Thermal entrance problem for blood flow inside an axisymmetric tube: The classical Graetz problem extended for Quemada’s bio-rheological fluid with axial conduction

Abstract

The heat-conducting nature of blood is critical in the human circulatory system and features also in important thermal regulation and blood processing systems in biomedicine. Motivated by these applications, in the present investigation, the classical Graetz problem in heat transfer is extended to the case of a bio-rheological fluid model. The Quemada bio-rheological fluid model is selected since it has been shown to be accurate in mimicking physiological flows (blood) at different shear rates and hematocrits. The steady two-dimensional energy equation without viscous dissipation in stationary regime is tackled via a separation of variables approach for the isothermal wall temperature case. Following the introduction of transformation variables, the ensuing dimensionless boundary value problem is solved numerically via MATLAB based algorithm known as bvp5c (a finite difference code that implements the four-stage Lobatto IIIa collocation formula). Numerical validation is also presented against two analytical approaches namely, series solutions and Kummer function techniques. Axial conduction in terms of Péclet number is also considered. Typical values of Reynolds number and Prandtl number are used to categorize the vascular regions. The graphical representation of mean temperature, temperature gradient, and Nusselt numbers along with detail discussions are presented for the effects of Quemada non-Newtonian parameters and Péclet number. The current analysis may also have potential applications for the development of microfluidic and biofluidic devices particularly which are used in the diagnosis of diseases in addition to blood oxygenation technologies.

Critical analysis of generalized Newtonian fluid flow past a non‐linearly stretched curved surface: A numerical solution for Carreau model

“…However, his results show close agreement with Massey and Clayton [9]. A few notable studies related to non‐Newtonian fluids were found in [16–25], heat transfer along with nanofluids [26–36], and stretching surfaces [37, 38].…”

Thermal entry problem for a tube with prescribed heat flux condition using viscoplastic fluid: An extended Graetz problem for Casson fluid with axial conduction and viscous dissipation

“…The widely utilized Homotopy method (OHAM) is applied to obtain the solution, and it is determined that the Casson fluid and normalized slip both increase entropy generation. Some interesting studies related to non-Newtonian fluids [18][19][20][21][22][23], Graetz problem [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] gliding bacteria [39][40][41][42] and robust numerical approaches [43][44][45] are worthwhile to mention. Motivated by above mention studies, we aim to elaborate on the impact of axial conduction and viscous dissipation on the Graetz problem inside the tube using Casson viscoplastic fluid under imposed heat flux at the wall.…”

Graetz problem for the Rabinowitsch fluid model: A comparative analysis for circular tube and flat channel

“…Few notable analyses related to non-Newtonian fluids and Graetz problems can be found in the literary works. [2][3][4][5][6][7][8][9][10][11][12][13][14] Nusselt 15 adopt the same approach and investigated this analysis in a more efficient way that's why few researchers called this problem as 'the Graetz Nusselt problem'. Seller et al 16 theoretically elaborated on this problem in terms of Bessel functions.…”

“…Few notable analyses related to non-Newtonian fluids and Graetz problems can be found in the literary works. 2–14…”