Solution Of Boundary Layer Flow Research Articles

Dual Solutions of Boundary Layer Flow and Heat Transfer in Hybrid Nanofluid over a Stretching/Shrinking Cylinder

The boundary layer flow over a stretching/shrinking cylinder in hybrid nanofluid with the effects of suction, partial slip and convective boundary condition is studied. Hybrid nanoparticles Al2O3 and TiO2 with water as based fluid are considered in the study. The partial differential equations are transformed to ordinary differential equations by employing the similarity variables. The numerical results are obtained using the bvp4c solver in MATLAB software. The influence of nanoparticles volume fraction (Al2O3-TiO2 in water-based fluid), curvature parameter, suction parameter, partial slip parameter and Biot number on the velocity profile, temperature profile, skin friction coefficient and heat transfer rate are discussed. The numerical results indicate that for shrinking surface case, the dual solutions exist for a certain range of curvature parameter and suction parameter.

On numerical solution of boundary layer flow of viscous incompressible fluid past an inclined stretching sheet in porous medium and heat transfer using spline technique

In this paper, the boundary layer flow of viscous incompressible fluid over an inclined stretching plate in porous media with body force and heat transfer has been studied. To solve this problem, we develop a suitable spline method which is used to calculate the velocity function of the flow problem. We proceed as follows:•With a suitable stream function, the concerned boundary layer equation is converted into non-linear third order ordinary differential equation together with appropriate boundary conditions in an infinite domain which has been further linearized by using quasi-linearization method.•Then, we develop a non polynomial quintic spline technique which has been used to find the numerical values of the velocity function of the flow problem. The convergence analysis of the developed spline technique has been discussed.•Later, the method developed so far has been applied to solve nonlinear boundary value problem for different angles of inclination and Froude number. The values obtained so far have been used to study heat flow problem. Finally, skin friction has been discussed.

Local Non-Similar Solutions for Boundary Layer Flow over a Nonlinear Stretching Surface with Uniform Lateral Mass Flux: Utilization of Third Level of Truncation

The present study aims to examine the effects of uniform lateral mass flux on the boundary layer flow induced by a non-linearly stretching surface. For uniform mass flux, the boundary layer flow does not conform to a similarity solution. The problem may be resolved by the similarity solution only when the transverse velocity at the boundary of the porous stretching surface is of the form vw∼xp−12. In other words, the flow becomes non-similar; to date, this has not been reported in the literature. That is why, in the current study, the local-similarity approximation up to the third level of truncation is utilized to solve the problem. The pseudo-similarity variable, stream function and transformed streamwise coordinate are defined such that the continuity equation is identically satisfied, and the momentum equation reduces to a non-similar dimensionless boundary layer equation. We derived the non-similar equations of the first, second and third levels of truncations and compared the numerical results obtained from different levels of truncations. In order to find numerical solutions to these equations, the built-in MATLAB routine, known as bvp4c, is used. Further, all non-similar terms that appear in the momentum equations are retained without any approximations. The approximations are introduced only in the subsidiary equations and relative boundary conditions. For the case of suction, the rate of increase in the numerical values of skin friction coefficient obtained from the first level of truncation with increasing velocity index parameter is found to be underestimated, while overestimation is found in the case of injection. The numerical results that were obtained from the third level of truncations are plotted against the embedding physical parameters and are then discussed.

Similarity solution for boundary layer flow near a moving vertical porous plate with combined effects of nonlinear thermal radiation and suction/injection having convective surface boundary condition

