thermo‐diffusion, and chemical reaction effect on wedge flow with melting. Using similarity transformation, the nonlinear PDEs regulate nanofluid flow is converted to nonlinear ODEs. The MATLAB solver is used to solve the boundary value problem numerically. The interaction of relevant physical entities on nanoparticle concentration, nanofluid temperature, nanofluid velocity, skin friction, rate of heat, and mass transfer is graphically portrayed. This study will aid in the development of cooling devices and various shapes in heat sinks, as well as improving the heat transfer characteristics of Casson flow and strengthening formerly industrial uses. In the limiting situation, current findings are compared to analysis of findings. Flow velocity and concentration compacts in association with enhancing values of chemical reaction factor while temperature increases with enhancing the values of chemical reaction parameter. An upsurge in the temperature of the fluid is seen with the increasing Eckert number. It is found that the melting process increases the thicknesses of Solutal, thermal, and momentum boundary layers while it reduces mass transfer rate, heat transfer rate, and Skin friction. The Casson fluid displays a superior heat transfer mechanism than the Newtonian fluid. This study would be valuable in designing cooling gadgets and heat sinks of various shapes which will enhance the heat transfer properties of Casson nanofluids thereby increasing their applications in industrial perspectives. Moreover, the study reveals the novel applications of Casson nanofluids in cooling devices and heat sinks.

Analysis of the stress-strain state of a radially inhomogeneous transversely isotropic sphere with a fixed side surface

Introduction. The paper considers an axisymmetric problem of elasticity theory for a radially inhomogeneous transversally isotopic nonclosed sphere containing none of the 0 and 𝜋 poles. It is believed that the elastic moduli are linear functions of the radius of the sphere. It is assumed that the side surface of the sphere is fixed, and arbitrary stresses are given on the conic sections, leaving the sphere in equilibrium. The work objective is an asymptotic analysis of the problem of elasticity theory for a radially inhomogeneous transversally isotropic sphere of small thickness, and a study of a three-dimensional stress-strain state based on this analysis.Materials and Methods. The three-dimensional stress-strain state is investigated on the basis of the equations of elasticity theory by the method of homogeneous solutions and asymptotic analysis.Research Results. After the homogeneous boundary conditions set on the side surfaces of the sphere are met, a characteristic equation is obtained, and its roots are classified with respect to a small parameter characterizing the thickness of the sphere. The corresponding asymptotic solutions depending on the roots of the characteristic equation are constructed. It is shown that the solutions corresponding to a countable set of roots have the character of a boundary layer localized in conic slices. The branching of the roots generates new solutions that are characteristic only for a transversally isotropic radially inhomogeneous sphere. A weakly damping boundary layer solution appears, which can penetrate deep away from the conical sections and change the picture of the stress-strain state.Discussion and Conclusions. Based on the solutions constructed, it is possible to determine the applicability areas of existing applied theories and propose a new more refined applied theory for a radially inhomogeneous transversally isotropic spherical shell.

Linear active disturbance rejection control of servo systems via IMC principle with active damping and sliding mode techniques

In this paper, a new linear active disturbance rejection control (LADRC) is proposed for servo systems based on active damping (AD), internal model control (IMC) and sliding mode control (SMC). The IMC rules are applied to tune the controller gains for a prescribed tracking performance. The AD method injects a virtual damping force into the motion system to actively attenuate disturbances. By deriving the sliding dynamics of IMC, a boundary layer solution of SMC is employed to enhance the robustness of control system. As the improved robust IMC scheme is applied as the controller of LADRC, the uncompensated disturbance of linear extended state observer can be better suppressed. With this method, a higher tracking control accuracy is expected to be obtained in the presence of uncertain dynamics. Theoretical stability was analyzed based on the Lyapunov method. Comparative experiments were conducted on a permanent magnet synchronous motor to validate the superiority of the proposed approach.

Exact solutions for 2D boundary layer flow of two types of viscoelastic fluids and heat transfer on a permeable shrinking sheet with thermal radiation and variable surface temperature: existence of multiple solutions

A study with existence of multiple exact solutions has its own importance, and if the study is performed for a viscoelastic fluid, then the attraction becomes very high. So, here an attempt is performed, where the flow of two types of viscoelastic fluids and heat transfer due to shrinking of a permeable sheet is described. In the energy flow, the impacts of thermal radiation and variable temperature of the wall are simultaneously undertaken. By using similarity approach, in addition to single solution, the dual and triple closed-form solutions of the flow field and temperature are obtained in some specific flow situations of two fluids, namely, second-grade and Walters liquid B. Dual solutions for the flow of second-grade fluid are detected only for stronger mass suction with smaller viscoelastic parameter of second-grade fluid and for both weaker and stronger mass suctions (for two ranges) with larger viscoelastic parameter and unique solution in case of mass injection. However, for the flow of Walters liquid B, boundary layer solution is attained for mass suction only, and most importantly, triple solutions for some mass suction range are found. Now for heat flow, an exceptional result is witnessed: if the wall temperature distribution is assumed suitably, then the heat transfer rate is not dependent on the fluid rheology. Also, influences of involved physical parameters on velocity, temperature and other important quantities are described through some graphs.

Experimental and CFD study on influence of viscosity on layer melt crystallization

• Layer melt crystallization of aqueous sucrose solutions with different viscosities. • Viscosity influence on freezing based on ice crystal yield and purity results. • CFD study on heat transfer inside melt crystallizer prior to ice crystallization. • Temperature distributions and thermal boundary layers of undercooled solutions. In the present work, the influence of solution viscosity on growth kinetics and purification efficiency in layer melt crystallization was investigated. Melt crystallization experiments were conducted for three different types of aqueous sucrose solution as they are ideal solutions and a relatively wide viscosity range can be investigated with a moderate change of freezing points. The aqueous 10 wt%, 23 wt%, and 30 wt% sucrose solutions have a dynamic viscosity value of 2.01 mPas, 4.74 mPas, and 7.21 mPas at their respective freezing points of −0.63 °C, −1.78 °C, and −2.64 °C. The solution temperature distribution was predicted by computational fluid dynamics (CFD) simulations run in COMSOL Multiphysics 5.6 software. Experimental results showed that a higher solution viscosity caused a higher crystal layer impurity and lower crystal yields in static layer melt crystallization. The cooling process of different solutions predicted by a CFD heat transfer study showed that the supersaturation region is wider for less concentrated solutions as cooling proceeds more rapidly. Hence, the temperature gradients obtained follow the boundary layer theory, i.e., the thinner the boundary layer, the faster the heat transfer.