Thermal Boundary Layer Equations Research Articles

Two-dimensional turbulent thermal classical far wake

The two-dimensional turbulent thermal classical far wake is investigated. The turbulence is described by the Boussinesq hypothesis for the Reynolds stresses with Prandtl’s mixing length model for the eddy viscosity and eddy thermal conductivity. Two conservation laws are derived for the thermal boundary layer equations for the far wake using the multiplier method and two conserved quantities are obtained from the conserved vectors and boundary conditions. The Lie point symmetry associated with the momentum and thermal conserved vectors is derived. It is found that the momentum and thermal mixing lengths are proportional. In obtaining the invariant solution the cases ν≠0, κ≠0 and ν=0, κ=0 needed to be treated separately, where ν is the kinematic viscosity and κ is the thermal conductivity of the fluid. When ν≠0, κ≠0 the associated Lie point symmetry is determined in full but an analytical solution of the reduced ordinary differential equations could not be derived. The invariant solution is obtained numerically using a shooting method with the two conserved quantities as targets. The width of the wake is infinite. When ν=0, κ=0 the width of the wake is finite. In order to obtain the associated Lie point symmetry in full, Prandtl’s hypothesis that the mixing length is proportional to the width of the wake was imposed. The invariant solution is obtained analytically. The lines of constant mean temperature difference and the streamlines for the mean velocity deficit are plotted. The numerical and analytical solutions agree in the limit ν→0, κ→0.

Local well-posedness to the thermal boundary layer equations in Sobolev space

<abstract><p>In this paper, we study the local well-posedness of the thermal boundary layer equations for the two-dimensional incompressible heat conducting flow with nonslip boundary condition for the velocity and Neumann boundary condition for the temperature. Under Oleinik's monotonicity assumption, we establish the local-in-time existence and uniqueness of solutions in Sobolev space for the boundary layer equations by a new weighted energy method developed by Masmoudi and Wong.</p></abstract>

Numerical and sensitivity computations of three-dimensional flow and heat transfer of nanoliquid over a wedge using modified Buongiorno model

Numerical investigation of the three-dimensional flow and heat transfer of 36 nm Al2O3–H2O nanoliquid over a wedge surface is carried out by utilizing the modified Buongiorno model (MBM). The boundary layer approximation is assumed to be valid. The thermophysical properties of Al2O3–H2O nanoliquid are deliberated in the study by modeling them through the use of correlations based on experimental data. The thermal boundary layer equation comprises the Brownian motion and thermo-migration effects caused by nanoparticles. Zero mass flux boundary condition is also accounted. Optimization of the heat transfer rate of nanoliquid is made using the Response surface methodology (RSM). Two stream functions are used to derive the similarity transformations and they are employed to arrive nonlinear ordinary differential system from the governing partial differential system, and then the subsequent nonlinear problem is treated numerically using the Finite Element Method (FEM). The heat flow features are scrutinized using two-dimensional, surface, and streamline plots. The results of flow due to a moving wedge in the same direction of free stream velocity are compared with those of a moving wedge in the opposite direction of free stream velocity. The thermal layer and nanoparticles volume fraction layers are enlarged due to the chaotic movement of nanoparticles. The thermophoresis mechanism improves the thermal layer at the wedge surface. The high level of pressure gradient factor, high level of shear-to-strain rate and low level of Lewis number, were found to be the optimal operating condition that would maximize the heat transfer rate.

Well‐posedness of thermal boundary layer equation in two‐dimensional incompressible heat conducting flow with analytic datum

Well‐posedness of thermal boundary layer equation in two‐dimensional incompressible heat conducting flow with analytic datum

Two-Phase Microscopic Heat Transfer Model for Three-Dimensional Stagnation Boundary-Layer Flow in a Porous Medium

Abstract This work examines the steady three-dimensional forced convective thermal boundary-layer flow of laminar and incompressible fluid in a porous medium. In this analysis, it is assumed that the solid phase and the fluid phase, which is immersed in a porous medium are subjected to local thermal nonequilibrium (LTNE) conditions, which essentially leads to one thermal boundary-layer equation for each phase. Suitable similarity transformations are introduced to reduce the boundary-layer equations into system of nonlinear ordinary differential equations, which are analyzed numerically using an implicit finite difference-based Keller-box method. The numerical results are further confirmed by the asymptotic solution of the same system for large three-dimensionality parameter, and the corresponding results agree well. Our results show that the thickness of boundary layer is always thinner for all permeability parameters tested when compared to the nonporous case. Also, it is noticed that the temperature of solid phase is found to be higher than the corresponding fluid phase for any set of parameters. There is a visible temperature difference in the two phases when the microscopic interphase rate is quite large. The physical hydrodynamics to these parameters is studied in some detail.

