Prescribed Wall Heat Flux Research Articles

Flow due to a porous stretching/shrinking sheet with thermal radiation and mass transpiration

AbstractThis study investigates the behavior of carbon nanotubes (CNT) approaching an unsteady flow of a Newtonian fluid over a stagnation point on a stretching surface employing porous media. It flows when the liquid begins to move with the progression of time. Heat exchange with the environment has an impact on the flow. The implicitly limited component technique is used to solve the nondimensional partial differential equation with an associated boundary layer, which is an unstable system. Analytically, the solutions, as well as the required boundary conditions, are obtained. The effects of mass transpiration, volume fraction, and heat radiation on Newtonian fluid flow through porous media are explored. Single‐ and multi‐walled CNTs are used as well as water, as base fluids in the experiment. The impact of thermal radiation and heat source/sink is shown in the energy equation, which is solved under four different cases: uniform heat flux case, constant wall temperature case, general power‐law wall heat flux case, and general power‐law wall temperature case. By supplying distinct physical characteristics, a theoretical analysis of the existence and nonexistence of unique and dual solutions may be explored. These physical parameters determine the velocity distribution and temperature distribution. Prescribed surface temperature (PST) and prescribed wall heat flux (PHF) heat transfer solutions can be written using confluent hypergeometric equations, and generic power‐law PST and PHF situations can also be expressed using confluent hypergeometric equations. The graphical representations assist in the discussion of the current study's findings.

Mathematical modelling of classical Graetz–Nusselt problem for axisymmetric tube and flat channel using the Carreau fluid model: a numerical benchmark study

Abstract The thermal entrance problem (also known as the classical Graetz problem) is studied for the complex rheological Carreau fluid model. The solution of two-dimensional energy equation in the form of an infinite series is obtained by employing the separation of variables method. The ensuing eigenvalue problem (S–L problem) is solved for eigenvalues and corresponding eigenfunctions through MATLAB routine bvp5c. Numerical integration via Simpson’s rule is carried out to compute the coefficient of series solution. Current problem is also tackled by an alternative approach where numerical solution of eigenvalue problem is evaluated via the Runge–Kutta fourth order method. This problem is solved for both flat and circular confinements with two types of boundary conditions: (i) constant wall temperature and (ii) prescribed wall heat flux. The obtained results of both local and mean Nusselt numbers, fully developed temperature profile and average temperature are discussed for different values of Weissenberg number and power-law index through graphs and tables. This study is valid for typical range of Weissenberg number W e ≤ 1 $\left(We\le 1\right)$ and power-law index n < 1 $\left(n{< }1\right)$ for shear-thinning trend while n > 1 $\left(n{ >}1\right)$ for shear-thickening behaviour. The scope of the present study is broad in the context that the solution of the said problem is achieved by using two different approaches namely, the traditional Graetz approach and the solution procedure documented in M. D. Mikhailov and M. N. Ozisik, Unified Analysis and Solutions of Heat and Mass Diffusion, New York, Dover, 1994.

GASEOUS SLIP FLOW FORCED CONVECTION IN A MICROPIPE AND PARALLEL-PLATE MICROCHANNEL WITH A PIECEWISE UNIFORM WALL HEAT FLUX AND AXIAL HEAT CONDUCTION

A forced convective heat transfer problem inside a micropipe as well as inside a microchannel with parallel walls in the laminar gaseous slip flow regime, a with finite heating region and a prescribed wall heat flux, including the axial heat conduction effect, is analytically investigated. The temperature field and Nusselt number are derived under the assumption that the flow is hydrodynamically fully developed in the finite-length heating region. The solution is found by applying the functional analysis method, by decomposing the energy equation into a pair of first-order partial differential equations. The first-order slip boundary conditions are imposed at the gas–wall interface. The analytical solution is compared with available calculations. The results show that the thermal characteristics in the heating region are significantly affected by the axial heat conduction, rarefaction effects, and finite length of the heating region.

