Mixed Convection Boundary Layer Research Articles

Marangoni Mixed Convection Boundary Layer Flow in a Nanofluid

The problem of Marangoni mixed convection boundary layer flow and heat transfer that can be formed along the interface of two immiscible fluids in a nanofluid is studied using different types of nanoparticles. Numerical solutions of the similarity equations are obtained using the shooting method. Three types of metallic or nonmetallic nanoparticles, namely copper (Cu), alumina (23AlO) and titania (2TiO) are consideredby using a water-based fluid to investigate the effect of the solid volume fraction or nanoparticle volume fraction parameter ϕ of the nanofluid. The influences of the interest parameters on the reduced velocity along the interface, velocity profiles as well as the reduced heat transfer at the interface and temperature profiles were presented in tables and figures.

G-Jitter mixed convective slip flow of nanofluid past a permeable stretching sheet embedded in a Darcian porous media with variable viscosity.

The unsteady two-dimensional laminar g-Jitter mixed convective boundary layer flow of Cu-water and Al2O3-water nanofluids past a permeable stretching sheet in a Darcian porous is studied by using an implicit finite difference numerical method with quasi-linearization technique. It is assumed that the plate is subjected to velocity and thermal slip boundary conditions. We have considered temperature dependent viscosity. The governing boundary layer equations are converted into non-similar equations using suitable transformations, before being solved numerically. The transport equations have been shown to be controlled by a number of parameters including viscosity parameter, Darcy number, nanoparticle volume fraction, Prandtl number, velocity slip, thermal slip, suction/injection and mixed convection parameters. The dimensionless velocity and temperature profiles as well as friction factor and heat transfer rates are presented graphically and discussed. It is found that the velocity reduces with velocity slip parameter for both nanofluids for fluid with both constant and variable properties. It is further found that the skin friction decreases with both Darcy number and momentum slip parameter while it increases with viscosity variation parameter. The surface temperature increases as the dimensionless time increases for both nanofluids. Nusselt numbers increase with mixed convection parameter and Darcy numbers and decreases with the momentum slip. Excellent agreement is found between the numerical results of the present paper with published results.

The effect of unsteadiness on mixed convection boundary-layer stagnation-point flow over a vertical flat surface embedded in a porous medium

The unsteady mixed convection boundary-layer flow towards a stagnation point on a heated vertical surface embedded in a porous medium is considered. The governing partial differential equations are reduced to a system of nonlinear ordinary differential equations using a similarity transformation. These are then solved numerically using the Keller-box method. These numerical results are complemented by asymptotic solutions for large values of A. The influence of dimensionless parameters λ (mixed convection), A (time variation), n (wall temperature variation) on the flow field and heat transfer is analyzed and discussed, dual solutions being found for both assisting and opposing buoyant flows.

Solar Radiation Assisted Mixed Convection Flow Along a Vertical Plate

ABSTRACTSolar radiation assisted mixed convection boundary layer flow of Newtonian fluid along a non- reflecting, non-absorbing and ideally transparent semi-infinite vertical plate is studied here. Beer's law is used to express the solar radiation term. It is convenient to transform the non-linear dimensionless parabolic partial differential equations into (i) primitive variable formulation (PVF) and (ii) stream function formulation (SFF) before applying the numerical schemes. Coupled equations thus obtained from PVF are integrated numerically through implicit finite difference method together with the Gaussian elimination technique whereas block tridiagonal Keller-box technique is adopted to simulate the system of equations obtained from SFF. Numerical results from these two methods are also compared graphically in order to test the validation of the two schemes. However, due to less computation time and accuracy numeric results of shear stress, local Nusselt number coefficient, velocity and temperature profiles are obtained via SFF. It is found that velocity as well as temperature of the non-absorbing fluid enhances owing to the increase in solar radiation parameter.