This piece of work explored the impact of nonlinear thermal radiation on natural convection flow in the vicinity of a constant moving vertical porous plate with suction/injection. The moving vertical porous plate is considered to be either move in the same direction with fluid in the free stream or move in the negative x-direction while the fluid move in the positive x-direction. The partial differential equations (PDEs) associated with the flow properties are simplified to ordinary differential equations (ODEs) using similarity variables. Then, the resulting ODEs and the boundary condition are translated into the IVP through the shooting method. Furthermore, the final IVP and the reduced boundary conditions are solved numerically through the Runge–Kutta method. The influence of the emerging dimensionless quantities is analyzed through tables and line graphs. The numerical results obtained revealed that with suction/injection, the shear stress could be enhanced by propagating the local convective heat transfer parameter Bix, nonlinear thermal radiation R and the local Grashof number Grx for the two cases of plate motion considered. The result also revealed that in the presence of suction/injection, the heat transfer rate decreases with Bix and Grx, whereas enhanced with R augment for the two plate motions considered. Furthermore, for a constant moving vertical porous plate and constant moving freestream velocity, the convective heat transfer parameter at any point x is proportional to the corresponding heat transfer coefficient related to hot fluid. The nonlinear thermal radiation enhances the velocity and temperature profiles.

Impact of nonlinear thermal radiation on nonlinear mixed convection flow near a vertical porous plate with convective boundary condition

This study presents similarity solution for boundary layer flow near a vertical porous plate with combined effects of nonlinear density variation with temperature and nonlinear thermal radiation. To accurately predict the flow phenomenon near the porous plate, the convective boundary condition is considered at the plate surface. The two-dimensional partial differential equations are transformed to ordinary differential equations through the similarity transformation. The resulting ordinary differential equations are solved numerically in Maple software using the Runge–Kutta–Ferhlberg fourth-fifth order (RKF45) algorithm. The influence of the inherit parameters like the nonlinear thermal radiation parameter, suction/injection parameter, nonlinear Boussinesq approximation parameters, local convective heat transfer parameter, local Grashof number, and Prandtl number governing the fluid behaviour is discussed. We found that the rate of heat transfer improves with the injection and nonlinear thermal radiation parameter whereas decreases with suction, local convective heat transfer parameter and local Grashof number when air and mercury are used as the working fluids. Furthermore, with the growth in the values of local Grashof number, convective heat transfer parameter and nonlinear thermal radiation parameter and in the presence of suction/injection, the porous plate surface friction witnessed an observable growth. Suction growth plays a supportive role on the velocity curve near the porous plate but a contrary trend is seen in the free stream. The temperature distribution also decays with suction augment. Injection growth is inversely proportional to the velocity profile near the porous plate but we recorded the opposite phenomenon in the free stream.

Numerical Solution for Boundary Layer Flow of a Dusty Micropolar Fluid Due to a Stretching Sheet with Constant Wall Temperature

This paper aims to present the numerical study of a dusty micropolar fluid due to a stretching sheet with constant wall temperature. Using the suitable similarity transformation, the governing partial differential equations for two-phase flows of the fluid and the dust particles are reduced to the form of ordinary differential equations. The ordinary differential equations are then numerically analysed using the bvp4c function in the Matlab software. The validity of present numerical results was checked by comparing them with the previous study. The results graphically show the numerical solutions of velocity, temperature and microrotation distributions for several values of the material parameter K, fluid-particle interaction parameter and Prandtl number for both fluid and dust phase. The effect of microrotation is investigated and analysed as well. It is found that the distributions are significantly influenced by the investigated parameters for both phases.

MHD Slip Flow for Casson Nano Fluid over a Radically Surface with Zero Mass Flux at the Surface, Non-liner Radiation and Temperature Jump

The impact of second order velocity slip with zero mass flux over a malting radially surface is deliberated in the present study to explore the solution of boundary layer flow for MHD Casson nano-fluid heat and mass transfer utilizing nonlinear thermal radiation, heat source and velocity and temperature jumping effect. The temperature and mass equation effect of cross-diffusion is set. Scheme of R.K. using the shooting method, the fourth order is used to derive the correct transformation of the nonlinear differential equation (ODE) mathematical solution from the partial differential equation (PDE) model. The visual definition is used to explain the effect of fluid flow variable such as temperature change, first and second order velocity slip parameter, magnetic parameter, heat source and radiation parameter, surface melting parameter, temperature ratio parameter on dimensional density, temperature and mass profiles. The numerous Nussels, local skin tension and the amount of Sherwoods is expressed in table form. The results obtained confirm that there is an excellent agreement in open literature with those available. In the case of first and second order slip conditions, the skin frication is magnified. The contrary effect can also be seen in the amount of Nusselt.

