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Article

Y.Y. Lok, I. Pop, D.B. Ingham and N. Amin

The purpose of this paper is to study theoretically the steady two‐dimensional mixed convection flow of a micropolar fluid impinging obliquely on a stretching vertical…

Abstract

Purpose

The purpose of this paper is to study theoretically the steady two‐dimensional mixed convection flow of a micropolar fluid impinging obliquely on a stretching vertical sheet. The flow consists of a stagnation‐point flow and a uniform shear flow parallel to the surface of the sheet. The sheet is stretching with a velocity proportional to the distance from the stagnation point while the surface temperature is assumed to vary linearly. The paper attempts also to show that a similarity solution of this problem can be obtained.

Design/methodology/approach

Using a similarity transformation, the basic partial differential equations are first reduced to ordinary differential equations which are then solved numerically using the Keller box method for some values of the governing parameters. Both assisting and opposing flows are considered. The results are also obtained for both strong and weak concentration cases.

Findings

These results provide information about the effect of a/c (ratio of the stagnation point velocity and the stretching velocity), σ (shear flow parameter) and K (material parameter) on the flow and heat transfer characteristics in mixed convection flow near a non‐orthogonal stagnation‐point on a vertical stretching surface. The results show that the shear stress increases as K increases, while the heat flux from the surface of the sheet decreases with an increase in K.

Research limitations/implications

The results in this paper are valid only in the small region around the stagnation‐point on the vertical sheet. It is found that for smaller Prandtl number, there are difficulties in the numerical computation due to the occurrence of reversed flow for opposing flow. An extension of this work could be performed for the unsteady case.

Originality/value

The present results are original and new for the micropolar fluids. They are important in many practical applications in manufacturing processes in industry.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 19 no. 3/4
Type: Research Article
ISSN: 0961-5539

Keywords

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Article

S. Kubacki and E. Dick

This paper aims to provide improvements to the newest version of the k‐ ω turbulence model of Wilcox for convective heat transfer prediction in turbulent axisymmetric jets…

Abstract

Purpose

This paper aims to provide improvements to the newest version of the k‐ ω turbulence model of Wilcox for convective heat transfer prediction in turbulent axisymmetric jets impinging onto a flat plate.

Design/methodology/approach

Improvements to the heat transfer prediction in the impingement zone are obtained using the stagnation flow parameter of Goldberg and the vortex stretching parameter of Wilcox. The third invariant of the strain rate tensor in the form of Shih et al. and the blending function of Menter are applied in order make negligible the influence of the impingement modifications in the benchmark flows for turbulence models. Further, it is demonstrated that for two‐dimensional jets impinging onto a flat plate the stagnation region Nusselt number predicted by the original k‐ ω model is in good agreement with direct numerical simulation (DNS) and experimental data. Also for two‐dimensional jets, the proposed modification is deactivated.

Findings

The proposed modification has been applied to improve the convective heat transfer predictions in the stagnation flow regions of axisymmetric jets impinging onto a flat plate with nozzle‐plate distances H/D = 2, 6, 10 and Reynolds numbers Re = 23,000 and 70,000. Comparison of the predicted and experimental mean and fluctuating velocity profiles is performed. The heat transfer rates along a flat plate are compared to experimental data. Significant improvements are obtained with respect to the original k‐ ω model.

Originality/value

The proposed modification is simple and can be added to the k‐ ω model without causing stability problems in the computations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 19 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

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Article

Giulia Giantesio, Anna Verna, Natalia C. Roşca, Alin V. Rosca and Ioan Pop

This paper aims to study the problem of the steady plane oblique stagnation-point flow of an electrically conducting Newtonian fluid impinging on a heated vertical sheet…

Abstract

Purpose

This paper aims to study the problem of the steady plane oblique stagnation-point flow of an electrically conducting Newtonian fluid impinging on a heated vertical sheet. The temperature of the plate varies linearly with the distance from the stagnation point.

Design/methodology/approach

The governing boundary layer equations are transformed into a system of ordinary differential equations using the similarity transformations. The system is then solved numerically using the “bvp4c” function in MATLAB.

