Solution Of Boundary Layer Equations Research Articles

Numerical Simulation of Thermally Radiative and EMHD Influenced Viscoplastic Fluid Flow: A Second Law Investigation

"This work rigorously presents the findings of entropy generation in Casson fluid flow on a Riga plate, exposed in a radiative environment. The conceptual basis for the investigation of fluid flow is the system of governing boundary layer equations. By means of implicit finite difference procedure the solutions of boundary layer equations are acquired. Primary goal of this study manifests the optimization of entropy generation in Casson fluid flow associated with EMHD (Electro Magneto Hydro Dynamics) Lorentz force and thermal radiation. Additionally, the contours of entropy generation and Bejan number are demonstrated. Validation of present results with previously obtained results from literature is accomplished conscientiously."

Singularity Formation in an Incompressible Boundary Layer on an Upstream Moving Wall under Given External Pressure

The two-dimensional laminar flow of a viscous incompressible fluid over a flat surface is considered at high Reynolds numbers. The influence exerted on the Blasius boundary layer by a body moving downstream with a low velocity relative to the plate is studied within the framework of asymptotic theory. The case in which a small external body modeled by a potential dipole moves downstream at a constant velocity is investigated. Formally, this classical problem is nonstationary, but, after passing to a coordinate system comoving with the dipole, it is described by stationary solutions of boundary layer equations on the wall moving upstream. The numerically found solutions of this problem involve closed and open separation zones in the flow field. Nonlinear regimes of the influence exerted by the dipole on the boundary layer with counterflows are calculated. It is found that, as the dipole intensity grows, the dipole-induced pressure acting on the boundary layer grows as well, which, after reaching a certain critical dipole intensity, gives rise to a singularity in the flow field. The asymptotics of the solution near the isolated singular point of the flow field is studied. It is found that, at this point, the vertical velocity grows to infinity, viscous stress vanishes, and no solution of the problem exists at higher dipole intensities.

Singular Nonlinear Problems for Phase Trajectories of Some Self-Similar Solutions of Boundary Layer Equations: Correct Formulation, Analysis, and Calculations

We study a singular initial value problem for a nonlinear non-autonomous ordinary differential equation of the second order, defined on a semi-infinite interval and degenerating in the initial data for the phase variable. The problem arises in the dynamics of a viscous incompressible fluid as an auxiliary problem in the study of self-similar solutions of the boundary layer equations for a stream function with a zero pressure gradient (plane-parallel laminar flow in a mixing layer). It is also of independent mathematical interest. Using the previously obtained results on singular nonlinear Cauchy problems and parametric exponential Lyapunov series, a correct formulation and a complete mathematical analysis of this singular initial value problem are given. Restrictions on the “self-similarity parameter” for the global existence of solutions are formulated, two-sided estimates of solutions, and results of calculations of the phase trajectories of solutions for different values of this parameter are given.

Singular Nonlinear Problems for Phase Trajectories of Some Self-Similar Solutions of Boundary Layer Equations: Correct Formulation, Analysis, and Calculations

Singular Nonlinear Problems for Phase Trajectories of Some Self-Similar Solutions of Boundary Layer Equations: Correct Formulation, Analysis, and Calculations

A thermal management of Darcy-Forchheimer SWCNT–MWCNT Cross hybrid nanofluid flow due to vertical stretched cylinder with and without inertia effects

Heat transfer from a surface enshrouded in a porous medium through which non-Newtonian hybrid nanofluid is flowing is of great practical importance in several fields of engineering. Cross fluid model is implemented due to its significant properties like yield stress features, flow in regions of both low and high shear rates, finite viscosity and time constant. Especially, convective flows in a porous medium are of considerable interest in view of the present and potential utilization of geothermal energy, solar energy, etc. for power production. The purpose of this paper is to present the numerical solutions of boundary-layer equations describing the steady free convection flow of non-Newtonian hybrid nanofluid in a saturated porous medium described by Forchheimer extended Darcy’s law. Flow and heat transfer characteristics with and without inertia effects are explored. Percentage of heat transfer rate (HTR) enhancement is obtained. The major outcomes of the study are that velocity of Cross hybrid nanofluid exhibits opposite effects in response to amplification of Weissenberg number and porosity parameter in both presence and absence of inertia. The heat transfer rate for quadratic radiation contributes to moderate response. Further, percentage enhancement of heat transfer rate is greater subject to disparate Weissenberg number compared to that of Brinkman number.

Exact Solutions of Boundary Layer Equations in Polymer Solutions

The paper presents new exact solutions of equations derived earlier. Three of them describe unsteady motions of a polymer solution near the stagnation point. A class of partially invariant solutions with a wide functional arbitrariness is found. An invariant solution of the stationary problem in which the solid boundary is a logarithmic curve is constructed.