Non-Newtonian Mixed Convection Flow along an Isothermal Horizontal Circular Cylinder

The mixed convective laminar two-dimensional boundary-layer flow of non-Newtonian pseudo-plastic fluids is studied along an isothermal horizontal circular cylinder using a modified power-law viscosity model. In this model, there are no unrealistic limits of zero or infinite viscosity; consequently, no irremovable singularities are introduced into boundary-layer formulations for such fluids. Therefore, the boundary-layer equations can be solved numerically by using marching order implicit finite difference method with double sweep technique. Numerical results are presented for the case of shear-thinning fluids in terms of the fluid velocity and temperature distributions, shear stresses and rate of heat transfer in terms of the local skin-friction and local Nusselt number respectively. Here, it is found that heating the cylinder delays separation and if the cylinder is warm enough, suppress it completely. Cooling the cylinder brings the separation point nearer to the lower stagnation point and for a very cold cylinder there will not be boundary-layer on the cylinder.

Mixed convection flow from a horizontal circular cylinder embedded in a porous medium filled by a nanofluid: Buongiorno–Darcy model

Steady mixed convection boundary layer flow from an isothermal horizontal circular cylinder embedded in a porous medium filled with a nanofluid has been studied for both cases of a heated and cooled cylinder using the Buongiorno–Darcy mathematical nanofluid model. The resulting system of nonlinear partial differential equations is solved numerically using an implicit finite-difference scheme. The solutions for the flow and heat transfer characteristics are evaluated numerically for various values of the governing parameters, namely the constant mixed convection parameter λ, the traditional Lewis number Le, the buoyancy ratio parameter Nr, the Brownian motion parameter Nb and the thermophoresis parameter Nt. It is found that in the present case of the porous medium flow, the separation is always suppressed at negative values of λ. When λ changes from −2.1 to 0, one has a “heating” of the cylinder, but a heating in the negative range of λ (λ < 0). However, for a clear (Newtonian) fluid, Merkin (1977) found that heating the cylinder (λ > 0) delays the separation of the boundary layer and if the cylinder is hot enough (large values of λ > 0), then it is suppressed completely at a positive value of λ, somewhere between 0.88 and 0.89. On the other hand, cooling the cylinder (λ < 0) brings the boundary layer separation point nearer to the lower stagnation point and for a sufficiently cold cylinder (large values of λ < 0) there will not be a boundary layer on the cylinder.

Mixed Convection in Viscoelastic Boundary Layer Flow and Heat Transfer Over a Stretching Sheet

AbstractAn investigation is carried out on mixed convection boundary layer flow of an incompressible and electrically conducting viscoelastic fluid over a linearly stretching surface in which the heat transfer includes the effects of viscous dissipation, elastic deformation, thermal radiation, and non-uniform heat source/sink for two general types of non-isothermal boundary conditions. The governing partial differential equations for the fluid flow and temperature are reduced to a nonlinear system of ordinary differential equations which are solved analytically using the homotopy analysis method (HAM). Graphical and numerical demonstrations of the convergence of the HAM solutions are provided, and the effects of various parameters on the skin friction coefficient and wall heat transfer are tabulated. In addition it is demonstrated that previously reported solutions of the thermal energy equation given in [1] do not converge at the boundary, and therefore, the boundary derivatives reported are not correct.

Soret and Dufour effects on mixed convection unsteady MHD boundary layer flow over stretching sheet in porous medium with chemically reactive species

This paper studies the thermal-diffusion and diffusion thermo-effects in the hydro-magnetic unsteady flow by a mixed convection boundary layer past an impermeable vertical stretching sheet in a porous medium in the presence of chemical reaction. The velocity of the stretching surface, the surface temperature, and the concentration are assumed to vary linearly with the distance along the surface. The governing partial differential equations are transformed into self-similar unsteady equations using similarity transformations and solved numerically by the Runge-Kutta fourth order scheme in association with the shooting method for the whole transient domain from the initial state to the final steady state flow. Numerical results for the velocity, the temperature, the concentration, the skin friction, and the Nusselt and Sherwood numbers are shown graphically for various flow parameters. The results reveal that there is a smooth transition of flow from unsteady state to the final steady state. A special case of our results is in good agreement with an earlier published work.

