Unsteady stagnation flow and heat transfer towards a shrinking sheet

The unsteady stagnation flow towards a shrinking sheet is investigated. With assumptions that the sheet is shrunk impulsively from rest, and simultaneously the surface temperature is suddenly increased from that of surrounding fluid, the boundary layer equations are transformed to a set of nonlinear partial differential equations by means of a similarity transformation. The highly accurate analytical approximations are given, which match the numerical results given by the Keller–Box scheme.     

Stagnation point flow and heat transfer over a stretching/shrinking sheet in a porous medium

The steady stagnation point flow and heat transfer over a shrinking sheet in a porous medium is studied. A similarity transformation is used to reduce the governing system of partial differential equations to a set of nonlinear ordinary differential equations which are then solved numerically using the Keller-box method. The behavior of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. 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. The results indicate that dual solutions exist for the shrinking case.     

Dual solutions in boundary layer stagnation-point flow and mass transfer with chemical reaction past a stretching/shrinking sheet

In this paper, an analysis is presented to study dual nature of solution of mass transfer with first order chemical reaction in boundary layer stagnation-point flow over a stretching/shrinking sheet. The governing equations are transformed into a set of self-similar ordinary differential equations by similarity transformations. The transformed equations are solved numerically using very efficient shooting method. The study reveals that the dual solutions of velocity and concentration exist for certain values of velocity ratio parameter (the ratio of stretching/shrinking rate and straining rate). The concentration boundary layer thickness decreases with increasing values of Schmidt number and reaction-rate parameter for both solutions.     

Unsteady MHD flow and heat transfer with viscous dissipation past a stretching sheet

In this paper a study is carried out to analyze the unsteady heat transfer effects of viscous dissipation on the steady boundary layer flow past a stretching sheet with prescribed constant surface temperature in the presence of a transverse magnetic field. The sheet is assumed to stretch linearly along the direction of the fluid flow. The assumed initial steady flow and temperature field neglecting dissipation effects becomes transient by accounting dissipation effects when time t′ > 0. The temperature and the Nusselt number are computed numerically using an implicit finite difference method. The obtained steady temperature field with dissipation is of practical importance.     

Influence of heat transfer on the nonorthogonal stagnation point flow of a third‐order fluid towards a stretching surface

The present article looks at the theoretical analysis of a steady stagnation‐point flow with heat transfer of a third‐order fluid towards a stretching surface. The formulation of the problem has been carried out for a third order fluid and constructed partial differential equations are rehabilitated into ordinary differential equations. The consequential ordinary differential equations are solved analytically using the homotopy analysis method (HAM). Graphical illustrations are shown for various parameters involved in the flow equations. Numerical values of skin friction coefficients and heat flux are computed and presented through tables. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21042     

Mixed convection flow near a non-orthogonal stagnation point towards a stretching vertical plate

An analysis of the steady two-dimensional mixed convection flow of an incompressible viscous fluid near an oblique stagnation point on a heated or cooled stretching vertical flat plate has been studied. It is assumed that the plate is stretched with a velocity proportional to the distance from a fixed point and the temperature of the plate is constant. Both the cases of the assisting and opposing flows are considered. It is shown that the velocity increases as the shear parameter γ increases with the increase of the straining parameter a/c. These flows have a boundary layer structure near the stagnation region. It is also found that the flow has an inverted boundary layer structure when the stretching velocity of the surface exceeds the stagnation velocity of the free stream (a/c < 1). It is shown that the position of the point x_s of zero skin friction (shear stress on the wall) is shifted to the left or to the right of the origin and it depends upon the balance between obliqueness, straining motion and buoyancy effects.     

On impulsively started convection: The case of stagnation point flow

Transient convection in incompressible planar and axisymmetric point flow is analyzed numerically in this work, and the thermal boundary layer response to surface sudden heating and cooling in the two settings is presented and compared over a range of Prandtl number between 0.5 and 100. A comparison between surface sudden cooling and heating is performed and different criteria are established as to when surface sudden heating and cooling are equivalent in terms of the transition time. With no initial thermal boundary layer (surface and fluid are at the same temperature), the transition time from the initial steady state to the final steady state upon surface sudden cooling or heating is found to be a constant regardless of the surface heating or cooling extent above or below the initial surface temperature, and is dependent only on the Prandtl number. With the existence of an initial thermal boundary layer, the transition time is dependent upon the heating or cooling extent, the initial surface temperature, the Prandtl number and whether heating/cooling is towards building-up or demolishing the thermal boundary later. It takes longer time when surface sudden heating or cooling is towards demolishing the thermal boundary layer than building it up. With symmetric surface sudden cooling or heating above or below the far-field fluid temperature, the transition time is independent on the surface cooling or heating extent and is a function of only the Prandtl number. A considerable difference in the thermal boundary layer response in the two settings is found. The transition time from the initial to the final steady state in axisymmetric stagnation point flow is less than that in plane stagnation flow under the same conditions.     