On The Numerical Solutions of Boundary Layer Equations of Williamson Fluid Past a Moving Plate

Laminar boundary layer flow of Williamson fluid over a moving plate is discussed in this paper. The governing equations of the flow problem are transformed into similarity equations using similarity technique. The reduced equations are numerically solved by finite difference method. The graphical presentation is discussed.

Numerical solution of Boundary Layer Equations based on optimization: The Ostrach and Blasius models

The numerical solutions for the Blasius equation and for the Ostrach system where investigated. A combination of optimization procedure and Shooting Method where systematized in order to produce a powerful method for solving nonlinear systems of differential equations, namely Initial Value Problem Approximation by Sequential Parameter Optimization (IVASO). Using the IVASO method, it was shown to be possible and easy to obtain an accurate solution for the Blasius equation. It was also demonstrated that the Ostrach system can be solved by IVASO. Thissystem has a large sensibility to initial conditions, and that its solution for near zero (η≈0) is strongly correlated with its solution far from the origin (η≫0), and consequently, an accurate solution for the boundary layer demands a highly accurate solution of the initial values problem, this is similar to the butterfly effect usually studied in chaotic systems.

Accuracy of Calculating the Momentum-Loss Thickness in a Laminar Boundary Layer on the Surface of a Hemisphere in a Supersonic Air Flow

An engineering method of calculating the momentum-loss thickness in a laminar boundary layer on the surface of a hemisphere is presented based on the approximation of the results of systematic numerical solutions of boundary layer equations.

Stagnation point flow of hyperbolic tangent fluid with Soret-Dufour effects

Combined effects of Soret (thermal-diffusion) and Dufour (diffusion-thermo) in MHD stagnation point flow of tangent hyperbolic fluid by a stretching sheet are discussed in the present article. The laws of conservation of mass, momentum, energy and concentration are employed to develop the mathematical model of physical phenomenon. Suitable transformations lead to convert the nonlinear partial differential equations into the ordinary differential equations. The series solutions of boundary layer equations along with boundary conditions are obtained. Convergence of the developed series solutions is discussed via plots and numerical values. The behaviors of different physical parameters on the velocity, temperature and concentration fields are plotted and analyzed. Numerical values of skin friction coefficient, local Nusselt and Sherwood numbers are computed and analyzed. It is found that Dufour and Soret numbers result in the enhancement of temperature and concentration distributions, respectively. Furthermore a comparison is presented with the previous published results in a limiting way to justify the present solutions.

Nonclassical Symmetry Analysis of Heated Two-Dimensional Flow Problems

Abstract This article analyses the nonclassical symmetries and group invariant solution of boundary layer equations for two-dimensional heated flows. First, we derive the nonclassical symmetry determining equations with the aid of the computer package SADE. We solve these equations directly to obtain nonclassical symmetries. We follow standard procedure of computing nonclassical symmetries and consider two different scenarios, ξ 1≠0 and ξ 1=0, ξ 2≠0. Several nonclassical symmetries are reported for both scenarios. Furthermore, numerous group invariant solutions for nonclassical symmetries are derived. The similarity variables associated with each nonclassical symmetry are computed. The similarity variables reduce the system of partial differential equations (PDEs) to a system of ordinary differential equations (ODEs) in terms of similarity variables. The reduced system of ODEs are solved to obtain group invariant solution for governing boundary layer equations for two-dimensional heated flow problems. We successfully formulate a physical problem of heat transfer analysis for fluid flow over a linearly stretching porous plat and, with suitable boundary conditions, we solve this problem.

Asymptotic accelerated boundary layer over the permeable wall

The paper reports on numerical and analytical investigation of the laminar boundary layer with a favorable pressure gradient over a permeable wall. Researchers have obtained an analytical solution of boundary layer equations for asymptotic flow conditions. This solution allows proposing a relative asymptotic skin-friction function, which determines a degree of influence of flow acceleration and permeable wall on the flow. There are ranges of this function, where effects of permeable wall and streamwise pressure gradient have to be considered only in combination. Numerical simulation has showed that such combined influence of favorable pressure gradient and permeable wall extends the asymptotic flow. The study of a strong gas blowing into the accelerated flow has revealed that favorable pressure gradient impedes separation of the boundary layer. At that asymptotic flow starts from the point, where the separation would occur at gas injection into the zero-pressure gradient flow.

