The Graetz–Brinkman problem in a plane-parallel channel with adiabatic-to-isothermal entrance

An analysis of laminar hydrodynamically developed forced convection in the thermal entrance region of a plane-parallel channel with isothermal walls is presented. The effect of viscous dissipation is taken into account in a self-consistent way, i.e. by assuming that this effect is important not only in the region downstream of the entrance section, but also in the region upstream. In the latter region, the channel is assumed to be thermally insulated. As a consequence, the thermal entrance temperature distribution is non-uniform. The temperature field and the local Nusselt number are evaluated.     

The extended Graetz problem with piecewise constant wall temperature for pipe and channel flows

Axial heat conduction effects within the fluid can be important for duct flows if the Prandtl number is relatively low (liquid metals). In addition, axial heat conduction effects within the flow might also be important, if the heating zone is relatively short in length. The present paper shows an entirely analytical solution to the extended Graetz problem with piecewise constant wall temperature boundary conditions. The solution is based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation into a pair of first order partial differential equations. The obtained analytical solution is as simple to compute as the one without axial heat conduction. The analytical results are compared to available numerical calculations and good agreement is found.     

Analytical solution of Graetz problem in pipe flow with periodic inlet temperature variation

The paper reports an analytical solution for the temperature field in a fully developed pipe flow subject to periodic (of any shape) inlet temperature variation. The solution is given in term of a series of Kummer functions for the cases of uniform and constant wall temperature and wall heat flux, thus comprising also the adiabatic wall case. A “fully developed” region for the fluctuating component of the fluid temperature is also evidenced and closed-form solutions are given. An interpretation of the temperature field as superposition of travelling thermal waves is presented and discussed.     

The effect of wall conduction for the extended Graetz problem for laminar and turbulent channel flows

Axial heat conduction effects within the fluid can be important for duct flows if either the Prandtl number is relatively low (liquid metals) or if the dimensions of the duct are small (micro heat exchanger). In addition, axial heat conduction effects in the wall of the duct might be of importance. The present paper shows an entirely analytical solution to the extended Graetz problem including wall conduction (conjugate extended Graetz problem). The solution is based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation into a pair of first order partial differential equations. The obtained analytical solution is relatively simple to compute and valid for all Péclet numbers. The analytical results are compared to own numerical calculations with FLUENT and good agreement is found.     

Solutions of the extended Graetz problem

In this paper laminar forced convective heat transfer problems inside ducts, with axial conduction, subjected to the three main types of boundary conditions are solved exactly. The general method of solution involves a change of the dependent variable leading to a square integrable function in the real line. A complete basis for the vector space of these functions is used to generate an infinite expansion for the solution. The form of solutions is presented for the flows inside a circular pipe, the annular space between pipes, and between parallel plates.     

Extensions to the solution of the Graetz problem

In the original work of Graetz on heat transfer in laminar flow, the simplifying assumption was made that axial diffusion could be ignored in comparison with the other terms in the equation. This gives results which are sufficiently accurate for the large values of the Péclét number occurring with classical fluids, but the assumption cannot be applied to heat transfer in liquid metals where axial diffusion plays a significant role. Previous papers taking diffusion into account have involved separate computations for each value of the Péclét number, whereas in this paper the eigenvalues are given in the form of an asymptotic expansion so that the required values can be calculated in a simple fashion. The solution given here also takes into account preheating of the incoming fluid, and shows that this has a significant effect.     

Thermal entry flow for a viscoelastic fluid: the Graetz problem for the PTT model

A theoretical study of the entrance thermal flow problem is presented for the case of a fluid obeying the Phan-Thien and Tanner (PTT) constitutive equation. This appears to be the first study of the Graetz problem with a viscoelastic fluid. The solution was obtained with the method of separation of variables and the ensuing Sturm-Liouville system was solved for the eigenvalues by means of a freely available solver, while the ordinary differential equations for the eigenfunctions and their derivatives were calculated numerically with a Runge-Kutta method.The scope of the present study was quite wide: it encompassed both the plane and axisymmetric geometries for channel and tube flows; two types of thermal boundary conditions with either an imposed wall temperature or an applied heat flux; inclusion of viscous dissipation; and elastic (through the Weissenberg number) and elongational (through the PTT parameter ?) effects. The main underlying assumptions were those of constant physical properties, negligible axial heat conduction, and fully developed hydrodynamic conditions. The results are discussed in terms of the main effects brought about by viscoelasticity and viscous dissipation on the Nusselt number variation and the bulk temperature.     

An efficient green element formulation for the Graetz problem

We propose an efficient Green element method (GEM) technique for the solution of the generalized Graetz problem. The main point is to illustrate how GEM concepts can be adapted to handle heat or mass transport in tube flow; with axial conduction first ignored but later included. Several numerical examples are tested to demonstrate this numerical approach; for all cases, it is seen that GEM offers an elegant and in comparison with the problem difficulty, a reliable and straightforward approach.     