Slip effects on boundary layer stagnation-point flow and heat transfer towards a shrinking sheet

In the present study, we analyze the effects of partial slip on steady boundary layer stagnation-point flow of an incompressible fluid and heat transfer towards a shrinking sheet. Similarity transformation technique is adopted to obtain the self-similar ordinary differential equations and then the self-similar equations are solved numerically using shooting method. This investigation explores the conditions of the non-existence, existence, uniqueness and duality of the solutions of self-similar equations numerically. Due to the increase of slip parameter (δ), the range of velocity ratio parameter (c/a) where the similarity solution exists, increases.     

Viscoelastic boundary layer flow and heat transfer over an exponential stretching sheet

Viscoelastic boundary layer flow and heat transfer over an exponential stretching continuous sheet have been examined in this paper. Approximate analytical similarity solution of the highly non-linear momentum equation and confluent hypergeometric similarity solution of the heat transfer equation are obtained. Accuracy of the analytical solution for stream function is verified by numerical solutions obtained by employing Runge-Kutta fourth order method with shooting. These solutions involve an exponential dependent of stretching velocity, prescribed boundary temperature and prescribed boundary heat flux on the flow directional coordinate. The effects of various physical parameters like viscoelastic parameter, Prandtl number, Reynolds number, Nusselt number and Eckert number on various momentum and heat transfer characteristics are discussed in detail in this work.     

Unsteady mixed convection flow of a micropolar fluid near the stagnation point on a vertical surface

The unsteady mixed convection boundary-layer flow of a micropolar fluid near the region of the stagnation point on a double-infinite vertical flat plate is studied. It is assumed that the unsteadiness is caused by the impulsive motion of the free stream velocity and by sudden increase or sudden decrease in the surface temperature from the uniform ambient temperature. The problem is reduced to a system of non-dimensional partial differential equations, which is solved numerically using the Keller-box method. This method may present well-behaved solutions for the transient (small time) solution and those of the steady-state flow (large time) solution. It was found that there is a smooth transition from the small-time solution (initial unsteady-state flow) to the large-time solution (final steady-state flow). Further, it is shown that for both assisting and opposing cases and a fixed value of the Prandtl number, the reduced steady-state skin friction and the steady-state heat transfer from the wall (or Nusselt number) decrease with the increase of the material parameter. On the other hand, it is shown that with the increase of the Prandtl number and a fixed value of the material parameter, the reduced steady-state skin friction decreases when the flow is assisting and it increases when the flow is opposing.     

Three-dimensional viscous flow and heat transfer over a permeable shrinking sheet

This paper presents a numerical analysis of a steady three-dimensional fluid flow and heat transfer towards a permeable shrinking sheet. The governing nonlinear partial differential equations are transformed into a system of ordinary differential equations by a similarity transformation, which are then solved numerically by a shooting method. The effects of the governing parameters on the skin friction and heat transfer from the surface of the shrinking surface are illustrated graphically. It is found that dual solutions exist for the shrinking case. A comparison with known results from the open literature has been done and it is shown to be in excellent agreement.     

Mixed convection in the stagnation point flow adjacent to a vertical surface in a viscoelastic fluid

An analytic technique, namely the homotopy analysis method (HAM), is applied to study the steady mixed convection in two-dimensional stagnation flows of a viscoelastic fluid around heated surfaces for the case when the temperature of the wall varies linearly with the distance from the stagnation point. The two-dimensional boundary layer equations governing the flow and thermal fields are reduced by appropriate transformations to a set of two ordinary differential equations. These equations are solved analytically using the HAM in the buoyancy assisting and opposing regions. It is found that, as for the Newtonian flow case, a reverse flow region develops in the buoyancy opposing flow region, and dual solutions are found to exist in that flow regime for a certain ranges of the buoyancy and viscoelastic parameters.     

Lattice Boltzmann simulation of mixed convection heat transfer in a corrugated wall cavity utilizing water‐based nanofluids

In the present study, the effects of Cu and CuO nanoparticles' presence on mixed convection heat transfer in a lid‐driven cavity with a corrugated wall are investigated using the lattice Boltzmann method. The boundary fitting method with second‐order accuracy at both velocity and temperature fields is used to simulate the curved boundaries in the LBM. The problem is investigated for different Richardson numbers (0.1–10), volume fractions of nanoparticles (0–0.05), curve amplitudes (0.05–0.25), and phase shifts of corrugated wall (0–270) when the Reynolds number is equal to 25. The volume fraction of added nanoparticles to the water‐based fluid is less than 0.05 to make dilute suspensions. Results show that adding nanoparticles enhances the rate of heat transfer. It is found that nanoparticles have significant effects on both fluid flow and heat transfer of the mixed convection, especially for low Richardson numbers. A comparison between Cu and CuO nanoparticles shows the Cu nanoparticles have a better effect on heat transfer enhancement for all tested conditions. The results also represent the effective role of a corrugated wall on the rate of nanofluid heat transfer. It is observed that increasing the wavy wall's amplitude leads to a decrease of the average Nusselt numberfor a high Richardson number. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21024     

Natural convection in a parabolic enclosure with an internal vertical heat source filled with Cu–water nanofluid

