Theoretical analysis of thermal entrance problem for blood flow: An extension of classical Graetz problem for Casson fluid model using generalized orthogonality relations

Khan, Muhammad Waris Saeed; Ali, Nasir

doi:10.1016/j.icheatmasstransfer.2019.104314

Cited by 37 publications

(16 citation statements)

References 27 publications

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“…These results also reveal that the asymptotic local Nusselt number is independent of Brinkman number. For a Newtonian liquid in a flat channel,

{Nu}_{fd} = 17.5

was found by Khan and Ali 29 and the current study also validates this value. The deviation in the local Nusselt number is expounded from the values of nondimensional mean temperature and temperature gradient at the wall (Equations (35) and (36)).…”

Section: Resultssupporting

confidence: 89%

“…For the solution of Equation (20) with constrained boundary conditions (18), a outlined procedure (see details in References [28] and [29]) is followed, which leads us to the following fully developed temperature profiles:

\begin{matrix} lefttrue \\ θ_{\infty} = B r true{\begin{matrix} trueleft \\ true(4 true(\frac{3 n + 1}{1 + n} true) (1 - c *^{3 + 1 / n} - c * \frac{(3 + 1 / n)}{3} + c *^{4} \frac{(3 + 1 / n)}{3} true) true) true(\frac{1 + n}{n} true) \frac{c *}{9} ({\overset{r}{true}}^{3} - 1) - \\ true(\frac{1 + n}{n} true) \frac{1}{{true(3 + \frac{1}{n} true)}^{2}} true({\overset{r}{true}}^{(3 + \frac{1}{n})} - 1 true) + true(true(\frac{3 n + 1}{1 + n} true) {(\begin{array}{l} 1 - c *^{3 + 1 / n} - c * \frac{true(3 + 1 / n true)}{3} + \\ c *^{4} \frac{true(3 + 1 / n)}{3} \end{array})}^{2} true(\frac{1}{true(3 + 1 / n true)} true) true) \\ true(- c *^{(3 + 1 / n)} true(\frac{2 n + 1}{2 (3 n + 1)} true) - \frac{c *^{4}}{6} true(\frac{1 + n}{n} true) true) \ln \bar{r} \end{matrix} true} \\ c ... \end{matrix}

…”

Section: Thermal Analysismentioning

confidence: 99%

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

2020

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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.

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“…These results also reveal that the asymptotic local Nusselt number is independent of Brinkman number. For a Newtonian liquid in a flat channel,

{Nu}_{fd} = 17.5

Section: Resultssupporting

confidence: 89%

\begin{matrix} lefttrue \\ θ_{\infty} = B r true{\begin{matrix} trueleft \\ true(4 true(\frac{3 n + 1}{1 + n} true) (1 - c *^{3 + 1 / n} - c * \frac{(3 + 1 / n)}{3} + c *^{4} \frac{(3 + 1 / n)}{3} true) true) true(\frac{1 + n}{n} true) \frac{c *}{9} ({\overset{r}{true}}^{3} - 1) - \\ true(\frac{1 + n}{n} true) \frac{1}{{true(3 + \frac{1}{n} true)}^{2}} true({\overset{r}{true}}^{(3 + \frac{1}{n})} - 1 true) + true(true(\frac{3 n + 1}{1 + n} true) {(\begin{array}{l} 1 - c *^{3 + 1 / n} - c * \frac{true(3 + 1 / n true)}{3} + \\ c *^{4} \frac{true(3 + 1 / n)}{3} \end{array})}^{2} true(\frac{1}{true(3 + 1 / n true)} true) true) \\ true(- c *^{(3 + 1 / n)} true(\frac{2 n + 1}{2 (3 n + 1)} true) - \frac{c *^{4}}{6} true(\frac{1 + n}{n} true) true) \ln \bar{r} \end{matrix} true} \\ c ... \end{matrix}

…”

Section: Thermal Analysismentioning

confidence: 99%

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

2020

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“…Recently, Ali and Khan 17 provided an analytical treatment of the Graetz-Nusselt problem for Ellis fluids with both prescribed heat flux and imposed constant wall temperature boundary conditions. Blood flow analysis in entrance regions of both tube and channel confinements was investigated for a Casson viscoplastic fluid with isothermal wall conditions by Khan and Ali (2019). 18 Khan et al 19 studied the Graetz problem with a Robertson-Stiff fluid model for both tube and channel geometries under constant wall temperature boundary condition.…”

Section: Introductionmentioning

confidence: 99%

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

Khan

Ali

Bég

2022

Proc Inst Mech Eng H

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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.

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“…Researchers have exhibited great interest in Casson fluid flow on an elongated surface which can be seen in Refs. [15][16][17][18][19][20][21][22][23][24][25][26] .…”

Section: Reaction Rate Constantmentioning

confidence: 99%

Impact of Newtonian heating and Fourier and Fick’s laws on a magnetohydrodynamic dusty Casson nanofluid flow with variable heat source/sink over a stretching cylinder

Ramzan

Shaheen

Chung

et al. 2021

Sci Rep

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The present investigation aims to deliberate the magnetohydrodynamic (MHD) dusty Casson nanofluid with variable heat source/sink and modified Fourier’s and Fick’s laws over a stretching cylinder. The novelty of the flow model is enhanced with additional effects of the Newtonian heating, activation energy, and an exothermic chemical reaction. In an exothermic chemical reaction, the energy of the reactants is higher than the end products. The solution to the formulated problem is attained numerically by employing the MATLAB software function bvp4c. The behavior of flow parameters versus involved profiles is discussed graphically at length. For large values of momentum dust particles, the velocity field for the fluid flow declines, whereas an opposite trend is perceived for the dust phase. An escalation is noticed for the Newtonian heating in the temperature profile for both the fluid and dust-particle phase. A comparison is also added with an already published work to check the validity of the envisioned problem.

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Theoretical analysis of thermal entrance problem for blood flow: An extension of classical Graetz problem for Casson fluid model using generalized orthogonality relations

Cited by 37 publications

References 27 publications

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

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

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

Impact of Newtonian heating and Fourier and Fick’s laws on a magnetohydrodynamic dusty Casson nanofluid flow with variable heat source/sink over a stretching cylinder

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