Extended Graetz problem accompanied by Dufour and Soret effects

Simultaneous heat and mass transfer between parallel plates are analyzed taking into account the Soret and Dufour effects. Both heat and mass transport are examined considering conduction in the axial and transverse directions plus longitudinal advection. The equations differ from the classical heat and mass transfer ones in considering the effect of the temperature gradient upon the mass flux, and conversely the effect of the concentration gradient upon heat flux, in accordance with the dictates of thermodynamics of irreversible processes. The special problems solved evaluate the effect of an imposed temperature difference between the confining walls upon the solute concentration distribution of a multisolute which diffuses against the concentration gradient forced by the prevailing temperature gradient. Details and numerical results are presented only for binary solutions.The asymptotic concentration difference, for a specified temperature difference, depends on the Soret and Dufour coefficients. The approach to the asymptotes is determined by the complete solution of the governing equations.     

Extended Graetz problem including streamwise conduction and viscous dissipation in microchannel

Extended Graetz problem in microchannel is analyzed by using eigenfunction expansion to solve the energy equation. The hydrodynamically developed flow is assumed to enter the microchannel with uniform temperature or uniform heat flux boundary condition. The effects of velocity and temperature jump boundary condition on the microchannel wall, streamwise conduction and viscous dissipation are all included. From the temperature field obtained, the local Nusselt number distributions are shown as the dimensionless parameters (Peclet number, Knudsen number, Brinkman number) vary. The fully developed Nusselt number for each boundary condition is obtained also in terms of these parameters.     

The effect of gaseous slip on microscale heat transfer: An extended Graetz problem

On the basis of Langmuir’s theory of adsorption of gases on solids, the effect of temperature jump on microscale heat transfer is investigated. A mathematical model, extended from the classical Graetz problem, is developed to analyze convective heat transfer in a microtube in various slip-flow regimes. The surface slip corrections are made by employing the Langmuir model, as well as the conventional Maxwell model. The effects of axial heat conduction are also investigated by extending the finite integral transform technique to the slip-flow case. We show that the Langmuir model always predicts a reduction in heat transfer with increasing rarefaction, as does the Maxwell model, except when the energy accommodation coefficient is relatively much smaller than that for momentum accommodation. This implies that, for most physical applications, the Reynolds analogy between heat transfer and momentum transfer is preserved in slip-flow regimes with low Mach numbers.     

Heat transfer by laminar Hartmann flow in thermal entrance region with a step change in wall temperatures: the Graetz problem extended

Thermally developing laminar Hartmann flow through a parallel-plate channel, including both viscous dissipation, Joule heating and axial heat conduction with a step change in wall temperatures, has been studied analytically. Expressions for the developing temperature and local Nusselt number in the entrance region are obtained in terms of Peclet number Pe, Hartmann number M, Brinkman number Br, under electrically insulating wall conditions, χ=−1 and perfectly conducting wall conditions, χ=0. The associated eigenvalue problem is solved by obtaining explicit forms of eigenfunctions and related expansion coefficients. We show that the nonorthogonal eigenfunctions correspond to Mathieu's functions. We propose a new asymptotic solution for the modified Mathieu's differential equation. The asymptotic eigenfunctions for large eigenvalues are also obtained in terms of Pe and M. Results show that the heat transfer characteristics in the entrance region are strongly influenced by Pe, M, Br and χ.     

Magnetohydrodynamic peristaltic flow of unsteady tangent‐hyperbolic fluid in an asymmetric channel

The aim of the present numerical investigation is to explore the impact of magnetic field on peristaltic flow of an incompressible tangent‐hyperbolic fluid in an asymmetric channel. The present physical model is developed based on the considered flow configuration and with the help of small Reynolds number approximations. The current flow problem is revealed under the influence of applied magnetic field. The asymmetric channel has been considered to narrate the present physical problem. Considered physical situation in the current investigation gives the unsteady coupled highly nonlinear system of partial differential equations. Also, the simplified equations for pressure, pressure gradient, and streamlines have been obtained with the help of suitable transformations. A regular perturbation scheme is employed to produce the semi‐analytical results of the present problem. The influence of various physical parameters on pressure, pressure gradient, and streamlines are illustrated with the help of graphs. From the present analysis, it is observed that the increasing magnetic number decreases the pressure and pressure gradient in the channel. Also, the size of trapping bolus increases with increasing values of Weissenberg number.     