The numerical investigation of the natural convection in concave and convex parabolic enclosures with a nanofluid consisting of water and copper nanoparticles is carried out by using the finite volume method. The upper and lower walls of the enclosures are adiabatic while the sidewalls are isothermal at a cold temperature. An internal heat source of constant length (ε = 0.2) and negligible thickness is placed at various vertical positions along the center of the enclosure. It was found that the increase in the location of the heat source leads to a drop in the water and nanofluid flow circulation in both types of enclosures. For both considered Cases I and II, the average Nusselt number increases when the Rayleigh number and solid volume fraction increase. Moreover, it was concluded that Case I with δ = 0.8 is the optimum case for heat transfer enhancement for Ra = 10³ and Ra = 10⁴. Case II with δ = 0.5 is optimum for Ra = 10⁵. Both cases are satisfied when the nanofluid is used with ? = 0.2.     

Numerical investigation on the use of nanofluids in natural convection heat exchangers using a mixture‐phase approach

In the present research, the behavior of a Newtonian nanofluid (water–Al₂O₃) in a mixture phase model approach is numerically examined. The process of heating is done in two different ways. Deterioration was found in the mean Nusselt number of a nanofluid in the mixture‐phase model approach when compared to the mean Nusselt number of pure water. However, in the single‐phase model there was an increase in the Nusselt number when compared to the Nusselt number of pure water. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20383     

An Exact Solution for Heat Transfer of a Jet Co-Axially Impinging on a Rotating Disk and Its Comparisons with Stagnation Point Experiments

IntroductionJet impingement is a widely used high-efficiencytechnique fOr cooling rotating disks, which are end-wallsurfaces of gas turbine rotors, comPuter disk drives etc.Fluid f'low, heat trallsfer and geometric arrangement inthe case of a single round jet impinging co-axially in anorthogonal mode on a rotatng disk are characterized byFig. l.Many peculiarihes of fluid fIow and heat transfer ofreal impinging jets under comPlicated conditions(different system geometry, impinging flow proper…     

Enhancement of mixed convection heat transfer in a three-dimensional horizontal channel flow by insertion of a moving block

Enhancement of mixed convection heat transfer rate of a heat surface in a three dimensional horizontal channel with insertion of a moving block is studied numerically. Bossinique assumption is not adopted, then methods of Roe scheme, preconditioning, and dual time stepping are needed to solve the governing equations. Contributions of important parameters of Gr/Re² and moving block velocity to the heat transfer rate are validated. Due to the consideration of natural convection, under situations of large magnitude of Gr/Re² a counter-effect for promotion of heat transfer rate is observed. Oppositely, under situations of small magnitude of Gr/Re², the enhancement of heat transfer rate is remarkably achieved.     

A numerical investigation on the heat and fluid flow of various nanofluids on a stretching sheet

Following the necessity of investigating fluid flow and heat transfer in the stretching sheet problem and effect of nanofluids on them, performance of various nanofluids were investigated in the present study. Three base fluids (deionized water, ethylene glycol, and engine oil) in combination with 18 nanoparticles (metals and their oxides) were investigated. While experimental methods are preferable, a mathematical model was developed and solved by applying differential quadrature method due to lack of such experimental data. With the results obtained in the real dimensions, the error caused by the cancellation of the viscosity effect due to the dimensionless variables was omitted. Effects of magnetic field and volume fraction of nanoparticle on the fluid flow and heat transfer characteristics were investigated. Highest heat transfer rate as well as small amounts of shear stress was obtained for deionized water–Al and deionized water–Mg nanofluids. Increasing volume fraction of nanoparticle was observed to increase both heat transfer and shear stress rates, while presence of a magnetic field caused an increase in shear stress and decrease in heat transfer rate.     

Nonsimilarity boundary layer flow of copper–water nanofluid near stagnation point in a porous medium

The effects of nanoparticle shape are first introduced to study the nonsimilar solutions of stagnation point boundary layer flow of water–copper nanofluid saturated in a porous medium. Two cases of solid matrix of porous medium, including glass balls and aluminum foam, are considered. By using a new empirical correlation for the heat capacitance, thermal conductivity, and thermal diffusivity of the nanofluid saturated in a porous medium, the governing equations of the problem are constructed and reduced by dimensionless variables and nonsimilar transformations, and the homotopy analysis method is adopted to solve the partial differential equations. The results indicate that the heat transfer is significantly enhanced with the increase of permeability of the porous medium on the surface of the stagnation point boundary layer flow. In addition, it is found that the empirical shape of the nanoparticle has an impact on the heat transfer.     

Micropolar fluid flow towards a stretching/shrinking sheet in a porous medium with suction

This paper presents a numerical analysis of a micropolar fluid flow towards a permeable stretching/shrinking sheet in a porous medium. The governing nonlinear partial differential equations are transformed into a system of ordinary differential equations by a similarity transformation, before being solved numerically by a finite-difference scheme known as the Keller-box method. The effects of the governing parameters on the fluid flow and heat transfer characteristics are illustrated graphically. It is found that dual solutions exist for the shrinking case, whereas for the stretching case, the solution is unique.