Hybrid Nanofluid Flow Past a Shrinking Cylinder with Prescribed Surface Heat Flux

This numerical study was devoted to examining the occurrence of non-unique solutions in boundary layer flow due to deformable surfaces (cylinder and flat plate) with the imposition of prescribed surface heat flux. The hybrid Al2O3-Cu/water nanofluid was formulated using the single phase model with respective correlations of hybrid nanofluids. The governing model was simplified by adopting a similarity transformation. The transformed differential equations were then numerically computed using the efficient bvp4c solver with the ranges of the control parameters 0.5%≤ϕ1,ϕ2≤1.5% (Al2O3 and Cu volumetric concentration), 0≤K≤0.2 (curvature parameter), 2.6<S≤3.2 (suction parameter) and −2.5<λ≤0.5 (stretching/shrinking parameter). Dual steady solutions are presentable for both a cylinder (K>0) and a flat plate (K=0) with the inclusion of only the suction (transpiration) parameter. The real and stable solutions were mathematically validated through the stability analysis. The Al2O3-Cu/water nanofluid with ϕ1=0.5% (alumina) and ϕ2=1.5% (copper) has the highest skin friction coefficient and heat transfer rate, followed by the hybrid nanofluids with volumetric concentrations (ϕ1=1%,ϕ2=1%) and (ϕ1=1.5%,ϕ2=0.5%), respectively. Surprisingly, the flat plate surface abates the separation of boundary layer while it enhances the heat transfer process.

Dual solutions for boundary layer flow and heat transfer of biomagnetic fluid over a stretching/shrinking sheet in presence of a magnetic dipole and a prescribed heat flux

This paper analyzes the steady boundary layer flow and heat transfer of biomagnetic fluid over a stretching/shrinking sheet with prescribed surface heat flux under the influence of a magnetic dipole. The governing equations are transformed into a set of ordinary differential equations (ODEs) by using similarity transformations. Numerical results are obtained using the boundary value problem solver bvp4c of MATLAB. The effects of various physical parameters on the velocity and temperature profiles as well as skin friction coefficient are discussed. The paper shows that dual solutions exist for certain values of stretching/shrinking sheet and suction parameters. Stability analysis is performed to determine which solution is stable and physically valid. Results of the stability analysis depict that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable.

Stability analysis for multiple solutions of boundary layer flow towards a shrinking sheet: Analytical solution by using least square method

The Purpose:In some cases, most of the boundary value problems contain multiple solutions in which one of the solution is stable or physical reliable and other solutions are unstable. In this study, the problem of hydromagnetic flow due to shrinking sheet is considered and calculated dual solutions against velocity ratio parameter in case of shrinking sheet. A stability analysis is performed for the checking, which solution is stable. The main materials and methods used:Initially, due to more than one independent variable, the flow is governed by partial differential equations (PDEs). For simplicity the conversion of these PDEs were made in form of ODEs (ordinary differential equations) and solved analytically by using weighted residual method named as least square method. The resulting nonlinear system of equations is linearized by using Newton method (Burden and Faires, 1991). For the accuracy of this method, a comparison is made with the previous published work (Wang, 2008). The algorithm of least square method is constructed in Mathematica software and is very easy to apply as compare to analytical method namely: homotopy analysis method. In case of shrinking sheet there exists more than one solution. To check which solution is physical reliable or stable and unstable, a present problem is converted into time dependent problem and then stability analysis is performed and BVP4c method is used for calculating the eigenvalues. The main results obtained:In case of shrinking sheet there exists dual solutions and dual solutions are calculated for some values of velocity ratio parameter λand magnetic parameterM by using least square method. The range of dual solutions is shown by plotting the graphs of skin friction coefficient and local Nusselt number. After performing the stability analysis, the smallest eigenvalues are calculated for various values of λ and M. The main conclusions of the work:It is observed that positive eigenvalue corresponds to stable solution or physical reliable i.e. the disturbance in the solution decay initially and negative eigenvalue corresponds to unstable solution i.e. it produces initial growth of disturbance. It is also observed that eigenvalues become higher in the presence of Lorentz force.

