Aspect ratio effect on natural convection in water near its density maximum temperature

The effect of the aspect ratio on natural convection in water subjected to density inversion has been investigated in this study. Numerical simulations of the two-dimensional, steady state, incompressible flow in a rectangular enclosure with a variety of aspect ratios, ranging from 0.125 to 100, have been accomplished using a finite element model. Computations cover Rayleigh numbers from 10³ to 10⁶. Results reveal that the aspect ratio, A, the Rayleigh number, Ra, and the density distribution parameter, R, are the key parameters to determine the heat transfer and fluid flow characteristics for density inversion fluids in an enclosure. A new correlation for predicting the maximum mean Nusselt number is proposed in the form of , with the constants a and b depending on density distribution number R. It is demonstrated that the aspect ratio has a strong impact on flow patterns and temperature distributions in rectangular enclosures. The stream function ratio Ψ_inv/|Ψ_reg| is introduced to describe quantitatively the interaction between inversional and regular convection. For R=0.33, the density inversion enhancement is observed in the regime near A=3.     

Steady mixed convection flow of Maxwell fluid over an exponentially stretching vertical surface with magnetic field and viscous dissipation

The steady mixed convection flow and heat transfer from an exponentially stretching vertical surface in a quiescent Maxwell fluid in the presence of magnetic field, viscous dissipation and Joule heating have been studied. The stretching velocity, surface temperature and magnetic field are assumed to have specific exponential function forms for the existence of the local similarity solution. The coupled nonlinear ordinary differential equations governing the local similarity flow and heat transfer have been solved numerically by Chebyshev finite difference method. The influence of the buoyancy parameter, viscous dissipation, relaxation parameter of Maxwell fluid, magnetic field and Prandtl number on the flow and heat transfer has been considered in detail. The Nusselt number increases significantly with the Prandtl number, but the skin friction coefficient decreases. The Nusselt number slightly decreases with increasing viscous dissipation parameter, but the skin friction coefficient slightly increases. Maxwell fluid reduces both skin friction coefficient and Nusselt number, whereas buoyancy force enhances them.     

Melting heat transfer in steady laminar flow over a moving surface

The steady laminar boundary layer flow and heat transfer from a warm, laminar liquid flow to a melting surface moving parallel to a constant free stream is studied in this paper. The continuity, momentum and energy equations, which are coupled nonlinear partial differential equations are reduced to a set of two nonlinear ordinary differential equations, before being solved numerically using the Runge–Kutta–Fehlberg 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. Effects of the melting parameter, moving parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. It is found that the problem admits dual solutions.     

Effects of viscous dissipation and heat generation (absorption) in a thermal boundary layer of a non-Newtonian fluid over a continuously moving permeable flat plate

The problem of boundary-layer flow and heat transfer of a non-Newtonian power-law fluid over a moving porous infinite flat plate in the presence of viscous dissipation and heat generation or absorption is investigated analytically. It is assumed that both the momentum and the energy equations are coupled by the stress friction factor, and an assumption is introduced regarding the heat-transfer index. It is found that exact analytical solutions for velocity and temperature exist only for pseudoplastic fluids in the presence of suction at the surface. The effects of the suction parameter, Eckert number, and the heat generation or absorption parameter on the velocity and temperature profiles, as well as on the skin-friction coefficient and Nusselt number are discussed.     

Flow of couple stress fluid with variable thermal conductivity

The steady flow and heat transfer of a couple stress fluid due to an inclined stretching cylinder are analyzed. The thermal conductivity is assumed to be temperature dependent. The governing equations for the flow and heat transfer are transformed into ordinary differential equations. Series solutions of the resulting problem are computed. The effects of various interested parameters, e.g., the couple stress parameter, the angle of inclination, the mixed convection parameter, the Prandtl number, the Reynolds number, the radiation parameter, and the variable thermal conductivity parameter, are illustrated. The skin friction coefficient and the local Nusselt number are computed and analyzed. It is observed that the heat transfer rate at the surface increases while the velocity and the shear stress decrease when the couple stress parameter and the Reynolds number increase. The temperature increases when the Reynolds number increases.     