Findings

An exact similarity solution of the magnetohydrodynamic (MHD) Navier–Stokes equations under the Boussinesq approximation is obtained. Numerical solutions of the relevant functions and the structure of the flow field are presented and discussed for several values of the parameters which influence the motion: the Hartmann number, the parameter describing the oblique part of the motion, the Prandtl number (Pr) and the Richardson numbers. Dual solutions exist for several values of the parameters.

Originality value

The present results are original and new for the problem of MHD mixed convection oblique stagnation-point flow of a Newtonian fluid over a vertical flat plate, with the effect of induced magnetic field and temperature.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 12
Type: Research Article
ISSN: 0961-5539

Keywords

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Article

Saeed Dinarvand, Reza Hosseini and Ioan Pop

– The purpose of this paper is to do a comprehensive study on the unsteady general three-dimensional stagnation-point flow and heat transfer of a nanofluid by Buongiorno’s model.

Abstract

Purpose

The purpose of this paper is to do a comprehensive study on the unsteady general three-dimensional stagnation-point flow and heat transfer of a nanofluid by Buongiorno’s model.

Design/methodology/approach

In this study, the convective transport equations include the effects of Brownian motion and thermophoresis. By introducing new similarity transformations for velocity, temperature and nanoparticle volume fraction, the basic equations governing the flow, heat and mass transfer are reduced into highly non-linear ordinary differential equations. The resulting non-linear system has been solved both analytically and numerically.

Findings

The analysis shows that velocity, temperature and nanoparticle concentration profiles in the respective boundary layers depend on five parameters, namely unsteadiness parameter A, Brownian motion parameter Nb, thermophoresis parameter Nt, Prandtl number Pr and Lewis number Le. It is found that the thermal boundary layer thickens with a rise in both of the Brownian motion and the thermophoresis effects. Therefore, similar to the earlier reported results, the Nusselt number decreases as the Brownian motion and thermophoresis effects become stronger. A correlation for the Nusselt number has been developed based on a regression analysis of the data. This correlation predicts the numerical results with a maximum error of 9 percent for a usual domain of the physical parameters.

Originality/value

The stagnation point flow toward a wavy cylinder (with nodal and saddle stagnation points) that a little attention has been given to it up to now. The examination of unsteadiness effect on the general three-dimensional stagnation-point flow. The application of an interesting and global model (Boungiorno’s model) for the nanofluid that incorporates the effects of Brownian motion and thermophoresis. The study of the effects of Brownian motion and thermophoresis on the nanofluid flow, heat and mass transfer characteristics. The prediction of correlation for the Nusselt number based on a regression analysis of the data. General speaking, we can tell the problem with this geometry, characteristics, the applied model, and comprehensive results, was Not studied and analyzed in literature up to now.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 5
Type: Research Article
ISSN: 0961-5539

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Article

Khalid Mahmood, Muhammad Sajid, Nasir Ali and Tariq Javed

An attempt is made to study magnetohydrodynamic viscous fluid impinging orthogonally toward a stagnation point on a vertical surface lubricated with power law fluid. It…

Abstract

Purpose

An attempt is made to study magnetohydrodynamic viscous fluid impinging orthogonally toward a stagnation point on a vertical surface lubricated with power law fluid. It has been assumed that the surface temperature varies linearly with the distance from the stagnation point. The problem is governed by system of partial differential equations for both the base fluid and the lubricant. The continuity of velocity and shear stress is assumed at the interface layer between the base fluid and the lubricant. Dimensionless variables are introduced to transform original problem into ordinary differential equations. An implicit finite-difference scheme known as the Keller-Box method is implemented to obtain the numerical solutions. The influence of various important parameters is presented in the form of graphs and tables. The limiting cases for full and no-slip conditions are deduced from the present solutions. A comparison of the present results with the existing results in the special case validates the obtained numerical solutions. The purpose of this study is to see the behaviour of flow characteristics in the presence of lubrication.

Design/methodology/approach

The authors’ problem is governed by system of partial differential equations for both the base fluid and the lubricant. Dimensionless variables are introduced to transform original problem into ordinary differential equations. The obtained ordinary differential equation along with boundary conditions are highly nonlinear and coupled. An implicit finite-difference scheme known as the Keller-Box method is implemented to obtain the numerical solutions.

