Electro-osmotic flow of biological fluid in divergent channel: drug therapy in compressed capillaries

The multi-phase flow of non-Newtonian through a divergent channel is studied in this article. Jeffrey fluid is considered as the base liquid and tiny gold particles for the two-phase suspension. Application of external electric field parallel to complicated capillary with net surface charge density causes the bulk motion of the bi-phase fluid. In addition to, electro-osmotic flow with heat transfer, the simultaneous effects of viscous dissipation and nonlinear thermal radiation have also been incorporated. Finally, cumbersome mathematical manipulation yields a closed-form solution to the nonlinear differential equations. Parametric study reveals that more thermal energy is contributed in response to Brinkman number which significantly assists gold particles to more heat attain high temperature, as the remedy for compressed or swollen capillaries/arteries.


Scientific Reports
| (2021) 11:23652 | https://doi.org/10.1038/s41598-021-03087-0 www.nature.com/scientificreports/ They applied a thin shinning sheet/layer on the rotating disk, by using gold and silver particles. They inferred that ethanol suspension with gold particles yields a perfect coating on any rotating surface. Nazeer et al. 19 obtained an approximate solution for two different kinds of multiphase flows. Suspensions are formed by considering Third-grade fluid as the base liquid while, Hafnium and crystal particles are considered. The gravitational force causes the flow of an MHD multiphase flow through an inclined channel. In 16 , Couette flow of Couple stress fluid is simulated. The magnetized moving upper wall of the channel drives the two-phase flow. The heating effects at the boundary attenuates the shear thickening effects. Zeeshan et al. 20 have applied numerical techniques for a free-stream flow on an inclined sheet. Range-Kutta method with shooting technique is applied to obtain a numerical solution of nonlinear differential equations. Paul et al. 21 worked on modeling of industrial particle and multiphase flows using combinations of DEM for free surface fluid-particle flows. Internal flows through closed channels change the hydrodynamic structure of the flow. There can be rapid dynamical changes in the internal flows, for the uniform channels with the compressed portion or a divergent configuration. Zheng et al. 22 brought convergent-divergent slit ribs to improve internal cooling. They observed that there is a vivid thermal enhancement working with small-angle trapezoidal slits that increase heat transfer. Mekheimer et al. 23 use gold nanoparticles as drug agents for therapy and, suggest that gold nanoparticles effectively contribute to drug delivery. Intrauterine particle-fluid motion through a compliant asymmetric tapered channel with heat transfer is reported by Bhatti et al. 24 . In 25,26 , nano-blood flows through catharized tapered arteries are reported. Jeffrey fluid is treated as the physiological fluid by using gold nano-particles work as the remedy. Some important studies are listed in the Refs. [27][28][29][30][31] .
In view of fore-going literature, it is evident that no attention has been paid towards the two-phase flow of Jeffrey fluid with heat transfer in a convergent channel. Additional contributions of viscous dissipation and nonlinear radiative flux are taken into account, as well. The electro-osmotic flow of Jeffrey fluid suspended with gold particles is an innovative concept which addresses the blood transport through compressed/swollen capillaries or arteries.

Mathematical model
Consider a steady two-phase flow of a non-Newtonian fluid with heat transfer in a divergent channel as shown in Fig. 1. The multiphase suspension is composed of Jeffrey fluid as the base liquid suspended with tiny size confined by gold particles. Let V f = u f ⌢ ζ , ⌢ η ,0,0 and V p = u p ⌢ ζ , ⌢ η ,0 , 0 are the velocity profiles of liquid and particle phases, respectively.
For the fluid phase, Otherwise.
The following transformation is used to get the non-dimensional form of the above equations  The pressure is obtained from the above equation which is given by

Results and discussion
The objective of this section is to highlight the impact of important physical parameters on the velocity and temperature distribution through graphs. Figures 2, 3 , 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16  Unlike, the previous case velocity of each phase reduces while more thermal energy incorporates into the system by expediting the heat transfer. This phenomenon is the application of the electro-osmotic process which ionizes the charged groups on the surface or due to preferential adsorption of ions within the fluid. It acts transversely to the motion of the fluid due to Lorentz's force. Hartman number M is a dimensionless number that corresponds to magnetic field induction. Figures 8, 9 and 10 highlights  It is observed that the increasing values of U HS decline the velocity profiles in both fluid and particle phases. However, it can be observed in Fig. 16 variation in U HS supports the temperature profile by increasing the force of friction between the adjacent layers of the base fluid. Finally, the influence of heat conduction from the wall on the viscous fluid is given in Fig. 17. It is noticed that more energy comes into the system due to slow down the process of conduction of heat by viscous dissipation when Brinkman number B r is varied. Hence, the temperature of multiphase flow rises.

Concluded remarks
A closed-form solution is obtained for the heat of a non-Newtonian fluid suspended with gold particles. Electroosmotic multiphase flow is analyzed in a divergent channel under the influence of viscous dissipation and thermal radiation. The most noteworthy observations catalog as: • Jeffrey parameter corresponds to the rise of both velocity profiles.
• The electro-osmotic parameter m and Helmholtz-Smoluchowski velocity U HS act differently on thermal and momentum distribution. • Additional gold particles expedite the flow of both phases. • More energy is added to the system due to Brinkman number B r .