Problem Of Boundary Layer Flow Research Articles

A paired spectral-finite difference approach for solving boundary layer flow problems

This study presents an innovative numerical technique for obtaining solutions to systems of partial differential equations. Two previously introduced numerical methods, namely the spectral quasilinearization method (SQLM) (Motsa, in J Appl Math 2013:Article ID 423628, 2013) and the spectral local linearization method (SLLM) (Motsa 2013) have been shown to be efficient methods for solving systems of PDEs but both contain slight limitations. Our proposed method hereinafter referred to as the paired spectral quasilinearization method (PSQLM), seeks to simplify a large system of PDEs by decoupling it into pairs of sub-systems and applying quasilinearization on the pairs in order to linearize the pairs. Crank–Nicolson scheme and spectral collocation method are, respectively, applied to discretize the system of decoupled linearized pairs of PDEs. The PSQLM is described for a general system of PDEs and a numerical experiment is conducted on a system of four differential equations that contain three possible combinations. We compare the different combinations to ascertain the effect of performing different pairings on the accuracy and convergence speed of solutions. We also compare the PSQLM with the SQLM (which solves the system as coupled) and the SLLM (that decouples a system) to test the efficacy of the PSQLM. The observations made from the comparison proves the PSQLM converges faster than the SLLM with very few iterations. It is also shown to use significantly lesser computational time than the SQLM to generate convergent solutions. The accuracy of the PSQLM is also observed to better both the SQLM and SLLM.

Magnetohydrodynamic (MHD) Boundary Layer Flow Past a Wedge with Heat Transfer and Viscous Effects of Nanofluid Embedded in Porous Media

The problem of two‐dimensional steady laminar MHD boundary layer flow past a wedge with heat and mass transfer of nanofluid embedded in porous media with viscous dissipation, Brownian motion, and thermophoresis effect is considered. Using suitable similarity transformations, the governing partial differential equations have been transformed to nonlinear higher‐order ordinary differential equations. The transmuted model is shown to be controlled by a number of thermophysical parameters, viz. the pressure gradient, magnetic, permeability, Prandtl number, Lewis number, Brownian motion, thermophoresis, and Eckert number. The problem is then solved numerically using spectral quasilinearization method (SQLM). The accuracy of the method is checked against the previously published results and an excellent agreement has been obtained. The velocity boundary layer thickness reduces with an increase in pressure gradient, permeability, and magnetic parameters, whereas thermal boundary layer thickness increases with an increase in Eckert number, Brownian motion, and thermophoresis parameters. Greater values of Prandtl number, Lewis number, Brownian motion, and magnetic parameter reduce the nanoparticles concentration boundary layer.

Magnetohydrodynamic Boundary Layer Flow and Heat Transfer of Nanofluids Past a Bidirectional Exponential Permeable Stretching/Shrinking Sheet With Viscous Dissipation Effect

The problem of boundary layer flow and heat transfer of magnetohydrodynamic (MHD) nanofluids which consist of Fe3O4, Cu, Al2O3, and TiO2 nanoparticles and water as the base fluid past a bidirectional exponentially permeable stretching/shrinking sheet is studied numerically. The mathematical model of the nanofluid incorporates the effect of viscous dissipation in the energy equation. By employing a suitable similarity transformation, the conservative equations for mass, momentum, and energy are transformed into the ordinary differential equations. These equations are then numerically solved with the utilization of bvp4c function in matlab. The effects of the suction parameter, magnetic parameter, nanoparticle volume fraction parameter, Eckert number, Prandtl number, and temperature exponent parameter to the reduced skin friction coefficient as well as the local Nusselt number are graphically presented. Cu is found to be prominently good in the thermal conductivity. Nevertheless, higher concentration of nanoparticles leads to the deterioration of heat transfer rate. The present result negates the previous literature on thermal conductivity enhancement with the implementation of nanofluid. Stability analysis is conducted since dual solutions exist in this study, and conclusively, the first solution is found to be stable.