Solution of Boundary Layer and Thermal Boundary Layer Equation

Solution of Boundary Layer and Thermal Boundary Layer Equation

Entropy Generation Minimization in MHD Boundary Layer Flow over a Slendering Stretching Sheet in the Presence of Frictional and Joule Heating

In the present paper, we study the entropy analysis of boundary layer flow over a slender stretching sheet under the action of a non uniform magnetic field that is acting perpendicular to the flow direction. The effects of viscous dissipation and Joule heating are included in the energy equation. Using similarity transformation technique the momentum and thermal boundary layer equations to a system of nonlinear differential equations. Numerical solutions are obtained using the shooting and fourth-order Runge-Kutta method. The expressions for the entropy generation number and Bejan number are also obtained using a suggested similarity transformation. The main objective of this article is to investigate the effects of different governing parameters such as the magnetic parameter (M2), Prandtl number (Pr), Eckert number (Ec), velocity index parameter (m), wall thickness parameter (α), temperature difference parameter (Ω), entropy generation number (Ns) and Bejan number (Be). All these effects are portrayed graphically and discussed in detail. The analysis reveals that entropy generation reduces with decreasing wall thickness parameter and increasing temperature difference between the stretching sheet and the fluid outside the boundary layer. The viscous and magnetic irreversibilities are dominant in the vicinity of the stretching surface.

Effect of thermal radiation on boundary layer flow and heat transfer of dusty fluid over an unsteady stretching sheet

The aim of this paper is to study the boundary layer flow and heat transfer analysis of an unsteady viscous dusty fluid over a porous stretching surface. Momentum Boundary layer equation considers the effect of transverse magnetic field whereas thermal Boundary layer equation considers the effect of thermal radiation. The governing partial Differential equations are reduced to coupled non-linear ordinary differential equations using similarity transformation. Numerical solutions of these equations are obtained with the help of RKF-45 method. The solution is found to be dependent on different governing parameters. Some important findings reported in this work reveal that the effect of radiation has significant impact in controlling the rate of heat transfer in the boundary layer region.Keywords: Unsteady flow and heat transfer; Boundary layer flow; Stretching porous Surface; Dusty fluid; Thermal radiation; Numerical solution

Mean temperature profiles in turbulent thermal convection

To predict the mean temperature profiles in turbulent thermal convection, the thermal boundary layer (BL) equation including the effects of fluctuations has to be solved. In Shishkina et al., Phys. Rev. Lett. 114 (2015), the thermal BL equation with the fluctuations taken into account as an eddy thermal diffusivity has been solved for large Prandtl-number fluids for which the eddy thermal diffusivity and the velocity field can be approximated respectively as a cubic and a linear function of the distance from the plate. In the present work we make use of the idea of Prandtl's mixing length model and relate the eddy thermal diffusivity to the stream function. With this proposed relation, we can solve the thermal BL equation and obtain a closed-form expression for the dimensionless mean temperature profile in terms of two independent parameters for fluids with a general Prandtl number. With a proper choice of the parameters, our predictions of the temperature profiles are in excellent agreement with the results of our direct numerical simulations for a wide range of Prandtl numbers from 0.01 to 2547.9 and Rayleigh numbers from 10^7 to 10^9.

Effects of Radiation and Viscous Dissipation on Transient Free Convection Heat Transfer Flow Past a Hot Vertical Surface in Porous Media

The radiation effects on unsteady transient free convection flow of a viscous incompressible gray, absorbing-emitting but non-scattering, optically-thick fluid occupying a semi-infinite porous regime adjacent to an infinite moving hot vertical plate with constant velocity taking viscous dissipation onto account has been carried out. Thermal radiation effects are simulated via a radiation-conduction parameter (),rK based on the Rossseland diffusion approximation. We employ a Darcian viscous flow model for the porous medium. The momentum and thermal boundary layer equations are non-dimensionalzed using appropriate transformations and then solved subject to physically realistic boundary conditions using the Ritz finite element method. The computed numerical results for velocity(),u temperature(),θ shear stress function()τ and wall temperature gradient function()Nu are presented through the graphs and tables for air (0.71)rP= and water (7.00).rP= It has been found that increasing thermal radiation parameter ()rKcauses a considerable increase in the flow velocity .u Temperature θ is significantly increased within the boundary layer with a rise in.rK The velocity is found to decrease with an increase in inverse permeability parameter ()pK and increases with increase in the Grashof number ()rG and Eckert number ().