Numerical and linear regression analysis on MHD two-phase flow in an asymmetric nonuniform channel

The flow through asymmetric nonuniform (convergent) channels with the effect of the magnetic field have a pronounced impact in engineering and biological fields such as chemical and food industries, blood flow through capillaries, and arteries, and so forth. With this motivation, the present study focuses on convective hydromagnetic particulate suspension flow in an asymmetric convergent channel under the heat generation effect. The numerical method is applied to solve the nondimensionalized equations governing the transport process of fluid and particle flow and its heat. To check the convergence of the computational results, a grid independence test has been performed. A comparison test has been made to validate the results and an admirable agreement is noticed with published results. Computation results are reported for the influence of emerging parameters on the fluid as well as particle velocity and temperature profiles through graphs and tables. A method of slope linear regression through data points is presented to study the impact of various parameters on skin friction and Nusselt number. The study pioneers the investigation on the significance of the combined influence of cross-flow Reynolds number and magnetic field on fluid and particle in the convergent channel and also reports its importance on drag coefficient and rate of heat transfer at the walls. It is perceived that a reduction in fluid velocity takes place with an increment in Magnetic parameter, Grashof number, and Reynolds number. An augmentation in fluid temperature is noted with an increment in Prandtl number and heat source parameter.     

A note on heat transfer to MHD oscillatory flow in an asymmetric wavy channel

This note deals with the MHD oscillatory flow of an optically thin fluid in an asymmetric wavy channel filled with porous medium. Based on some simplifying assumptions, the governing momentum and energy equations are solved and analytical solutions for fluid velocity, temperature distribution, Nusselt number and skin friction are constructed. The effects of radiation parameter, Peclet number, Hartmann number, porous medium shape factor and geometric parameters on flow and heat transfer characteristics have been examined in detail.     

Thermal and rheological effects in a classical Graetz problem using a nonlinear Robertson-Stiff fluid model

The present article elaborates the Graetz problem for the Robertson-Stiff fluid model with imposed iso-thermal conditions. The closed-form expression of Robertson-Stiff fluid velocity is obtained. Employing the classical separation of variables approach, the energy equation of the said problem is reduced into an eigenvalue problem. The solution of the eigenvalue problem is developed numerically via the MATLAB built-in algorithm BVP4C. The constants appearing in series solutions are computed by Simpson's rule. The special case of this analysis with appropriate scaling is also applicable for the Bingham, power-law, and Newtonian fluid models. The impact of the dissipation function on Nusselt numbers and mean temperature is also considered. The pictorial representation of average temp7erature and Nusselt number are discussed in the presence of the plug radius, power-law index, and Brinkman number. It is observed that the presence of the plug radius and power-law index delay the prevalence of fully developed conditions for the Nusselt number. Moreover, the local Nusselt number for channel confinement attains higher values as compared with tube confinement. The present investigation has numerous applications in the field of engineering, nanotechnology, biomedical sciences, and development of several thermal types of equipment or microfluidic devices.     

A solution for the complex eigenvalues and eigenfunctions of the periodic Graetz problem

The periodic Graetz problem arising in laminar transient forced convection inside a parallel-plate channel is analyzed. Nayfeh's method is applied to the complex eigenvalue problem yielding a single uniformly valid closed form asymptotic solution for the complex eigenfunctions. The asymptotic solution presented is accurate for all normalized complex eigenfunctions except for those corresponding to the smaller eigenvalues. Another solution of the complex eigenvalue problem, which is accurate for the smaller eigenvalues, is presented based on the finite Fourier transform technique and is utilized for checking the accuracy of the closed form asymptotic solution.     

On laminar hydromagnetic mixed convection flow in a vertical channel with symmetric and asymmetric wall heating conditions

The problem of hydromagnetic fully developed laminar mixed convection flow in a vertical channel with symmetric and asymmetric wall heating conditions in the presence or absence of heat generation or absorption effects is considered. Through proper choice of dimensionless variables, the governing equations are developed and three types of thermal boundary conditions are prescribed. These thermal boundary conditions are isothermal-isothermal, isoflux-isothermal, and isothermal-isoflux for the left-right walls of the channel. Analytical solutions for the velocity and temperature profiles for various special cases of the problem are reported. In addition, closed-form expressions for the Nusselt numbers and reversal flow conditions at both the left and right channel walls are derived. The general problem which includes the effects of both viscous dissipation and Joule heating is solved numerically by an implicit finite-difference scheme. Favorable comparisons of special cases with previously published work are obtained. A selected set of graphical results illustrating the effects of the various parameters involved in the problem including viscous and magnetic dissipations on the velocity and temperature profiles as well as flow reversal situations and Nusselt numbers is presented and discussed.     

Peristaltic flow of a Williamson fluid in an inclined asymmetric channel with partial slip and heat transfer

The present article deals with the peristaltic flow of a Williamson fluid in an inclined asymmetric channel. The relevant equations have been modeled. Analysis has been carried out in the presence of velocity and thermal slip conditions. Expressions for stream function, temperature, pressure gradient and heat transfer coefficients are derived. The solutions are compared with the existing available results in a limiting sense. Numerical integration has been performed for pressure rise per wavelength. Plots are presented and analyzed for various embedded parameters into the problem. Comparison between the solutions is also shown.