On the Analytic Solution of Magnetohydrodynamic (MHD) Flow by a Moving Wedge in Porous Medium

This paper is devoted to find analytic approximate solution by optimal homotopy asymptotic method (OHAM) for the problem of nonlinear boundary layer flow. Two-dimensional magneto-hydrodynamic (MHD) flow of a viscous fluid over a moving wedge in porous medium with suction/injection is investigated. Governing equations are transformed by similarity method into a third order Falkner-Skan equation and solved analytically using OHAM. This approach is highly efficient, ensuring a very rapid convergence of the solution only after one iteration. Graphical results are presented to discuss the effects of various parameters on velocity profiles. Further, the skin friction coefficient is also tabulated and compared with the corresponding results available in literature. Our results were found in an excellent agreement.

Numerical and closed-form solutions for the Maxwell equations of incompressible flow

The equations of fluid dynamics governing an incompressible flow may be recast into the form of a set of Maxwell equations for the Lamb vector and vorticity, which play a role analogous to the electric and magnetic fields, respectively. The challenge in utilizing this approach is that analogous source terms, analogous to the charge density and current density, must be known in order to solve the Maxwell equations directly. In this paper, we explore the construction of source terms for laminar incompressible flow. To demonstrate the utility of this approach, we use the newly formed source terms and demonstrate that common mathematical techniques may be borrowed from classical electrodynamics and used to solve the fluid Maxwell system. We illustrate the approaches using the classic nonlinear problem of the incompressible Blasius boundary layer flow. Four different methods commonly used in electrostatics are applied to arrive at numerical and closed-form solutions of the Blasius boundary layer flow. The results compare very well to the accepted solution and to each other.

Mathematical Model of Mixed Convection Boundary Layer Flow over a Horizontal Circular Cylinder Filled in a Jeffrey Fluid with Viscous Dissipation Effect

This paper delves into the problem of mixed convection boundary layer flow from a horizontal circular cylinder filled in a Jeffrey fluid with viscous dissipation effect. Both cases of cooled and heated cylinders are discussed. The governing equations which have been converted into a dimensionless form using the appropriate non-dimensional variables are solved numerically through the Keller-box method. A comparative study is performed and authentication of the present results with documented outcomes from formerly published works is excellently achieved. Tabular and graphical representations of the numerical results are executed for the specified distributions, considering the mixed convection parameter, Jeffrey fluid parameters and the Prandtl and Eckert numbers. Interestingly, boundary layer separation for mixed convection parameter happens for some positive (assisting flow) and negative (opposing flow) values. Strong assisting flow means the cylinder is heated, which causes the delay in boundary layer separation, whereas strong opposing flow means the cylinder is cooled, which conveys the separation point close to the lower stagnation point. Contradictory behaviours of both Jeffrey fluid parameters are observed over the velocity and temperature profiles together with the skin friction coefficient and Nusselt number. The increase of the Prandtl number leads to the decrement of the temperature profile, while the increase of the Eckert number results in the slight increment of the skin friction coefficient and decrement of the Nusselt number. Both velocity and temperature profiles of Eckert number show no effects at the lower stagnation point of the cylinder.

Dual solutions for mixed convective stagnation-point flow of an aqueous silica–alumina hybrid nanofluid

Hybrid nanofluid as an extension of nanofluid is obtained by dispersing composite nano-powder or several different nanoparticles in the base fluid. Hybrid nanofluids are potential fluids that offer better heat transfer performance and thermophysical properties than convectional heat transfer fluids (oil, water and ethylene glycol) and nanofluids with single nanoparticles. Here, a kind of hybrid nanofluid including silicon dioxide (SiO2) and aluminum oxide (Al2O3) nano-size particles with water as base fluid is analytically modeled to develop the problem of the steady laminar MHD mixed convection boundary layer flow of a SiO2–Al2O3/water hybrid nanofluid near the stagnation-point on a vertical permeable flat plate. The flow of nanofluids near the stagnation point has recently attracted the attention of many investigators because of its wide applications in the local cooling/heating processes, especially in industries of electronic devices and nuclear reactors. In first, analytic modeling of hybrid nanofluid is presented and using appropriate similarity variables, the governing PDEs are transformed into nonlinear ODEs in the dimensionless stream function, which is solved numerically applying the function bvp4c from MATLAB. Our results demonstrate that the developed model can be used with great confidence to study the flow and heat transfer of hybrid nanofluids. Moreover, dual solutions of hybrid nanofluid flow for both assisting and opposing regimes are observed, where the range of the mixed convection parameter for which the solution exists, increases with the volume fraction of second nanoparticle and magnetic field. Finally, the heat transfer rate of nanofluids and hybrid nanofluids with different values of nanoparticles volume fraction have been compared that HNF3 (ϕSiO2=ϕAl2O3=0.1) has the largest heat transfer rate between all cases.