Significance of Rarefaction, Streamwise Conduction, and Viscous Dissipation on the Extended Graetz–Nusselt Problem: The Case of Finite-Length Microchannels with Prescribed Wall Heat Flux

The article addresses the extended Graetz–Nusselt problem in finite-length microchannels for prescribed wall heat flux boundary conditions, including the effects of rarefaction, streamwise conduction, and viscous dissipation. The analytical solution proposed, valid for low-intermediate Peclet values, takes into account the presence of the thermal development region. The influence of all transport parameters (PecletPe, KnudsenKn, and BrinkmanBr) and geometrical parameters (entry length and microchannel aspect ratio) is investigated. Performances of different wall heat flux functions have been analyzed in terms of the averaged Nusselt number. In the absence of viscous dissipationBr=0, the best heating protocol is a decreasing wall heat flux function. In the presence of dissipationBr>0, the best heating protocol is a uniform wall heat flux.

Semi-Analytical Solution of the Graetz Problem With Uniform Wall Heat Flux Utilizing the Transversal Method of Lines

Abstract This study addresses the second Graetz problem with prescribed wall heat flux employing the transversal method of lines (TMOL), which deviates significantly from the traditional mathematical procedures employed in the past. The wall heat flux is customarily provided by electrical, radiative or solar heating in engineering applications. The TMOL transforms the governing two-dimensional energy equation with temperature-invariant thermo-physical properties into a sequence of adjoint ordinary differential equations of second order with the radial variable as the independent variable. The singular feature in those equations is the embedded axial variable interval. For the implementation of TMOL, a special computational domain consists in a condensed set of transversal lines displayed in the cross section of the tube. An approximate, semi-analytical temperature distribution is obtained with the solution of the first adjoint ordinary differential equation of second order, which is expressed in terms of the Kummer function of first kind M(a,b,c). From here, the approximate, semi-analytical wall and center temperature distributions exhibit excellent quality because the two compare favorably with the exact, analytical wall and center temperature distributions given by the classical Graetz infinite series. As a beneficial consequence, usage of the second adjoint ordinary differential equation of second order having more complex structure becomes unnecessary.

Momentum and heat transfer of a special case of the unsteady stagnation-point flow

This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer, which is also an exact solution to the unsteady Navier-Stokes (NS) equations. The boundary layer energy equation is solved with the closed form solutions for prescribed wall temperature and prescribed wall heat flux conditions. The wall temperature and heat flux have power dependence on both time and spatial distance. The solution domain, the velocity distribution, the flow field, and the temperature distribution in the fluids are studied for different controlling parameters. These parameters include the Prandtl number, the mass transfer parameter at the wall, the wall moving parameter, the time power index, and the spatial power index. It is found that two solution branches exist for certain combinations of the controlling parameters for the flow and heat transfer problems. The heat transfer solutions are given by the confluent hypergeometric function of the first kind, which can be simplified into the incomplete gamma functions for special conditions. The wall heat flux and temperature profiles show very complicated variation behaviors. The wall heat flux can have multiple poles under certain given controlling parameters, and the temperature can have significant oscillations with overshoot and negative values in the boundary layers. The relationship between the number of poles in the wall heat flux and the number of zero-crossing points is identified. The difference in the results of the prescribed wall temperature case and the prescribed wall heat flux case is analyzed. Results given in this paper provide a rare closed form analytical solution to the entire unsteady NS equations, which can be used as a benchmark problem for numerical code validation.

On an equation arising in natural convection boundary layer flow in a porous medium

A generalization of a boundary value problem treated previously by Magyari et al. (ZAMP 53:782–793, 2002a; Transp Porous Medium 46:91–102, 2002b), Zhang (Math Anal Appl 417:361–375, 2014) and Paullet (Appl Math E Notes 14:123–126, 2014) for a prescribed wall temperature and by Zhang (Appl Math Comput 339:367–373, 2018) for a prescribed wall heat flux is considered. The problem involves a parameter lambda and an exponent p. Numerical solutions are obtained for representative values of lambda and p, a feature of which is the existence of critical values lambda _c of lambda with two solution branches in lambda >lambda _c, with lambda _c dependent on p. Asymptotic solutions for large lambda and large p are derived for both types of boundary condition. For large lambda , the nature of the solution is essentially different on upper and on the lower branches with similar feature being seen in the behaviour for p large.