Mixed convection boundary layer flow past a vertical flat plate embedded in a non-Darcy porous medium saturated by a nanofluid

Purpose – The purpose of this paper is to numerically solve the problem of steady mixed convection boundary layer flow past a vertical flat plate embedded in a fluid-saturated porous medium filled by a nanofluid. The non-Darcy equation model along with the mathematical nanofluid model proposed by Tiwari and Das (2007) has been used. Design/methodology/approach – Using appropriate similarity transformations, the basic partial differential equations are transformed into ordinary differential equations. These equations have been solved numerically for different values of the nanoparticle volume fraction, the mixed convection and the non-Darcy parameters using the bvp4c function from Matlab. A stability analysis has been also performed. Findings – Numerical results are obtained for the reduced skin-friction, heat transfer and for the velocity and temperature profiles. The results indicate that dual solutions exist for the opposing flow case (λ&lt;0). The stability analysis indicates that for the opposing flow case, the lower solution branch is unstable, while the upper solution branch is stable. In addition, it is shown that for a regular fluid (φ=0) a very good agreement exists between the present numerical results and those reported in the open literature. Research limitations/implications – The problem is formulated for three types of nanoparticles, namely, copper (Cu), alumina (Al2O3) and titania (TiO2). However, the paper present results here only for the Cu nanoparticles. The analysis reveals that the boundary layer separates from the plate. Beyond the turning point it is not possible to get the solution based on the boundary-layer approximations. To obtain further solutions, the full basic partial differential equations have to be solved. Practical implications – Nanofluids have many practical applications, for example, the production of nanostructured materials, engineering of complex fluids, for cleaning oil from surfaces due to their excellent wetting and spreading behavior, etc. Social implications – Nanofluids could be applied to almost any disease treatment techniques by reengineering the nanoparticle properties. Originality/value – The present results are original and new for the boundary-layer flow and heat transfer past a vertical flat plate embedded in a porous medium saturated by a nanofluid. Therefore, this study would be important for the researchers working in porous media in order to become familiar with the flow behavior and properties of such nanofluids.

Mixed convection boundary layer flow from a vertical truncated cone in a nanofluid

Purpose– The purpose of this paper is to investigate the steady mixed convection boundary layer flow from a vertical frustum of a cone in water-based nanofluids. The problem is formulated to incorporate three kinds of nanoparticles: copper, alumina and titanium oxide. The working fluid is chosen as water with the Prandtl number of 6.2. The mathematical model used for the nanofluid incorporates the particle volume fraction parameter, the effective viscosity and the effective thermal diffusivity. The entire regime of the mixed convection includes the mixed convection parameter, which is positive for the assisting flow (heated surface of the frustum cone) and negative for the opposing flow (cooled surface of the frustum cone), respectively.Design/methodology/approach– The transformed non-linear partial differential equations are solved numerically for some values of the governing parameters. The derivatives with respect to? were discretized using the first order upwind finite differences and the resulting ordinary differential equations with respect to? were solved using bvp4c routine from Matlab. The absolute error tolerance in bvp4c was 1e-9.Findings– The features of the flow and heat transfer characteristics for different values of the governing parameters are analysed and discussed. The effects of the particle volume fraction parameter \phi, the mixed convection parameter \lambda and the dimensionless coordinate? on the flow and heat transfer characteristics are determined only for the Cu nanoparticles. It is found that dual solutions exist for the case of opposing flows. The range of the mixed convection parameter for which the solution exists increases in the presence of the nanofluids.Originality/value– The paper models the mixed convection from a vertical truncated cone using the boundary layer approximation. Multiple (dual) solutions for the flow reversals are obtained and the range of existence of the solutions was found. Particular cases for ?=0 (full cone), ? &gt;&gt;1 and (free convection limit) \lambda&gt;&gt;1were studied. To the authors best knowledge this problem has not been studied before and the results are new and original.

Influence of thermal radiation on mixed convection heat and mass transfer stagnation-point flow in nanofluids over stretching/shrinking sheet in a porous medium with chemical reaction

We have investigated the mixed convection boundary layer flow of nanofluids on a stagnation-point flow over a permeable stretching/shrinking sheet subject to thermal radiation, heat source/sink, viscous dissipation and chemical reaction by using numerical method. Three types of nanofluids namely copper–water, alumina–water, titanium dioxide–water were considered in the present study. The governing boundary layer equations are transformed into a system of nonlinear ordinary differential equations by using similarity transformation which are then solved numerically using fifth-order Runge–Kutta–Fehlberg method with shooting technique. The effects of various physical parameters are analyzed and discussed in graphical and tabular form. The effects of some physical parameters such as mixed convection parameter, radiation parameter, Schimdt number, porous parameter, Eckert number, chemical reaction parameter are analyzed on velocity, temperature and concentration profiles as well as on skin-friction coefficient, local Nusselt number and Sherwood number. It is found that copper–water exhibits higher mass transfer rates compared to alumina–water and titanium dioxide–water nanofluids for stretching and shrinking sheets.