Design method of internal waverider inlet under non-uniform upstream for inlet/forebody integration

A novel Bump-integrated three-dimensional internal waverider inlet (IWI) design method is presented for high-speed inlet/forebody integration. The low-kinetic-energy (boundary layer) flow generated by a blunted leading-edge and forebody boundary layer represents an extreme challenge in the integration of aircraft forebody and inlet. In this method, such an inlet's flowpath is divided into the entrance shockwave segment, the isentropic compression segment and the isolator. First, a three-dimensional inverse method of characteristics (3D-IMOC) is developed to obtain a compression surface that can generate a requested entrance shock wave in non-uniform upstream flow. This configuration realizes the integration of IWI and aircraft fuselage by incorporating a Bump to remove most of the boundary layer flow. This is followed by a three-dimensional, isentropic compression flow-path with cross sectional areas conforming to the specified Mach number distribution. Finally, a new three-dimensional Bump-integrated IWI was tested in M = 6 wind tunnel, under a rather thick boundary layer upstream flow (37% height of inlet entrance). Both of the experimental data and numerical simulation results show that, the new method of IWI and Bump can overcome serious boundary layer flow problems and improve the inlet performance.

A locally modified single-phase model for analyzing magnetohydrodynamic boundary layer flow and heat transfer of nanofluids over nonlinearly stretching sheet with chemical reaction

The problem of boundary layer flow and heat transfer of nanofluids over nonlinear stretching of a flat sheet in the presence of a magnetic field and chemical reaction is investigated numerically. In this paper, a new locally modified single-phase model for the analysis is introduced. In this model, the effective viscosity, density and thermal conductivity of the solid-liquid mixtures (nanofluids) which are commonly utilized in the homogenous single-phase model, are locally combined with the prevalent single-phase model. Similarity transformation is used to convert the governing equations into three coupled nonlinear ordinary differential equations. These equations depend on five local functions of the nanoparticle volume fraction viz., local viscosity ratio, magnetic, Prandtl, Brownian motion and thermophoresis functions. The equations are solved using Newton’s method and a block tridiagonal matrix solver. The results are compared to the prevalent single-phase model. In addition, the effect of important governing parameters on the velocity, temperature, volume fraction distribution and the heat and mass transfer rates are examined.

Influence of variable thermal conductivity and thermal radiation on slip flow and heat transfer of MHD power-law fluid over a porous sheet

In this paper, the problem of boundary layer flow and heat transfer of MHD power-law fluid over a porous sheet in the presence of partial slip is investigated numerically. We assume a temperature dependent thermal conductivity and slip conditions are employed in terms of shear stress. The suitable similarity transformations are used, to transform the governing partial dif- ferential equations (PDEs) into a system of nonlinear ordinary differential equations (ODEs). The resulting system of ODEs is solved numerically using Matlab bvp4c solver. The numerical values obtained for the velocity and temperature depend on power-law index, slip parameters, permeability, suction/injection parameter, Prandlt number and Nusselt number. The effects of various parameters on the flow and heat transfer characteristics are presented through graphs and tables and discussed from physical point of view.

Hydromagnetic boundary layer flow of a dusty fluid in a porous medium over a stretching sheet with slip effect

This paper investigated the problem of hydromagnetic boundary layer flow and heat transfer of a dusty fluid over a stretching sheet through a porous medium. The velocity slip was considered instead of the no-slip condition at the boundary. The governing partial equations were reduced into a set of non-linear ordinary differential equations by using the suitable similarity transformation. The transformed equations were numerically integrated using bvp4c in Matlab. The effects of various physical parameters on the velocity and temperature profiles of both phases, such as fluid-particle interaction parameter, magnetic parameter, mass concentration parameter, porosity parameter and Prandtl number were obtained and analyzed through several plots. Useful discussions were carried out with the help of plotted graphs and tables. Under the limiting cases, the obtained numerical results were compared and found to be in good agreement with previously published results.