A computational work on turbulent flow and heat transfer in a channel fitted with inclined baffles

Numerical studies of momentum and heat transfer characteristics have been investigated of a steady incompressible turbulent flow of air through channel. The channel has inclined baffles which are arranged on the walls in a periodically staggered way. The governing equations, namely, continuity, Navier–Stokes and energy, based on k–ω turbulence model to describe the turbulence phenomenon are solved using the finite volume method and the SIMPLE algorithm. Calculations are performed for a Reynolds number between 12,000 and 38,000. The axial velocity profiles, the velocity fields, the local and average coefficient of friction and the Nusselt number distribution were obtained for all the geometry considered and for different sections selected, upstream, downstream and between the two inclined baffles. Simulation results which were obtained by the use of baffles are validated by an experimental study. Good agreement is observed between numerical and experimental results data in the literature.     

Active and passive controls of nanoparticles in Maxwell stagnation point flow over a slipped stretched surface

A steady stagnation-point flow of an incompressible Maxwell fluid towards a linearly stretching sheet with active and passive controls of nanoparticles is studied numerically. The momentum equation of the Maxwell nanofluid is inserted with an external velocity term as a result of the flow approaches the stagnation point. Conventional energy equation is modified by incorporation of nanofluid Brownian and thermophoresis effects. The condition of zero normal flux of nanoparticles at the stretching surface is defined to impulse the particles away from the surface in combination with nonzero normal flux condition. A hydrodynamic slip velocity is also added to the initial condition as a component of the entrenched stretching velocity. The governing partial differential equations are then reduced into a system of ordinary differential equations by using similarity transformation. A classical shooting method is applied to solve the nonlinear coupled differential equations. The velocity, temperature and nanoparticle volume fraction profiles together with the reduced skin friction coefficient, Nusselt number and Sherwood number are graphically presented to visualize the effects of particular parameters. Temperature distributions in passive control model are consistently lower than in the active control model. The magnitude of the reduced skin friction coefficient, Nusselt number and Sherwood number decrease as the hydrodynamic slip parameter increases while the Brownian parameter has negligible effect on the reduced heat transfer rate when nanoparticles are passively controlled at the surface. It is also found that the stagnation parameter contributes better heat transfer performance of the nanofluid under both active and passive controls of normal mass flux.     

Flow and heat transfer of nanofluid past stretching/shrinking sheet with partial slip boundary conditions

The boundary layer flow of a nanofluid past a stretching/shrinking sheet with hydrodynamic and thermal slip boundary conditions is studied. Numerical solutions to the governing equations are obtained using a shooting method. The results are found for the skin friction coefficient, the local Nusselt number, and the local Sherwood number as well as the velocity, temperature, and concentration profiles for some values of the velocity slip parameter, thermal slip parameter, stretching/shrinking parameter, thermophoresis parameter, and Brownian motion parameter. The results show that the local Nusselt number, which represents the heat transfer rate, is lower for higher values of thermal slip parameter, thermophoresis parameter, and Brownian motion parameter.     

Mixed convection boundary layers in the stagnation-point flow toward a stretching vertical sheet

An analysis is made for the steady mixed convection boundary layer flow near the two-dimensional stagnation-point flow of an incompressible viscous fluid over a stretching vertical sheet in its own plane. The stretching velocity and the surface temperature are assumed to vary linearly with the distance from the stagnation-point. Two equal and opposite forces are impulsively applied along the x-axis so that the wall is stretched, keeping the origin fixed in a viscous fluid of constant ambient temperature. The transformed ordinary differential equations are solved numerically for some values of the parameters involved using a very efficient numerical scheme known as the Keller-box method. The features of the flow and heat transfer characteristics are analyzed and discussed in detail. Both cases of assisting and opposing flows are considered. It is observed that, for assisting flow, both the skin friction coefficient and the local Nusselt number increase as the buoyancy parameter increases, while only the local Nusselt number increases but the skin friction coefficient decreases as the Prandtl number increases. For opposing flow, both the skin friction coefficient and the local Nusselt number decrease as the buoyancy parameter increases, but both increase as Pr increases. Comparison with known results is excellent.     