Mixed convection boundary layer flow along vertical thin needles in nanofluids

Purpose – The purpose of this paper is to theoretically study the problem of mixed convection boundary layer flow and heat transfer past a vertical needle with variable wall temperature using nanofluids. The similarity equations are solved numerically for copper nanoparticles in the based fluid of water to investigate the effect of the solid volume fraction parameter of the fluid and heat transfer characteristics. The skin friction coefficient, Nusselt number, and the velocity and temperature profiles and are graphically presented and discussed. Design/methodology/approach – The transformed system of ordinary differential equations was solved using the function bvp4c from Matlab. The relative tolerance was set to 1e-10. For the study of the stability the authors also used the bvp4c function in combination with chebfun package from Matlab. Findings – It is found that the solid volume fraction affects the fluid flow and heat transfer characteristics. The numerical results for a regular fluid and forced convection flow are compared with the corresponding results reported by Chen and Smith. The solutions exists up to a critical value of λ, beyond which the boundary layer separates from the surface and the solution based upon the boundary-layer approximations is not possible Originality/value – The paper describes how multiple (dual) solutions for the flow reversals are obtained. A stability analysis for this flow reversal has been also done showing that the lower solution branches are unstable, while the upper solution branches are stable.

Mixed convection stagnation flow towards a vertical shrinking sheet

The steady mixed convection stagnation-point boundary layer flow past a vertical stretching/shrinking sheet in a viscous fluid is numerically investigated. The numerical solutions are obtained by solving the similarity equations which are derived via the similarity transformation technique in order to reduce the nonlinear partial differential equations into a system of nonlinear ordinary differential equations. Numerical techniques, namely the Keller-box method, along with the shooting method are used to solve the transformed ordinary differential equations. Results have been presented and discussed for the effects of the governing parameters on the skin friction coefficient and the local Nusselt number as well as for the velocity and temperature profiles. The present results show quite interesting flow behavior for the shrinking sheet problem compared to the stretching sheet problem. On the other hand, it is also found that dual solutions exist for the present problem. The streamlines for the two dimensional flow have also been presented.

Mixed convection boundary layer flow of a casson fluid near the stagnation point over a permeable surface

The problem of steady mixed convection boundary layer flow of a Casson fluid near the stagnation-point on a vertical surface when the wall is permeable, where there is suction or injection effect, is considered. The governing partial differential equations are converted into ordinary differential equations by similarity transformation, which is then solved numerically using the shooting method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are presented for different values of the governing parameters. It is found that the imposition of suction is to increase the velocity profiles and to delay the separation of boundary layer, while the injection parameter decreases the velocity profiles.

Thermal radiation and chemical reaction effects on MHD mixed convective boundary layer slip flow in a porous medium with heat source and Ohmic heating

An analytical study for the problem of mixed convection with thermal radiation and first-order chemical reaction on magnetohydrodynamics boundary layer flow of viscous, electrically conducting fluid past a vertical permeable surface embedded in a porous medium has been presented. Slip boundary condition is applied at the porous interface. The heat equation includes the terms involving the viscous dissipation, radiative heat flux, Ohmic dissipation, the internal absorption and absorption of radiation, whereas the mass transfer equation includes the effects of chemically reactive species of first order. The dimensionless governing equations for this investigation are formulated and the non-linear coupled differential equations are solved analytically using the perturbation technique. Comparisons with previously published work on special cases of the problem are performed and results are found to be in excellent agreement. The results obtained show that the velocity, temperature and concentration fields are appreciably influenced by the presence of magnetic field, thermal radiation, chemical reaction and Ohmic dissipation. It is observed that the effect of magnetic field, heat source and thermal radiation is to decrease the velocity, temperature profiles in the boundary layer. The effect of increasing the values of rarefaction parameter is to increase the velocity in the momentum boundary layer. Further, it is found that increasing the value of the chemical reaction decreases the concentration of species in the boundary layer. Also, the effects of the various parameters on the skin-friction coefficient, local Nusselt number and local Sherwood number at the surface are discussed.