Chemical reaction and radiation effects on MHD flow past an exponentially stretching sheet with heat sink

In this study, the problem of MHD boundary layer flow past an exponentially stretching sheet with chemical reaction and radiation effects with heat sink is studied. The governing system of PDEs is transformed into a system of ODEs. Then, the system is solved numerically by using Runge-Kutta-Fehlberg fourth fifth order (RKF45) method available in MAPLE 15 software. The numerical results obtained are presented graphically for the velocity, temperature and concentration. The effects of various parameters are studied and analyzed. The numerical values for local Nusselt number, skin friction coefficient and local Sherwood number are tabulated and discussed. The study shows that various parameters give significant effect on the profiles of the fluid flow. It is observed that the reaction rate parameter affected the concentration profiles significantly and the concentration thickness of boundary layer decreases when reaction rate parameter increases. The analysis found is validated by comparing with the results previous work done and it is found to be in good agreement.

Viscous flow due to an exponentially shrinking permeable sheet in nanofluid in presence of slip

ABSTRACTClassical problem of steady boundary layer flow of nanofluid over an exponentially permeable shrinking sheet in presence of slip is investigated. The model used for nanofluid includes Brownian motion and thermophoresis effects. The governing equations for momentum, energy, and nanofluid solid volume fraction are transformed to ordinary differential equations with the help of similarity transformations and then solved numerically using fourth order Runge–Kutta method with shooting technique. It is found that the governing parameters, viz. the suction/blowing parameter, velocity slip, thermal and mass slip parameters, Brownian motion parameter, thermophoresis parameter, Prandtl number, and Lewis number significantly affect the flow field, heat, and mass transfer. The results obtained indicate that the dual solutions exist for certain values of the mass suction parameter. Velocity increases whereas the temperature and nanoparticle volume fraction decrease due to suction through the porous sheet. It is noted that with the increase in velocity slip fluid velocity increases whereas temperature and concentration decrease. Due to increase in thermal slip and mass slip both temperature and concentration decrease.

Effects of anisotropic slip on three-dimensional stagnation-point flow past a permeable moving surface

In this paper, the problem of a steady laminar three-dimensional stagnation point boundary layer flow (Homann stagnation flow) on a permeable moving surface with anisotropic slip in a viscous fluid is investigated. A similarity transformation reduces the governing system of nonlinear partial differential equations into the ordinary (similarity) differential equations. The resulting equations are then solved numerically by using the bvp4c function in Matlab. The effects of surface mass transfer parameter, slip parameter and moving parameter on the fluid flow characteristics are presented in the forms of tables and figures and are discussed in detail. Finally, it is concluded from the stability analysis that the first (upper branch) solution is stable and physically realizable, while the second (lower branch) solution is not stable.

Dual Solutions for Opposing Mixed Convection in Porous Media

The problem of steady mixed convection boundary layer flow on a cooled vertical permeable circular cylinder embedded in a fluid-saturated porous medium is studied. Here, we evaluate the flow and heat transfer characteristics numerically for various values of the governing parameters and demonstrate the existence of dual solutions beyond a critical point.

Haar wavelet quasilinearization approach for MHD Falkner–Skan flow over permeable wall via Lie group method

PurposeThis paper aims to deal with two-dimensional magneto-hydrodynamic (MHD) Falkner–Skan boundary layer flow of an incompressible viscous electrically conducting fluid over a permeable wall in the presence of a magnetic field.Design/methodology/approachUsing the Lie group approach, the Lie algebra of infinitesimal generators of equivalence transformations is constructed for the equation under consideration. Using these suitable similarity transformations, the governing partial differential equations are reduced to linear and nonlinear ordinary differential equations (ODEs). Further, Haar wavelet approach is applied to the reduced ODE under the subalgebra 4.1 for constructing numerical solutions of the flow problem.FindingsA new type of solutions was obtained of the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis.Originality/valueTo find a solution for the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis is a new approach for fluid problems.

