Analytical solutions to simultaneously developing laminar flow inside parallel-plate channels

Integral transform method for boundary layer equations in simultaneous heat and fluid flow problems

“…Table 2 brings a set of reference results for the fluid bulk temperature, with different values of Pr, along the duct axial coordinate. The exact fully converged results are then utilized to inspect the relative accuracy of the approximate analytic-type solutions 16 , and such comparison, not possible before due to the inexistence of a truly benchmark solution, shows that the approximate solution is reasonably accurate, offering two to three digits in agreement along most of the entry region. 16 , based on a linearization of the velocity problem and application of the generalized integral transform technique to the resulting linear temperature problem.…”

“…The exact fully converged results are then utilized to inspect the relative accuracy of the approximate analytic-type solutions 16 , and such comparison, not possible before due to the inexistence of a truly benchmark solution, shows that the approximate solution is reasonably accurate, offering two to three digits in agreement along most of the entry region. 16 , based on a linearization of the velocity problem and application of the generalized integral transform technique to the resulting linear temperature problem. The agreement is quite good, except in the region close to the duct inlet, when the approximation introduced on the velocity field affects more significantly the related temperature distributions and, consequently, the Nusselt number.…”

“…In a previous contribution 16 , the problem of simultaneous heat and fluid flow inside channels was analytically handled through the integral transform approach, by applying a linearization procedure to the velocity problem and yielding a linear non-separable energy equation. The approximate analytic expressions were then compared against finite difference results for the complete non-linear problem, offering a validation of this extended Graetz-type solution.…”

Hybrid solution for the laminar flow of power-law fluids inside rectangular ducts

“…This spectral-type approach is based on eigenfunction expansions yielding to solutions where the most features are: (i) an automatic and straightforward global error control and, (ii) an only mild cost increase in overall computational effort for multidimensional situations. Due to its hybrid nature, this scheme has been well indicated for benchmarking purposes and for the validation of different numerical methods in many classes of problems such as, non-linear heat and¯uid¯ow problems, including the Navier±Stokes equations (Pe Ârez Guerrero and Lima et al, 1997;Quaresma and Cotta, 1997), the laminar and turbulent boundary layer equations in duct¯ows (Cotta and Carvalho, 1991;Carvalho et al, 1993;Machado and Cotta, 1995;Figueira da Silva and Cotta, 1996;Cotta and Pimentel, 1998) and, convection-diffusion and eigenvalue problems (Campos Silva et al, 1992;Mikhailov and Cotta, 1994), as well as, laminar boundary layer equations in non-Newtonian¯ows in channels (Magno et al, 1999).…”

Heat transfer in laminar flow of non-Newtonian fluids in ducts of elliptical section

“…It is the Generalized Integral Transform Technique -GITT [3], a method which has been used successfully to solve several diffusive problems such as those dealing with the flow in ducts with irregular geometries [4,5], with time varying coefficients and problems involving space dependence for the boundary conditions [6,7], problems with thermally and hydrodynamically developing flows [8,9], diffusive problems involving moving boundaries [10,11], problems of non-Newtonian fluid flows [12,13], among others.…”