Self-similar analysis of fluid flow, heat, and mass transfer at orthogonal nanofluid impingement onto a flat surface

Momentum, heat, and mass transfer in the vicinity of a stagnation point at uniform impingement of a nanofluid onto a flat plate were investigated. The novelty of the work consists in obtaining self-similar forms for the Hiemenz flow of a nanofluid and the self-similar representation of the velocity, thermal, and diffusion boundary layer equations derived on the basis of symmetry analysis using discrete symmetries. Momentum, energy, and concentration equations in the self-similar form were solved numerically. In frames of this analysis, functional dependence of the physical properties of nanofluids (viscosity, thermal conductivity, and diffusion coefficient) on concentration and temperature profiles was included as a part of the mathematical model, whose form enables including different models for the thermophysical properties of the nanofluid. Also novel are numerical results that revealed the influence of the nanoparticle concentration on the velocity, temperature, and concentration profiles, as well as on the normalized Nusselt numbers and surface friction coefficients illustrated in the form of analytical relations and graphically. The focus was put not only on modeling of heating of a colder wall by a hotter nanofluid but also on cooling of a hotter wall by a colder nanofluid. For the latter case, theoretical results were validated against experimental data available in the literature. Effects of various dimensionless parameters on the Nusselt number and surface friction coefficient were elucidated.

Convective Poiseuille flow of Al2O3-EG nanofluid in a porous wavy channel with thermal radiation

In current article, convective Poiseuille boundary layer flow of ethylene glycol (C2H6O2)-based nanofluid with suspended aluminum oxide (Al2O3) nanoparticles through a porous wavy channel has been examined. The impact of thermal radiation, Ohmic dissipation, electric field, and magnetic fields are also considered. The flow is due to constant pressure gradient in a wavy frame of reference. The governed momentum and thermal boundary layer equations is system of PDE’s, which are converted to system of ODE’s via suitable similarity transformations. The homotopy analysis method is applied to solve the governed flow problem. Convergence of series solutions is inspected through h-curves and residual errors norm, whereas the optimal value of convergence control parameter is obtained by means of genetic algorithm Nelder–Mead approach. The influence of numerous involving parameters like Hartmann number, Grashof number, Eckert number, electric parameter, radiation parameter, and porosity parameter on flow, heat transfer, skin friction coefficient and Nusselt number are illustrated through graphs and discussed briefly.

Numerical study of heat source/sink effects on dissipative magnetic nanofluid flow from a non-linear inclined stretching/shrinking sheet

This paper numerically investigates radiative magnetohydrodynamic mixed convection boundary layer flow of nanofluids over a nonlinear inclined stretching/shrinking sheet in the presence of heat source/sink and viscous dissipation. The Rosseland approximation is adopted for thermal radiation effects and the Maxwell-Garnetts and Brinkman models are used for the effective thermal conductivity and dynamic viscosity of the nanofluids respectively. The governing coupled nonlinear momentum and thermal boundary layer equations are rendered into a system of ordinary differential equations via local similarity transformations with appropriate boundary conditions. The non-dimensional, nonlinear, well-posed boundary value problem is then solved with the Keller box implicit finite difference scheme. The emerging thermo-physical dimensionless parameters governing the flow are the magnetic field parameter, volume fraction parameter, power-law stretching parameter, Richardson number, suction/injection parameter, Eckert number and heat source/sink parameter. A detailed study of the influence of these parameters on velocity and temperature distributions is conducted. Additionally the evolution of skin friction coefficient and Nusselt number values with selected parameters is presented. Verification of numerical solutions is achieved via benchmarking with some limiting cases documented in previously reported results, and generally very good correlation is demonstrated. This investigation is relevant to fabrication of magnetic nanomaterials and high temperature treatment of magnetic nano-polymers.