A novel investigation of a micropolar fluid characterized by nonlinear constitutive diffusion model in boundary layer flow and heat transfer.

The rheological and heat-conduction constitutive models of micropolar fluids (MFs), which are important non-Newtonian fluids, have been, until now, characterized by simple linear expressions, and as a consequence, the non-Newtonian performance of such fluids could not be effectively captured. Here, we establish the novel nonlinear constitutive models of a micropolar fluid and apply them to boundary layer flow and heat transfer problems. The nonlinear power law function of angular velocity is represented in the new models by employing generalized "n-diffusion theory," which has successfully described the characteristics of non-Newtonian fluids, such as shear-thinning and shear-thickening fluids. These novel models may offer a new approach to the theoretical understanding of shear-thinning behavior and anomalous heat transfer caused by the collective micro-rotation effects in a MF with shear flow according to recent experiments. The nonlinear similarity equations with a power law form are derived and the approximate analytical solutions are obtained by the homotopy analysis method, which is in good agreement with the numerical solutions. The results indicate that non-Newtonian behaviors involving a MF depend substantially on the power exponent n and the modified material parameter [Formula: see text] introduced by us. Furthermore, the relations of the engineering interest parameters, including local boundary layer thickness, local skin friction, and Nusselt number are found to be fitted by a quadratic polynomial to n with high precision, which enables the extraction of the rapid predictions from a complex nonlinear boundary-layer transport system.

Heat transfer analysis in ferromagnetic viscoelastic fluid flow over a stretching sheet with suction

In this article, an investigation has been performed to explore the two-dimensional boundary layer flow problem and heat transfer characteristic of ferromagnetic viscoelastic fluid flow over a stretching surface with a linear velocity under the impact of magnetic dipole and suction. The governing PDEs are converted into a system of nonlinear ODEs by applying appropriate similarity approach. The modelled equations are then solved numerically by utilizing efficient Runge–Kutta–Fehlberg procedure based on shooting algorithm. Influence of pertinent flow parameter involved, such as ferromagnetic interaction parameter, suction parameter, viscoelastic parameter, Prandtl number on dimensionless velocity, temperature, skin friction, and Nusselt inside the boundary layer, are portrayed graphically and discussed. The results show that pressure profile and skin friction coefficient increase with the variation of ferromagnetic interaction parameter and opposite behaviour is noted for local Nusselt number.

Magnetohydrodynamic Cattaneo-Christov flow past a cone and a wedge with variable heat source/sink

In the present article, the problem of boundary layer flow of MHD electrically conducting fluid past a cone and a wedge with non-uniform heat source/sink along with Cattaneo-Christov heat flux is investigated numerically. At first, the flow equations are converted into ODE via appropriate self similarity transforms and the resulting equations are solved with the assistance of R.-K. and Newton’s methods. The influence of several dimensionless parameters on velocity and temperature fields in addition to the friction factor and reduced heat transfer coefficient has been examined with the support of graphs and numerical values. The heat transfer phenomenon in the flow caused by the cone is excessive when compared to the wedge flow. Also, the thermal and momentum boundary layers are not the same for the flow over a cone and wedge.

Unsteady viscous MHD flow over a permeable curved stretching/shrinking sheet

Purpose The purpose of this paper is to theoretically study the problem of the unsteady boundary layer flow past a permeable curved stretching/shrinking surface in the presence of a uniform magnetic field. The governing nonlinear partial differential equations are converted into ordinary differential equations by similarity transformation, which are then solved numerically. Design/methodology/approach The transformed system of ordinary differential equations was solved using a fourth-order Runge-Kutta integration scheme. Results for the reduced skin friction coefficient and velocity profiles are presented through graphs and tables for several sets of values of the governing parameters. The effects of these parameters on the flow characteristics are thoroughly examined. Findings Results show that for the both cases of stretching and shrinking surfaces, multiple solutions exist for a certain range of the curvature, mass suction, unsteadiness, stretching/shrinking parameters and magnetic field parameter. Originality/value The paper describes how multiple (dual) solutions for the flow reversals are obtained. It is shown that the solutions exist up to a critical value of the shrinking parameter, beyond which the boundary layer separates from the surface and the solution based upon the boundary layer approximations is not possible.