Effect of Chemical Reaction on Mixed Convective Nanofluid Flow on a Vertical Plate with Uniform Heat and Mass Fluxes

The purpose of this paper is to consider a two dimensional free convective flow of a nanofluid due to the combined effects of thermal and mass diffusion in the presence of a chemical reaction of first order. The objective of the present investigation is to analyze the free convective flow in the presence of prescribed wall heat flux and mass flux condition. The governing equations of the linear momentum, energy equation and concentration are obtained in a dimensionless form by introducing a suitable group of similarity transformations. The transformed coupled non-linear ordinary differential equations are solved numerically by using appropriate boundary conditions for the various values of physical parameters. Computations are performed for a wide range of values of the various governing flow parameters of the velocity, temperature and species concentration profiles and results are presented graphically. Numerical results for the skin friction coefficient and local Nusselt number are also presented and analyzed in detail. The obtained results are compared with previously published work and are found to be in excellent agreement. The results are a very useful source of information for researchers on the subject of a free convective flow of a nanofluid. This paper illustrates chemical reaction effects on free convective flow of a nanofluid from a vertical plate with uniform heat and mass fluxes.

Thermodynamic analysis of viscoelastic fluid in a porous meduim with prescribed wall heat flux over stretching sheet subjected to a transitive magnetic field

An analysis is performed for entropy generation in a steady laminar boundary-layer flow of an electrically conducting second grade fluid in a porous medium prescribed wall heat flux subject to a transverse uniform magnetic field past a semi-infinite stretching sheet, The effects of viscous dissipation, internal heat generation of absorption due to deformation are considered in the energy equation. Kummer?s functions are used to obtain temperature field. The velocity, temperature are used to compute the entropy generation number with a change in various dimensionless parameters.

Flow and Heat Transfer to a Newtonian Fluid Over Non-linear Extrusion Stretching Sheet

The paper presents flow and energy transfer to a Newtonian fluid over non-linear extrusion stretching sheet problem involving a Newtonian fluid and suction (injection). Two different temperature conditions are studied, namely (i) prescribed surface temperature and (ii) prescribed wall heat flux, while considering heat transfer. Similarity transformation is used on the governing equations to arrive at a system of non-linear ordinary differential equations. A new analytical approach is then followed for solving the equation. Analytical expression is obtained for stream function and temperature in terms of the stretching parameters. The result in respect of the stretching sheet problem without the boundary layer approximation is obtained using those obtained with the boundary layer approximation. An asymptotic analysis is presented for large and small Prandtl numbers.

Exact analysis of the transient free convection flow of nanofluids between two vertical parallel plates in the presence of radiation

The transient free convection flow of nanofluids between two long vertical parallel plates has been investigated analytically in the presence of thermal radiation by considering prescribed wall temperature (PWT) and prescribed wall heat flux (PWHF) at one boundary while the other boundary is maintained at the initial fluid temperature. The exact analytical solutions for the nanofluid velocity, temperature, skin friction, and Nusselt number are derived in the form of rapidly converging series with the help of Laplace transform technique. Five different types of water‐based nanofluids containing copper (Cu), silver (Ag), copper oxide (CuO), titanium oxide (TiO2), and aluminium oxide (Al2O3) are considered in the analysis. The effects of nanoparticle volume fraction, radiation parameter, and temporal variable on the velocity, temperature, skin friction, Nusselt number, volume flow rate, and vertical heat flux have been discussed in detail. Some new heat and fluid flow characteristics of nanofluids have been presented. The present results can be used as a benchmark to validate the numerical solutions of transient free convection flow of nanofluids in a vertical channel for limiting cases. Also, the results are useful in gaining a deeper insight into the relevant practical systems.