Similarity solutions for MHD thermosolutal Marangoni convection over a flat surface in the presence of heat generation or absorption effects

The problem of steady, laminar, thermosolutal Marangoni convection flow of an electrically-conducting fluid along a vertical permeable surface in the presence of a magnetic field, heat generation or absorption and a first-order chemical reaction effects is studied numerically. The general governing partial differential equations are converted into a set of self-similar equations using unique similarity transformations. Numerical solution of the similarity equations is performed using an implicit, iterative, tri-diagonal finite-difference method. Comparisons with previously published work is performed and the results are found to be in excellent agreement. Approximate analytical results for the temperature and concentration profiles as well as the local Nusselt and sherwood numbers are obtained for the conditions of small and large Prandtl and Schmidt numbers are obtained and favorably compared with the numerical solutions. The effects of Hartmann number, heat generation or absorption coefficient, the suction or injection parameter, the thermo-solutal surface tension ratio and the chemical reaction coefficient on the velocity, temperature and concentration profiles as well as quantitites related to the wall velocity, boundary-layer mass flow rate and the Nusselt and Sherwood numbers are presented in graphical and tabular form and discussed. It is found that a first-order chemical reaction increases all of the wall velocity, Nusselt and Sherwood numbers while it decreases the mass flow rate in the boundary layer. Also, as the thermo-solutal surface tension ratio is increased, all of the wall velocity, boundary-layer mass flow rate and the Nusselt and Sherwood numbers are predicted to increase. However, the exact opposite behavior is predicted as the magnetic field strength is increased.     

Steady nonlinear hydromagnetic flow and heat transfer over a stretching surface of variable temperature

The steady nonlinear hydromagnetic flow of an incompressible, viscous and electrically conducting fluid with heat transfer over a surface of variable temperature stretching with a power-law velocity in the presence of variable transverse magnetic field is analysed. Utilizing similarity transformation, governing nonlinear partial differential equations are transformed to nonlinear ordinary differential equations and they are numerically solved using fourth-order Runge–Kutta shooting method. Numerical solutions are illustrated graphically by means of graphs. The effects of magnetic field, stretching parameter and Prandtl number on velocity, skin friction, temperature distribution and rate of heat transfer are discussed.     

Magnetohydrodynamic (MHD) flow of a second grade fluid in a channel with porous wall

An analysis has been carried out to study the effect of magnetic field on an electrically conducting fluid of second grade in a parallel channel. The coolant fluid is injected into the porous channel through one side of the channel wall into the other heated impermeable wall. The combined effect of inertia, viscous, viscoelastic and magnetic forces are studied. The basic equations governing the flow and heat transfer are reduced to a set of ordinary differential equations by using appropriate transformations for velocity and temperature. Numerical solutions of these equations are obtained with the help of Runge-Kutta fourth order method in association with quasi-linear shooting technique. Numerical results for velocity field, temperature field, skin friction and Nusselt number are presented in terms of elastic parameter, Hartmann number, Prandtl number and Reynolds number. Special case of our results is in good agreement with earlier published work.     

MHD boundary layer flow and heat transfer over a stretching sheet with induced magnetic field

In this paper, the problem of steady magnetohydrodynamic boundary layer flow and heat transfer of a viscous and electrically conducting fluid over a stretching sheet is studied. The effect of the induced magnetic field is taken into account. The transformed ordinary differential equations are solved numerically using the finite-difference scheme known as the Keller-box method. Numerical results are obtained for various values of the magnetic parameter, the reciprocal magnetic Prandtl number and the Prandtl number. The effects of these parameters on the flow and heat transfer characteristics are determined and discussed in detail. When the magnetic field is absent, the closed analytical results for the skin friction are compared with the exact numerical results. Also the numerical results for the heat flux from the stretching surface are compared with the results reported by other authors when the magnetic field is absent. It is found that very good agreement exists.     