A computational approach for solution of boundary layer equations for the free convection along a vertical plate

In this letter, the shifted Legendre polynomials are employed for solving the natural heat convection heat transfer. The operational matrix of derivative for the shifted Legendre polynomial are derived. Then the shifted Legendre polynomials expansions along with this operational matrix are applied for solving the natural convection governing equations. The advantage of the proposed method is that the need for guessing and correcting the velocity and temperature slopes, f ′′(0), and θ′(0) during the solution procedure is eliminated. Moreover, by using the given boundary conditions in the problem, a stable solution can be derived. The proposed method is evaluated by obtaining the similarity solutions of free convection along a vertical isothermal flat plate. The results obtained through this direct innovative solution show higher efficiency and performance in comparison with the conventional shooting method.

MHD stagnation point flow by a permeable stretching cylinder with Soret-Dufour effects

Combined effects of Soret (thermal-diffusion) and Dufour (diffusion-thermo) in MHD stagnation point flow by a permeable stretching cylinder were studied. Analysis was examined in the presence of heat generation/absorption and chemical reaction. The laws of conservation of mass, momentum, energy and concentration are found to lead to the mathematical development of the problem. Suitable transformations were used to convert the nonlinear partial differential equations into the ordinary differential equations. The series solutions of boundary layer equations through momentum, energy and concentration equations were obtained. Convergence of the developed series solutions was discussed via plots and numerical values. The behaviors of different physical parameters on the velocity components, temperature and concentration were obtained. Numerical values of Nusselt number, skin friction and Sherwood number with different parameters were computed and analyzed. It is found that Dufour and Soret numbers result in the enhancement of temperature and concentration distributions, respectively.

Stability, separation and close body interactions. Introduction.

Stability, separation and close body interactions. Introduction.

A new boundary layer equation and its application to the flow in turbo‐machinery

AbstractIn this article, a new boundary layer equation (BLE) in turbomachinery flow is established using semigeodesic coordinate system (S‐coordinate) on the curved two‐dimensional surface of the blade. BLE are a nonlinear partial differential equations for unknown normal gradient of the velocity and pressure on the blade, which could be applied in the blade shape optimal control. Furthermore the existence of solution of BLE and a finite element algorithm based on approximate inertial manifold are provided and numerical example of flow in a centrifugal impeller is also presented. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1455–1483, 2014

The Time-Dependent Similarity Solutions of Boundary Layer Equations of Power-Law Fluids with Non-Isothermal Surface

Unsteady, two dimensional boundary layer flows over a heated surface of power-law fluids are investigated. Surface temperature is assumed to have o power-law variation with the time and the distance. Similarity transformation is applied to the partial differential equation system with three independent variables is reduced into an ordinary differential equations systems. Numerical solutions of non-linear differential equations are found by using a finite difference scheme. Solutions are obtained for boundary layer flow velocity and thermal boundary layer profile. Effects of flow behavior index, Prandtl number, suction-injection parameter and surface temperature exponent with the time and the distance are outlined in the figures.

Similarity Solutions for Boundary Layer Equations of a Powel-Eyring Fluid

Boundary layer equations are derived for the first time for the Powel-Eyring fluid model, a non-Newtonian model proposed for pseudoplastic behavior. Using a scaling symmetry of the equations, partial differential system is transferred to an ordinary differential system. Resulting equations are numerically solved using a finite difference algorithm. Effects of non-Newtonian parameters on the solutions are discussed.

Unsteady separated stagnation-point flow of an incompressible viscous fluid on the surface of a moving porous plate

Using group-theoretic method, an analysis is presented for a similarity solution of boundary layer equations which represents an unsteady two-dimensional separated stagnation-point (USSP) flow of an incompressible fluid over a porous plate moving in its own plane with speed u0(t). It is observed that the solution to the governing nonlinear ordinary differential equation for the USSP flow admits of two solutions (in contrast with the corresponding steady flow where the solution is unique): one is the attached flow solution (AFS) and the other is the reverse flow solution (RFS). A novel result of the analysis is that in the case of stationary plate (u0(t) = 0), after a certain value of the magnitude of the blowing d (<0) at the plate, only the AFS exists and the solution becomes unique. For a stationary plate (u0(t) = 0), the USSP flow is found to be separated for all values of d in both the cases of AFS and RFS. It is also observed that when u0(t) = 0, in the RFS flow with wall suction d (>0), there are two stagnation-points in the flow but in the presence of blowing d (<0), there is only one stagnation-point in the flow which moves further and further up with increase in |d|. Suction is shown to increase the wall shear stress while blowing has an opposite effect. Streamlines for an USSP flow when u0(t) ≠ 0 are also plotted. It is found that in this case, the USSP flow is not in general separated.

Axisymmetric convective torch in fluid with power dependence of density on temperature

The exact solution of boundary layer equations for laminar convective torch from point source of heat in fluid with power dependence of density on temperature was found.