Analysis of heat and mass transfer with MHD and chemical reaction effects on viscoelastic fluid over a stretching sheet

This article addresses the mass and heat transfer analysis over an electrically conducting viscoelastic (Walters B′) fluid over a stretching surface in presence of transverse magnetic field. The impact of chemical reaction, as well as non-uniform heat source, are also taken into account. Similarity transformations are employed to model the equations. The governing equations comprises of momentum, energy, and concentration which are modified to a set of non-linear differential equations and then solved by applying confluent hypergeometric function known as “Kummer’s function”. The exact solution for heat equation is obtained for two cases i.e. (1) Prescribed surface temperature, (2) Prescribed wall heat flux. Physical behavior of all the sundry parameters are against concentration, temperature, and velocity profile are presented through graphs. The inclusion of magnetic field is counterproductive in diminishing the velocity distribution whereas reverse effect is encountered for concentration and temperature profiles.

MHD viscoelastic fluid flow through porous medium over a stretching sheet in the presence of non-uniform heat source/sink

The boundary layer flow, heat and mass transfer of an electrically conducting viscoelastic fluid over a stretching sheet embedded in a porous medium has been studied. The effect of transverse magnetic field, non-uniform heat source and chemical reaction on the flow has been analyzed. The Darcy linear model has been applied to account for the permeability of the porous medium. The method of solution involves similarity transformation. The confluent hypergeometric function (Kummer’s function) has been applied to solve the governing equations. Two aspects of heat equation namely, (1) prescribed surface ure and (2) prescribed wall heat flux are considered. The study reveals that the loss of momentum transfer in the main direction of flow is compensated by increasing in transverse direction vis-a-vis the corresponding velocity components due to magnetic force density. The application of magnetic field of higher density produces low solutal concentration and a hike in surface temperature.

Mixed convection boundary layer flow past vertical flat plate in nanofluid: case of prescribed wall heat flux

An analysis is carried out to investigate the steady mixed convection boundary layer flow of a water based nanofluid past a vertical semi-infinite flat plate. Using an appropriate similarity transformation, the governing partial differential equations are transformed into the coupled, nonlinear ordinary (similar) differential equations, which are then solved numerically for the Prandtl number Pr = 6.2. The skin friction coefficient, the local Nusselt number, and the velocity and temperature profiles are presented graphically and discussed. Effects of the solid volume fraction ϕ and the mixed convection parameter λ on the fluid flow and heat transfer characteristics are thoroughly examined. Different from an assisting flow, it is found that the solutions for an opposing flow are non-unique. In order to establish which solution branch is stable and physically realizable in practice, a stability analysis is performed.

Unsteady Heat and Mass Transfer Near the Stagnation-point on a Vertical Permeable Surface: a Comprehensive Report of Dual Solutions

In this paper, the problem of unsteady mixed convection boundary layer flow of a viscous incompressible fluid near the stagnation-point on a vertical permeable plate with both cases of prescribed wall temperature and prescribed wall heat flux is investigated numerically. Here, both assisting and opposing buoyancy forces are considered and studied. The non-linear coupled partial differential equations governing the flow, thermal and concentration fields are first transformed into a set of non-linear coupled ordinary differential equations by a set of suitable similarity transformations. The resulting system of coupled non-linear ordinary differential equations is solved numerically by using the Runge–Kutta scheme coupled with a conventional shooting procedure. Numerical results are obtained for the skin-friction coefficient, Nusselt number and Sherwood number as well as for the velocity, temperature and concentration profiles for some values of the governing parameters, namely, the unsteadiness parameter A, permeability parameter f0 and mixed convection parameter λ. It is found that dual solutions exist for both assisting and opposing flows, and the range of the mixed convection parameter for which the solution exists, increases with suction and unsteadiness parameters.

