Peristaltic transfer of nanofluid with motile gyrotactic microorganisms with nonlinear thermic radiation

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TL;DR

This study models the peristaltic transport of a Jeffrey nanofluid containing motile gyrotactic microorganisms through an anisotropically stenosed endoscope, incorporating Hall current, Joule heating, Darcy-Forchheimer porous media, Soret and Dufour effects, nonlinear thermal radiation, and activation energy. Key findings show that increasing the local temperature Grashof number boosts velocity, while increasing the nanoparticle Grashof number and bioconvection Rayleigh number attenuates velocity; temperature rises with thermophoresis and Brownian motion parameters; and activation energy increases nanoparticle concentration. The work has clinical significance for targeted drug delivery through catheters and understanding blood flow in stenosed arteries.

Full summary

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**Background:** The study addresses the peristaltic transport of a Jeffrey nanofluid containing motile gyrotactic microorganisms through an anisotropically stenosed endoscope. Peristaltic movements are fundamental in biological systems (blood flow, urine transport, drug delivery). The authors incorporate multiple physical phenomena: Hall current and Joule heating from a strong magnetic field, Darcy-Forchheimer porous medium effects, Soret and Dufour schemes, nonlinear thermal radiation via the Rosseland approximation, and activation energy via the amended Arrhenius scheme with Buongiorno type nanofluid model. The clinical motivation is to model medication delivery to malignant cells and clogged heart arteries via catheter, and gastric juice movement in the small intestine during endoscopy.

**Methods:** The problem is formulated in cylindrical coordinates (R, θ, Z) with an inner rigid tube (radius r₁) moving at constant speed V₀ and an outer anisotropically stenosed tube with a sinusoidal wave traveling along its wall. The Jeffrey non-Newtonian model is used for the fluid. Governing equations include continuity, momentum (with Lorentz force, buoyancy, and Darcy-Forchheimer terms), energy (with nonlinear radiation and Dufour effect), concentration (with chemical reaction and Arrhenius activation energy), and motile microorganism transport. Boundary conditions include slip at the wall for velocity and convective conditions for temperature. The long wavelength and low Reynolds number approximations (δ << 1) are applied to transform the partial differential equations into ordinary differential equations. A perturbation expansion in δ (zero-order and first-order) combined with the Homotopy Perturbation Method (HPM) is used to obtain analytical solutions for velocity w, temperature T, concentration C, motile microorganism density N, and pressure gradient dP/dz. Solutions are expressed in terms of radial coordinate r and axial coordinate z. Dimensionless parameters include: Gr (local temperature Grashof number = 0.2), RN (nanoparticle Grashof number = 0.4), Rb (bioconvection Rayleigh number = 0.4), H² (Hartmann number = 10), m (Hall parameter = 0.2), Da (Darcy number = 0.02), Fr (Forchheimer number = 10), Nt (thermophoresis parameter = 0.4), Nb (Brownian motion parameter = 0.3), Pr (Prandtl number = 0.3), Br (Brinkman number = 0.3), Rn (radiation parameter = 0.05), Du (Dufour number = 0.1), Sr (Soret number = 0.5), α (chemical reaction parameter = 0.3), ξ (activation energy parameter = 0.1), Pe (bioconvection Peclet number = 0.3), Le (Lewis number = 0.2), Lm (microorganism Lewis number = 0.2), βh (Biot number = 0.1), γ* (slip parameter = 0.8), λ* (Weissenberg number = 0.3), λ₁ (ratio of relaxation to retardation time = 0.9), ϵ (amplitude ratio = 0.2), and δ = 0.01.

**Key Results:** (1) Velocity: Increasing Gr from baseline increases velocity due to enhanced thermal buoyancy force. Increasing RN attenuates velocity due to increased viscous force. Increasing stenosis height h reduces velocity in the stenosed region (-0.5 ≤ z ≤ 0.5), consistent with blood clot pathophysiology. Increasing λ* (Weissenberg number) enhances velocity by reducing viscous forces. Increasing Rb (bioconvection Rayleigh number) diminishes velocity. Increasing slip parameter γ* boosts velocity by reducing wall resistance. Increasing Fr (Forchheimer number) decelerates flow due to quadratic drag. (2) Temperature: Increasing Nt (thermophoresis) and Nb (Brownian motion) both increase temperature. Increasing βh (Biot number) decreases temperature due to reduced thermal conductivity. (3) Concentration: Increasing α (chemical reaction parameter) contracts the concentration profile. Increasing ξ (activation energy parameter) augments concentration due to weakened chemical reaction. Increasing βt (temperature ratio parameter) drops concentration by widening the concentration boundary layer. (4) Motile microorganisms: Increasing Pe (Peclet number) decreases microorganism density because robust Peclet number implies weaker Brownian coefficients. Increasing Ω (bioconvection constant) reduces microorganism density. (5) Pressure gradient: dP/dz shows periodic behavior with two peaks at z = -0.5, 0.5 and minima at z = -1, 0, 1. Increasing ϵ amplifies the pressure gradient; increasing λ₁ reduces it due to decreased fluid viscosity.

**Clinical Implications:** The study models drug delivery via catheter to malignant cells and clogged arteries, and gastric juice movement during endoscopy. The inclusion of activation energy (Arrhenius scheme) is relevant for controlled drug release. The finding that stenosis reduces velocity confirms the hemodynamic impact of blood clots. The model suggests that manipulating parameters like thermophoresis, Brownian motion, and chemical reaction could optimize nanofluid-based drug delivery systems. The comparison with prior studies (Table 1) shows the current model predicts significantly higher velocities (e.g., 38.9, 58.5, 157.4 at r = 0.5, 1.5, 1.7) compared to El-dabe et al. 2021 (-0.702, -1.2, -1.4), Rahman et al. 2016 (-1.4, 1, 2.1), and El-dabe et al. 2020 (-0.7, -1.1, -1.2), attributed to the inclusion of additional physical effects.