Influence of hydrodynamic slip on convective transport in flow past a circular cylinder

The presence of a finite tangential velocity on a hydrodynamically slipping surface is known to reduce vorticity production in bluff body flows substantially while at the same time enhancing its convection downstream and into the wake. Here, we investigate the effect of hydrodynamic slippage on the convective heat transfer (scalar transport) from a heated isothermal circular cylinder placed in a uniform cross-flow of an incompressible fluid through analytical and simulation techniques. At low Reynolds (Re > 1) numbers, our theoretical analysis based on Oseen and thermal boundary layer equations allows for an explicit determination of the dependence of the thermal transport on the non-dimensional slip length l(s). In this case, the surface-averaged Nusselt number, Nu transitions gradually between the asymptotic limits of Nu similar to Pe(1/3) and Nu similar to Pe(1/2) for no-slip (l(s) -> 0) and shear-free (l(s) -> infinity) boundaries, respectively. Boundary layer analysis also shows that the scaling Nu similar to Pe(1/2) holds for a shear-free cylinder surface in the asymptotic limit of Re >> 1 so that the corresponding heat transfer rate becomes independent of the fluid viscosity. At finite Re, results from our two-dimensional simulations confirm the scaling Nu similar to Pe(1/2) for a shear-free boundary over the range 0.1 <= Re <= 10(3) and 0.1 <= Pr <= 10. A gradual transition from the lower asymptotic limit corresponding to a no-slip surface, to the upper limit for a shear-free boundary, with l(s), is observed in both the maximum slip velocity and the Nu. The local time-averaged Nusselt number Nu(theta) for a shear-free surface exceeds the one for a no-slip surface all along the cylinder boundary except over the downstream portion where unsteady separation and flow reversal lead to an appreciable rise in the local heat transfer rates, especially at high Re and Pr. At a Reynolds number of 10(3), the formation of secondary recirculating eddy pairs results in appearance of additional local maxima in Nu(theta) at locations that are in close proximity to the mean secondary stagnation points. As a consequence, Nu exhibits a non-monotonic variationwith l(s) increasing initially from its lowermost value for a no-slip surface and then decreasing before rising gradually toward the upper asymptotic limit for a shear-free cylinder. A non-monotonic dependence of the spanwise-averaged Nu on l(s) is observed in three dimensions as well with the three-dimensional wake instabilities that appear at sufficiently low l(s), strongly influencing the convective thermal transport from the cylinder. The analogy between heat transfer and single-component mass transfer implies that our results can directly be applied to determine the dependency of convective mass transfer of a single solute on hydrodynamic slip length in similar configurations through straightforward replacement of Nu and Pr with Sherwood and Schmidt numbers, respectively.

On heat transfer analysis of axisymmetric flow of viscous fluid over a nonlinear radially stretching sheet

This article deals with the exact solutions regarding heat transfer of viscous fluid over a non-linear radially stretching sheet. The effects of viscous dissipation on the heat transfer characteristics are considered in account in the presence of a transverse magnetic field for two types of boundary heating process namely prescribed power law surface temperature (PST) and prescribed heat flux (PHF). Similarity transformations are used to reduce the governing non-linear momentum and thermal boundary layer equations into a set of ordinary differential equations. The exact solutions of the reduced ordinary differential equations are developed in the form of confluent hypergeometric function and graphically sketched to see the influence of pertinent parameters on the temperature profiles. In addition the results for the wall temperature gradient (Nusselt number) are also presented in graphical form and discussed in detail. It is found that Prandtl number decelerates the temperature profiles in both PST and PHF cases. Thermal boundary layer thickness increases by increasing the Eckert number.

Slip flow and variable properties of viscoelastic fluid past a stretching surface embedded in a porous medium with heat generation

This study examines theoretically and computationally the non-Newtonian boundary layer flow and heat transfer for a viscoelastic fluid over a stretching continuous sheet embedded in a porous medium with variable fluid properties, slip velocity, and internal heat generation/absorption. The flow in boundary layer is considered to be generated solely by the stretching of the sheet adjacent to porous medium with boundary wall slip condition. Highly nonlinear momentum and thermal boundary layer equations governing the flow and heat transfer are reduced to set of nonlinear ordinary differential equations by appropriate transformation. The resulting ODEs are successfully solved numerically with the help of shooting method. Graphical results are shown for non-dimensional velocities and temperature. The effects of heat generation/absorption parameter, the porous parameter, the viscoelastic parameter, velocity slip parameter, variable thermal conductivity and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction coefficient and Nusselt number are presented. Comparison of numerical results is made with the earlier published results under limiting cases.