Suspended Particle Effect on Nanofluid Boundary Layer Flow Past a Stretching Surface

This article presents a numerical investigation on effect of suspended particles on boundary layer flow and heat transfer characteristics of a viscous nanofluid immersed in a porous medium over a stretching sheet. Copper Cu as nanoparticle and water with suspended particles as its base fluid have been considered. Two different temperature conditions are studied, namely (i) the sheet with prescribed surface temperature (PST) and (ii) the sheet with prescribed wall heat flux (PHF). Similarity transformations are used to reduce the governing nonlinear boundary layer equations to coupled higher order non-linear ordinary differential equations. These equations were numerically solved using Runge–Kutta–Fehlberg fourth-fifth order method. The effect of governing physical parameters on velocity and temperature profiles for the both nanofluid phase and dust phase are computed, analyzed and discussed. Besides, the local skin friction coefficient and local Nusselt number are computed and presented in tabular form. It is found that permeability parameter and volume fraction of nanoparticle has a significant influence on increasing rate of heat transfer. And momentum and thermal boundary layer are thinner due to the influence of suspended particles. Comparisons with previously published work are performed and excellent agreement is obtained.

Measurement of local heat transfer coefficient on the endwall of a turbine blade cascade by liquid crystal thermography

This paper presents convective heat transfer measurements on the endwall of a turbine blade cascade by means of the liquid crystal thermography. The measurement method is based on the heating, at uniform heat flux, of the endwall by means of an electronic circuit board. The heated region of the endwall surface exposed to the airflow is covered with a thermochromic liquid crystal coating. The colour map of the coating during a given test, featured by an assigned mass flow rate, is measured by using an image processing system. With a prescribed wall heat flux, the convective heat transfer coefficient is obtained, at every pixel location, from the measured temperature maps recorded at the steady-state. Heat transfer results, presented for the endwall of a high-pressure turbine cascade, have been obtained for different values of the Reynolds number. Regions of the endwall surface associated with high rates of heat transfer are well identified: the leading edge of blades, pressure and suction side corners, the blade wake and cascade throat. The spatially-resolved heat transfer measurements presented here could provide a useful baseline for a condition-specific, optimized endwall film-cooling solution that reduces overall heat transfer rates.

Effects of nanoparticle migration on force convection of alumina/water nanofluid in a cooled parallel-plate channel

Force convective heat transfer of alumina/water nanofluid inside a cooled parallel-plate channel in the creeping flow regime and the presence of heat generation is investigated theoretically. A modified two-component four-equation non-homogeneous equilibrium model is employed for the alumina/water nanofluid that fully accounts for the effects of nanoparticles volume fraction distribution. To impose the temperature gradients across the channel, the upper wall is subjected to a prescribed wall heat flux while the bottom wall is kept adiabatic. Moreover, due to the nanoparticle migration in the fluid, the no-slip condition of the fluid–solid interface at the walls is abandoned in favor of a slip condition that appropriately represents the non-equilibrium region near the interface. The results indicated that nanoparticles move from the adiabatic wall (nanoparticles depletion) toward the cold wall (nanoparticles accumulation) and construct a non-uniform nanoparticle distribution. Moreover, the anomalous heat transfer rate occurs when the Brownian motion takes control of the nanoparticle migration (smaller nanoparticles).

MHD mixed convection flow of a viscoelastic fluid over an inclined surface with a nonuniform heat source/sink

The steady mixed convection boundary layer flow over an inclined stretching surface immersed in an incompressible viscoelastic fluid is considered in this paper. Employing suitable similarity transformations, the governing partial differential equations are transformed into ordinary differential equations, and the transformed equations are solved numerically using Runge–Kutta–Fehlberg method. Herein, two different types of heating processes are considered, namely, (i) prescribed surface temperature and (ii) prescribed wall heat flux. The effects of the governing parameters on the flow field and heat transfer characteristics are obtained and discussed. It is found that velocity decreases and temperature increases with an increase in the value of angle of inclination.

Free convective flow from a heated vertical wall immersed in a thermally stratified environment

Free convective flow along a heated vertical wall immersed in a thermally stratified environment is studied using Lie group theory. The symmetries admitted by the governing system of boundary layer equations are identified along with the admitted classes of temperature stratifications. The cases of both prescribed wall temperature and prescribed wall heat flux are considered. The resulting similarity variables, the reduced systems of equations and their corresponding solutions, are presented for each class of temperature stratification. Specific solutions illustrating the different classes are provided.