Findings

Some findings of this study are that the lubricant increases the velocity of the base fluid inside the boundary layer. In the case of full slip, the effects of viscosity are suppressed by the lubricant. The temperature of the base fluid decreases by increase in lubrication on the surface. By increasing the slip on the surface, the skin friction decreases and local Nusselt number increases, but the rate of increase or decrease is less in magnitude for the case of opposing flow. The similarity solutions only exist for n = 1/2. A non-similar solution is obtained for the other values of the power-law index n.

Originality/value

The study of flow phenomenon over a lubricated surface has important applications in machinery components such as fluid bearings and mechanical seals. Coating is another major application of lubrication including the preparation of thin films, printing, painting, etc. The authors hope that the current study will provide the roadmap for the future studies in this direction.

Details

Industrial Lubrication and Tribology, vol. 69 no. 4
Type: Research Article
ISSN: 0036-8792

Keywords

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Article

Mustafa Turkyilmazoglu, Kohilavani Naganthran and Ioan Pop

The purpose of this paper is to present both an analytical and a numerical analysis of the unsteady magnetohydrodynamic (MHD) rear stagnation-point flow over off-centred…

Abstract

Purpose

The purpose of this paper is to present both an analytical and a numerical analysis of the unsteady magnetohydrodynamic (MHD) rear stagnation-point flow over off-centred deformable surfaces.

Design/methodology/approach

The numerical MATLAB solver bvp4c suitable for routine boundary value problem is used for the set of ordinary differential equations reduced from the governing partial differential equations.

Findings

Multiple solutions are found for particular eigenvalues. The physical solution is computed by the help of a linear stability analysis. The authors have succeeded in discovering the second solutions, and it is suggested that these solutions are unstable and not physically realisable in practice. The current findings add to a growing body of literature on MHD stagnation-point flow problems. It is also found that the governing parameters have different effects on the flow characteristics.

Practical implications

Even though problems of steady MHD flows have been extensively studied for stagnation-point flows, limited findings can be found on the unsteady MHD rear stagnation-point flow over off-centred deformable surfaces.

Originality/value

The originality of this work is the application of a magnetic field on a time-dependent MHD rear stagnation-point flow over off-centred deformable surfaces.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

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Article

Y.Y. Lok, A. Ishak and I. Pop

The purpose of this paper is to theoretically investigate the steady two‐dimensional magnetohydrodynamic (MHD) boundary layer flow over a shrinking sheet. The effects of…

Abstract

Purpose

The purpose of this paper is to theoretically investigate the steady two‐dimensional magnetohydrodynamic (MHD) boundary layer flow over a shrinking sheet. The effects of stretching and shrinking parameter as well as magnetic field parameter near the stagnation point are studied.

Design/methodology/approach

A similarity transformation is used to reduce the governing partial differential equations to a set of nonlinear ordinary differential equations which are then solved numerically using Keller‐box method.

Findings

The solution is unique for stretching case; however, multiple (dual) solutions exist for small values of magnetic field parameter for shrinking case. The streamlines are non‐aligned and a reverse flow is formed near the surface due to shrinking effect.

Practical implications

The flow due to a stretching or shrinking sheet is relevant to several practical applications in the field of metallurgy, chemical engineering, etc. For example, in manufacturing industry, polymer sheets and filaments are manufactured by continuous extrusion of the polymer from a die to a windup roller, which is located at a finite distance away. In these cases, the properties of the final product depend to a great extent on the rate of cooling which is governed by the structure of the boundary layer near the stretching surface.