Numerical solution of boundary layer flow of MHD nanofluid over a stretching surface with chemical reaction and viscous dissipation effects

Numerical solution of boundary layer flow of MHD nanofluid over a stretching surface with chemical reaction and viscous dissipation effects

Finite Element Method Solution of Boundary Layer Flow of Powell-Eyring Nanofluid over a Nonlinear Stretching Surface

The nonlinear convective flow of Eyring-Powell nanofluid using Catteneo-Christov model with heat generation or absorption term and chemical reaction rate over nonlinear stretching surface is analyzed. The simultaneous nonlinear partial differential equations governing the boundary layer flow are transformed to the corresponding nonlinear ordinary differential equations using similarity solution and then solved using Galerkin finite element method (GFEM). The impacts of pertinent governing parameters like Brownian diffusion, thermophoresis, mixed convection, heat generation or absorption, chemical reaction rate, Deborah numbers, Prandtl number, magnetic field parameter, Lewis number, nonlinear stretching sheet, and Eyring-Powell fluid parameters on velocity field, temperature, and nanoparticle concentration are given in both figures and tabular form. The result shows that the rise in chemical reaction rate will improve mass transfer rate and reduce heat transfer rate and local buoyancy parameter has quit opposite effect. The attributes of local skin friction coefficient, Nusselt number, and Sheer wood number are investigated and validated with existing literatures.

Numerical Solution of Boundary Layer Flow of Viscous Fluid Via Successive Linearization Method

In the recent years, a great deal of interest has been gained to fluids applications. Some fluids not easy to expressed by particular constitutive relationship between shear rates and stress and which is totally different than the viscous fluids [1,2]. These fluids including many home items namely, toiletries, paints, cosmetics certain oils, shampoo, jams, soups etc.

A Stability Analysis of Solutions in Boundary Layer Flow and Heat Transfer of Carbon Nanotubes over a Moving Plate with Slip Effect

The flow and heat transfer characteristics of both single-wall and multi-wall carbon nanotubes (CNTs) with water and kerosene as base fluid on a moving plate with slip effect are studied numerically. By employing similarity transformation, governing equations are transformed into a set of nonlinear ordinary equations. These equations are solved numerically using the bvp4c solver in Matlab which is a very efficient finite difference method. The influence of numerous parameters such as nanoparticle volume fraction, velocity ratio parameter and first order slip parameter on velocity, temperature, skin friction and heat transfer rate are further explored and discussed in the form of graphical and tabular forms. The results reveal that dual solutions exist when the plate and free stream move in the opposite direction and slip parameter was found to widen the range of the possible solutions. However, skin friction coefficients decrease, whereas the heat transfer increases in the presence of slip parameter. Single-wall carbon nanotubes (SWCNTs) give higher skin friction and heat transfer compared to multi-wall carbon nanotubes (MWCNTs) due to the fact that they have higher density and thermal conductivity. A stability analysis is carried out to determine the stability of the solutions obtained.

Multiple similarity solutions in boundary-layer flow on a moving surface

The similarity equations that arise when there is a power-law outer flow, characterized by the parameter $$\beta $$ , over a surface moving with the same power-law speed, described by the dimensionless parameter $$\lambda $$ , are considered. The critical values $$\lambda _c$$ of $$\lambda $$ are calculated in terms of $$\beta $$ , except in a range $$0.139 \lesssim \beta \lesssim 0.5$$ where there are no critical points. The behaviour of the solution with $$\lambda $$ for representative values of $$\beta $$ is examined, including cases where there are no critical points and one or two critical points leading to two and three solution branches. The asymptotic behaviour for large $$\lambda $$ is derived. For $$-2<\beta <-1$$ , the solution proceeds to large negative values of $$\lambda $$ with this asymptotic limit derived. Aiding flow, $$\lambda >0$$ , shows the existence of additional critical points, with a range $$-2.6583<\beta <-2$$ over which $$\lambda _c$$ takes all values both positive and negative. Relatively weak, $$\lambda =-0.5$$ , and stronger, $$\lambda =-5.0$$ , cases of opposing are treated. The weak case shows two disjoint sections of the solution. For the larger value of $$|\lambda |$$ , one section of the solution in which $$f''(0)$$ decreases monotonically as $$\beta $$ is increased and another section where there is a critical point with two solution branches is seen. In all the cases considered, the solution became singular as $$\beta \rightarrow -2$$ , this limit being discussed.