Thermal analysis of anti-icing systems in aeronautical velocity sensors and structures

This work reviews theoretical–experimental studies undertaken at COPPE/UFRJ on conjugated heat transfer problems associated with the transient thermal behavior of heated aeronautical Pitot tubes and wing sections with anti-icing systems. One of the main objectives is to demonstrate the importance of accounting for the conduction–convection conjugation in more complex models that attempt to predict the thermal behavior of the anti-icing system under adverse atmospheric conditions. The experimental analysis includes flight tests validation of a Pitot tube thermal behavior with the military aircraft A4 Skyhawk (Brazilian Navy) and wind tunnel runs (INMETRO and NIDF/COPPE/UFRJ, both in Brazil), including the measurement of spatial and temporal variations of surface temperatures along the probe through infrared thermography. The theoretical analysis first involves the proposition of an improved lumped-differential model for heat conduction along a Pitot probe, approximating the radial temperature gradients within the metallic and ceramic (electrical insulator) walls. The convective heat transfer problem in the external fluid is solved using the boundary layer equations for compressible flow, applying the Illingsworth variables transformation considering a locally similar flow. The nonlinear partial differential equations are solved using the Generalized Integral Transform Technique in the Mathematica platform. In addition, a fully local differential conjugated problem model was proposed, including both the dynamic and thermal boundary layer equations for laminar, transitional, and turbulent flow, coupled to the heat conduction equation at the sensor or wing section walls. With the aid of a single-domain reformulation of the problem, which is rewritten as one set of equations for the whole spatial domain, through space variable physical properties and coefficients, the GITT is again invoked to provide hybrid numerical–analytical solutions to the velocity and temperature fields within both the fluid and solid regions. Then, a modified Messinger model is adopted to predict ice formation on either wing sections or Pitot tubes, which allows for critical comparisons between the simulation and the actual thermal response of the sensor or structure. Finally, an inverse heat transfer problem is formulated aimed at estimating the heat transfer coefficient at the leading edge of Pitot tubes, in order to detect ice accretion, and estimating the relative air speed in the lack of a reliable dynamic pressure reading. Due to the intrinsic dynamical behavior of the present inverse problem, it is solved within the Bayesian framework by using particle filter.

A note on convective heat transfer of an MHD Jeffrey fluid over a stretching sheet

This article focuses on the exact solution regarding convective heat transfer of a magnetohydrodynamic (MHD) Jeffrey fluid over a stretching sheet. The effects of joule and viscous dissipation, internal heat source/sink and thermal radiation on the heat transfer characteristics are taken in account in the presence of a transverse magnetic field for two types of boundary heating process namely prescribed power law surface temperature (PST) and prescribed heat flux (PHF). Similarity transformations are used to reduce the governing non-linear momentum and thermal boundary layer equations into a set of ordinary differential equations. The exact solutions of the reduced ordinary differential equations are developed in the form of confluent hypergeometric function. The influence of the pertinent parameters on the temperature profile is examined. In addition the results for the wall temperature gradient are also discussed in detail.

Melting heat transfer on MHD convective flow of a nanofluid over a stretching sheet with viscous dissipation and second order slip

In this study, we investigate the effects of viscous dissipation and second order slip on MHD boundary layer flow of an incompressible, electrically conducting water-based nanofluid over a stretching sheet. The governing momentum boundary layer and thermal boundary layer equations with the boundary conditions are transformed into a system of nonlinear ordinary differential equations which are then solved numerically by using the Runge–Kutta–Fehlberg method. The effects of the flow parameters on the velocity, temperature, nanoparticle concentration, shearing stress, rate of heat transfer, and rate of mass transfer are analyzed, and illustrations are provided by the inclusion of figures and tables for various values of different parameters. We determine that the skin friction increases in magnitude, whereas the rate of heat transfer and rate of mass transfer decrease in magnitude as the strength of the magnetic field increases. In addition, the magnitudes of skin friction, rate of heat transfer, and rate of mass transfer decrease as the melting heat transfer and first-order slip parameter both increase.

Thermal boundary layer equation for turbulent Rayleigh-Bénard convection.

We report a new thermal boundary layer equation for turbulent Rayleigh-Bénard convection for Prandtl number Pr>1 that takes into account the effect of turbulent fluctuations. These fluctuations are neglected in existing equations, which are based on steady-state and laminar assumptions. Using this new equation, we derive analytically the mean temperature profiles in two limits: (a) Pr≳1 and (b) Pr≫1. These two theoretical predictions are in excellent agreement with the results of our direct numerical simulations for Pr=4.38 (water) and Pr=2547.9 (glycerol), respectively.