DNS of heat transfer in turbulent and transitional channel flow obstructed by rectangular prisms

Direct numerical simulation (DNS) of heat transfer in a channel flow obstructed by rectangular prisms has been performed for Re_τ = 80–20, where Re_τ is based on the friction velocity, the channel half width and the kinematic viscosity. The molecular Prandtl number is set to be 0.71. The flow remains unsteady down to Re_τ = 40 owing to the disturbance induced by the prism. For Re_τ = 30 and 20, the flow results in a steady laminar flow. In the vicinity of the prism, the three-dimensional complex vortices are generated and heat transfer is enhanced. The Reynolds number effect on the time-averaged vortex structure and the local Nusselt number are investigated. The mechanism of the heat transfer enhancement is discussed. In addition, the mean flow parameters such as the friction factor and the Nusselt number are examined in comparison with existing DNS and experimental data.     

Nonsimilar boundary-layer flow over a stationary permeable plate in a rotating fluid with a magnetic field

Summary  The nonsimilar boundary-layer flow and heat transfer over a stationary permeable surface in a rotating fluid in the presence of magnetic field, mass transfer and free stream velocity are studied. The parabolic partial differential equations governing the flow have been solved numerically by using a difference–differential method. For small streamwise distance, these partial differential equations are also solved by a perturbation technique with Shanks transformation. For uniform mass transfer, analytical solutions are obtained. The surface skin friction coefficients and the Nusselt number increase with the magnetic field, suction and streamwise distance from the leading edge of the plate except the skin friction coefficient in the y-direction which decreases with the increasing magnetic field. Received 4 December 2001; accepted for publication 24 September 2002     

Experimental investigation of heat transfer and pressure drop characteristics of flow through spirally fluted tubes

Spirally fluted tubes are used extensively in the design of tubular heat exchangers. In previous investigations, results for tubes with flute depths e/D_vi < 0.2 were reported, with most correlations applicable for Re ≥ 5000. This paper presents the results of an experimental investigation of the heat transfer and pressure drop characteristics of spirally fluted tubes with the following tube and flow parameter ranges: flute depth e/D_vi = 0.1−0.4, flute pitch p/D_vi = 0.4−7.3, helix angle θ/90° = 0.3−0.65, Re = 500−80,000, and Pr = 2−7. The heat transfer coefficients inside the fluted tube were obtained from measured values of the overall heat transfer coefficient using a nonlinear regression scheme. The friction factor data obtained consisted of 507 data points. The proposed correlation for the friction factor predicts 96% of the database within ±20%. The heat transfer correlation for the range 500 ≤ Re ≤ 5000 predicts 76% of the database (178 data points) within ±20%, and the correlation for the higher Re range predicts 97% of the 342 data points within ±20%. Comparison of heat transfer and friction data show that these tubes are most effective in the laminar and transition flow regimes. The present results show that the increase of flute depth in the range considered does not improve heat transfer.     

Effect of non-uniform heat source on MHD heat transfer in a liquid film over an unsteady stretching sheet

An analysis has been carried out to study the magnetohydrodynamic boundary layer flow and heat transfer characteristics of a laminar liquid film over a flat impermeable stretching sheet in the presence of a non-uniform heat source/sink. The basic unsteady boundary layer equations governing the flow and heat transfer are in the form of partial differential equations. These equations are converted to non-linear ordinary differential equations using similarity transformation. Numerical solutions of the resulting boundary value problem are obtained by the efficient shooting technique. The effects of magnetic and the non-uniform heat source/sink parameters on the dynamics are discussed. Findings of the paper reveal that non-uniform heat sinks are better suited for effective cooling of the stretching sheet. Skin friction coefficient and the local Nusselt number are also explored for typical values of magnetic and non-uniform heat source/sink parameters. The results are in excellent agreement with the earlier published works, under some limiting cases.     

MHD non-Darcian flow through a non-isothermal vertical surface embedded in a porous medium with radiation

The effect of MHD on steady two-dimensional laminar mixed flow about a vertical porous surface is numerically analyzed. Also the effects of radiation and heat generation and absorption are considered. A power law variation of temperature along the vertical wall is assumed. The nonlinear boundary-layer equations were transformed and the resulting differential equations were solved by an implicit finite difference scheme (Keller box method). Numerical results for the velocity distribution and the temperature distribution are presented for various values of Prandtl number Pr, magnetic parameter, porous medium parameter and internal heat generation or absorption coefficient. Further validation with previous works is carried out.