Series Solution of Boundary Layer Flow of a Nanofluid Over a Moving Semi-Infinite Plate

Series Solution of Boundary Layer Flow of a Nanofluid Over a Moving Semi-Infinite Plate

On the Solution of Boundary Layer Flow of Prandtl Fluid Past a Flat Surface

On the Solution of Boundary Layer Flow of Prandtl Fluid Past a Flat Surface

Stability of Dual Solutions in Boundary Layer Flow and Heat Transfer over an Exponentially Shrinking Cylinder

An investigation has been conducted to study the flow field of boundary layer and heat transfer passing shrinking cylinder using an exponent similarity variables. The shooting technique is used to solved the numerical computation but before that the governing equations in the form of partial differential equation will be transformed to nonlinear ordinary differential equations using suitable similarity variables. The skin friction coefficient, local Nusselt number and profiles are presented graphically. The effects of mass suction parameter and curvature parameter on flow and heat transfer behaviour are also presented. We notice the dual similarity solutions are available in certain range of the suction parameter which depend on curvature parameter. The stability analysis result is run by applying the bvp4c solver, in order to validate which solution is in stable flow and physically meaningful.

Unsteady boundary layer flow and heat transfer over an exponentially shrinking sheet with suction in a copper-water nanofluid

An analysis of unsteady boundary layer flow and heat transfer over an exponentially shrinking porous sheet filled with a copper-water nanofluid is presented. Water is treated as a base fluid. In the investigation, non-uniform mass suction through the porous sheet is considered. Using Keller-box method the transformed equations are solved numerically. The results of skin friction coefficient, the local Nusselt number as well as the velocity and temperature profiles are presented for different flow parameters. The results showed that the dual non-similar solutions exist only when certain amount of mass suction is applied through the porous sheet for various unsteady parameters and nanoparticle volume fractions. The ranges of suction where dual non-similar solution exists, become larger when values of unsteady parameter as well as nanoparticle volume fraction increase. So, due to unsteadiness of flow dynamics and the presence of nanoparticles in flow field, the requirement of mass suction for existence of solution of boundary layer flow past an exponentially shrinking sheet is less. Furthermore, the velocity boundary layer thickness decreases and thermal boundary layer thickness increases with increasing of nanoparticle volume fraction in both non-similar solutions. Whereas, for stronger mass suction, the velocity boundary layer thickness becomes thinner for the first solution and the effect is opposite in the case of second solution. The temperature inside the boundary layer increases with nanoparticle volume fraction and decreases with mass suction. So, for the unsteadiness and for the presence of nanoparticles, the flow separation is delayed to some extent.

New similarity solution of boundary layer flow along a continuously moving convectively heated horizontal plate by deductive group method

A mathematical model is presented and analyzed for steady two-dimensional non-isothermal laminar free convective boundary layer flow along a convectively heated moving horizontal plate. New similarity transformations are developed using one parameter deductive group transformations and hence the governing transport equations are reduced to a system of coupled, nonlinear ordinary differential equations with associated boundary conditions. The reduced equations are then solved numerically by an implicit finite difference numerical method. The effects of pertinent parameters on the non-dimensional velocity, temperature, friction factor and heat transfer rates are investigated and presented graphically. It is found that friction factor decreases with the free convective parameter and rate of heat transfer increases with the convection-conduction parameter.