Melting Effect on Mixed Convection Boundary Layer Flow About a Vertical Surface Embedded in a Porous Medium: Opposing Flows Case

The effect of melting phenomenon on the steady mixed convection boundary layer flow about a vertical surface embedded in a fluid-saturated porous medium is numerically studied. The boundary layer partial differential equations are transformed into a set of nonlinear ordinary differential equations by using similarity transformation and then are solved using the shooting method for different values of the mixed convection and melting parameters. Results show that dual solutions exist for a certain range of these parameters. The results also indicate that the melting phenomenon reduces the heat transfer rate and expedites the boundary layer separation.

Effect of double stratification on mixed convection boundary layer flow of a nanofluid past a vertical plate in a porous medium

The effect of thermal and mass stratification on mixed convection boundary layer flow over a vertical flat plate embedded in a porous medium saturated by a nanofluid has been investigated. The vertical plate is maintained at uniform and constant heat, mass and nanoparticle fluxes, and the behavior of the porous medium is described by the Darcy model. The model considered for nanofluids incorporates the effects of Brownian motion and thermophoresis. In addition, the thermal energy equations include regular diffusion and cross-diffusion terms. A suitable coordinate transformation is introduced, and the obtained system of non-similar, coupled and non-linear partial differential equations is solved numerically. The influence of pertinent parameters on the non-dimensional velocity, temperature, concentration and nanoparticle volume fraction are discussed. In addition, the variation of heat, mass and nanoparticle transfer rates at the plate are exhibited graphically for different values of physical parameters.

Finite element solution of mixed convection flow of a nanofluid over a vertical stretching sheet with power law containing metal oxide nanoparticles

Steady, two dimensional, mixed convection laminar boundary layer flow of an incompressible nanofluid over a semi-infinite vertical power law stretching sheet has been investigated numerically. The velocity and temperature of the sheet are assumed to vary in a power-law form. The resulting nonlinear governing equations (obtained with the Boussinesq approximation) have been solved, using a robust, extensively validated, Galerkin finite element method (FEM) for spherical shaped nanoparticles with volume fraction up to 4%, with associated boundary conditions. Different water-based nanofluids containing metal oxide nanoparticles, i.e., CuO, Al2O3 and TiO2 are taken into consideration. Nanofluids effectively enhance the heat transfer and reduce the skin friction as compared to pure water in the present problem. The effects of the various other parameters are discussed to achieve better control on the rate of heat transfer.

A study on the mixed convection boundary layer flow and heat transfer over a vertical slender cylinder

In this investigation, the series solutions of mixed convection boundary layer flow over a vertical permeable cylinder are constructed. Two types of series as well numerical solutions are presented by choosing exponential and rational bases. The resulting differential system are solved by employing homotopy analysis method (HAM) and Pade technique which have been proven to be successful in tackling nonlinear problems. We offer various verifications of the solutions by comparing to existing, documented results and also mathematically, through reduction of sundry parameters. The convergence of the series solutions have been discussed explicitly. Comparison with existing results reveal that the series solutions are not only valid for large (aiding flow) but also for small values (opposing flow) of ? and the dual solutions do not obtain in both cases.

A Mixed Convection Boundary Layer Flow over a Vertical Wall in a Porous Medium, with Exponentially Varying Fluid Viscosity

This study investigates a mixed convection boundary layer flow over a vertical wall embedded in a highly porous medium. The fluid viscosity is assumed to decrease exponentially with temperature. The boundary layer equations are transformed into a non-similar form using an appropriate non-similar variable ξ and a pseudo-similar variable η. The non-similar equations are solved using an efficient local non-similarity method. The effect of viscosity variation parameter on the heat transfer, skin friction and the velocity and temperature distribution within the boundary layer is investigated. The viscosity variation parameter, the viscous dissipation parameter and non-simi-larity variable are shown to have a significant effect on velocity and thermal boundary layer and also on the skin friction coefficient and heat transfer at the wall.