Originality/value

The present results are original and new for the MHD flow near the stagnation‐point on a shrinking sheet. For shrinking case, the velocity on the boundary is towards a fixed point which would cause a velocity away from the sheet. Therefore, this paper is important for scientists and engineers in order to become familiar with the flow behaviour and properties of such MHD flow and the way to predict the properties of this flow for the process equipments.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

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Article

M M Rahman, Teodor Grosan and Ioan Pop

The laminar two-dimensional stagnation-point flow and heat transfer of a viscous incompressible nanofluid obliquely impinging on a shrinking surface is formulated as a…

Abstract

Purpose

The laminar two-dimensional stagnation-point flow and heat transfer of a viscous incompressible nanofluid obliquely impinging on a shrinking surface is formulated as a similarity solution of the Navier-Stokes, energy and concentration equations. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The effect of the dimensionless strain rate, shrinking parameter, Brownian motion parameter and thermophoresis parameter on the flow, temperature and nanoparticle volume fraction is investigated in details. The paper aims to discuss these issues.

Design/methodology/approach

The transformed system of ordinary differential equations was solved using the function bvp4c from Matlab. The relative tolerance was set to 10−10.

Findings

It is found that dimensionless strain rate and shrinking parameter causes a shift in the position of the point of zero skin friction along the stretching sheet. Obliquity of the flow toward the surface increases as the strain rate intensifies. The results indicate that dual solutions exist for the opposing flow case.

Research limitations/implications

The problem is formulated for an incompressible nanofluid with no chemical reactions, dilute mixture, negligible viscous dissipation and negligible radiative heat transfer assuming nanoparticles and base fluid are locally in thermal equilibrium. Beyond the critical point λ c to obtain further solutions, the full basic partial differential equations have to be solved.

Originality/value

The present results are original and new for the oblique stagnation-point flow of a nanofluid past a shrinking sheet. Therefore, this study would be important for the researchers working in the relatively new area of nanofluids in order to become familiar with the flow behavior and properties of such nanofluids.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

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Article

Alessandra Borrelli, Giulia Giantesio and Maria Cristina Patria

This paper aims to analyze the steady two-dimensional stagnation-point flow of an electrically conducting Newtonian or micropolar fluid when the obstacle is uniformly heated.

Abstract

Purpose

This paper aims to analyze the steady two-dimensional stagnation-point flow of an electrically conducting Newtonian or micropolar fluid when the obstacle is uniformly heated.

Design/methodology/approach

The governing boundary layer equations are transformed into a system of ordinary differential equations using appropriate similarity transformations. Some analytical considerations about existence and uniqueness of the solution are obtained. The system is then solved numerically using the bvp4c function in MATLAB.

Findings

If the temperature of the obstacle Tw coincides with the environment temperature T0, then the motion reduces to the usual orthogonal stagnation-point flow; if Tw = T0, then it is necessary to include in the similarity function describing the velocity an oblique part due to the temperature. Also, the presence of a uniform external magnetic field orthogonal to the obstacle is examined. In all cases, the motion is reduced to a system of nonlinear ordinary differential equations with boundary conditions, whose solution is discussed numerically when the Prandtl and the Hartmann number varies.

Originality/value

The present results are original and new for the problem of magnetohydrodynamic mixed convection in the plane stagnation-point flow of a Newtonian or a micropolar fluid over a vertical flat plate. At infinity, the motion approaches the orthogonal stagnation-point flow of an inviscid fluid; the effect of an uniform external magnetic field is considered, and the obstacle has a uniform temperature.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 28 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

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Article

Siti Hidayah Muhad Saleh, Norihan Md. Arifin, Roslinda Nazar and Ioan Pop

The purpose of this paper is to present the results of an analysis performed to study unsteady mixed convection at the stagnation point flow over a plate moving along the…

Abstract

Purpose

The purpose of this paper is to present the results of an analysis performed to study unsteady mixed convection at the stagnation point flow over a plate moving along the direction of flow impingement. The similarity transformations are used to transform the governing nonlinear partial differential equation to a system of an ordinary differential equation.

Design/methodology/approach

The transformed equations are then solved numerically by a shooting technique together with bvp4c function.

Findings

The numerical results are compared with the corresponding results from previous researchers. The effects of the unsteadiness Parameter A, Prandtl number Pr, mixed convection parameter λ for plane (m = 0) and axisymmetric (m = 1) flow on the shear stress or the skin friction and heat transfer coefficients, as well as the velocity and temperature profiles, are presented and discussed.

Originality/value

Dual solutions for the opposing flow and multiple solutions for the assisting flow are found.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

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