Unsteady pulsatile fractional Maxwell viscoelastic blood flow with Cattaneo heat flux through a vertical stenosed artery with body acceleration

Abstract

In the current problem, we aim to discuss the heat transfer of pulsatile unsteady fractional Maxwell fluid (blood) flow through a vertical stenosed artery with body acceleration. The concept of fractional Cattaneo model will modify the energy equation. We will get the solutions using Laplace and finite Hankel transformations. The inverse of the transformed functions will be calculated numerically. It is observed that, the heat relaxation time causes a delay in the heat transfer until a critical time. In addition, the heat transfer increases sharply to take its maximum value at a critical value of time then it decreases to reach the steady state. Moreover, the blood velocity, the flow rate, and the shear stress continue to fluctuate during the time period due to the pulsatile phenomenon and body acceleration.

Appendix A

The constitutive relation of fractional Maxwell fluid is [9, 10]

$$\begin{aligned}&\left( 1+\lambda '^{\alpha }\frac{\partial ^{\alpha }}{\partial t'^{\alpha }}\right) \sigma '=\mu \lambda '^{\beta -1}\frac{\partial ^{\beta -1}{\dot{\Upsilon }}'}{\partial t'^{\beta -1}}, \end{aligned}$$

(A.1)

$$\begin{aligned}&{\tau }'= -p'\mathbf{I }+\sigma ', \end{aligned}$$

(A.2)

where ${\dot{\Upsilon }}'$ is the shear rate, $\sigma '$ is the shear stress and ${\tau }'$ denotes the stress tensor.

The momentum equation is

$$\begin{aligned} \rho \frac{D \mathbf{U }'}{D t'}=\nabla \cdot {\tau }'+\rho \mathbf{g }\varrho (T'-T_{0})+F'(t')\mathbf{e}_{\mathrm{z}}, \end{aligned}$$

(A.3)

where $\mathbf{e}_{\mathrm{z}}$ is the unit vector in z-direction. The combination between Eqs. (A.1), (A.2) and (A.3) when $\mathbf{U }'=(u',0,w')$ gives Eqs. (6) and (7).