Modulated complexed stenosed region consequences under the electroosmotic stimulation

The present study analyzes the theoretical consequences of slip effects in a complex stenosed region. The flow of blood in a stenosed region is incorporated with hybrid nanofluid features which are being prepared with copper and copper oxide nanoparticles. The flow is also intensified by applying an electric field in the axial direction. The governing equations for the proposed paradigm are solved and the corresponding closed-form solutions are obtained for the cases of mild stenosis. Parameters such as Electro-osmotic, velocity slip and Helmholtz–Smoluchowski are specially focused in this study. The heat transfer, hemodynamic velocity, wall shear stress and resistance impedance for the flow are precisely determined. The various parameters that influence the physical characteristics of flow are plotted, and their effects are discussed in detail. The present model has the potential application in medical pumps for drug delivery systems.


Mathematical formulation
The above Fig. 1, we have some representations to describe the blood flow across the diseased artery such as, ( r, z) represents cylindrical coordinates,( u, w) as velocities components along radial and axial direction,z d as axial displacement,R 1 for Non-stenotic radius of outer tube, s l used for stenosis length at l = 1, 2, 3 and stenosis location is represented by d l at l = 1, 2, 3. Geometry, describing the blood flow across an artery with multiple stenosis at outer wall is considered as www.nature.com/scientificreports/ In above, we have where δ * l represents the utmost height attained by stenosis for n = 2 the case is for symmetric and for n > 2 the case is considered for non-symmetric stenosis.Also, h(z) represents exterior boundary, i.e. (wall having multiple stenosis) is in their dimensional form.

Mathematically governed flow problem
The experimental and theoretical analysis of two-dimensional viscous fluid model of diseased artery with electric field and homogenous mixture of hybrid nanofluid with blood as base fluid governs the following mathematical equations, which are in dimensional form is given as 11 where in above ρ hnf , k hnf and µ hnf respectively the attributes of hybrid nanofluid particles for copper (Cu) and copper-oxide (CuO) cases and their mathematical form are listed as below To approximate the viscosity of hybrid nanofluid, Brinkman's viscosity model is used as 12 Thermal conductivity of hybrid nanofluid mixture is approximated by Maxwell and Hamilton-crosser's model given as 13 where In above ∅ * 1 and ∅ * 2 represents the volumetric fraction.Moreover, the thermophysical significance of hybrid nanofluid particle with base fluid are given as 13 Poison-Boltzman equation for electric potential distribution in presence of the electric double layer is given as 14 where ρ e the case of binary fluid stands which for net charge density.andconsists of two types of ions with equal and opposite charges, the net charge density is specified as, otherwise.
(2) ψ K A T * represents positive and negative charges in bulk concentration is the electric charge,z * is the valence of ions, K A is the Boltzmann constant, T * is the local absolute temperature of electrolyte solution and n 0 is the average concentration of ions.By assuming the symmetricity of electrolytes, the net charge density described in equation can be computed as is known as Debye-Huckel supposition, with this assumption the above expression may be computed as Utilizing non-dimensional parameters r = � r R 1 and ψ = ψ ξ and putting the value of ρ e in Eq. ( 11), we get By using ∂ψ ∂r = 0 at r = 0 and ψ = 1 at r = h(z).Above Eq. ( 15) can be written as Dimensionless parameters are given as By substituting Eqs. ( 16) and (17) in Eqs.(3-5), we obtained the following system of equations and considering the assumption of mild stenosis Constants are described as below ( 13) The relevant dimensionless boundary conditions are as [27][28][29] where By utilizing dimensionless parameters in Eq. ( 17), we get the corresponding dimensionless stenosis boundary conditions as

Exact solution of mathematical model
Above evaluated system of non-dimensional equations from ( 18)-( 20) are solved with associated non dimensional boundary conditions mentioned in Eqs.(26-27) with the help of computational software Mathematica.We get the following solutions.

Velocity profile
By solving Eq. ( 19) with dimensionless boundary conditions mentioned in Eq. ( 26), we get the velocity profile as The evaluated velocity profile in Eq. ( 30) is used to fined rate of flow as The expression of pressure gradient is obtained from above Eq.( 32) and defined as where The expression for wall shear stress is defined as Using Eq. ( 30) in above equation we get  www.nature.com/scientificreports/

Temperature profile
The temperature profile is evaluated by considering Eq. ( 20) with dimensionless boundary conditions given in Eq. ( 27) as

Graphical configuration and findings
This section facilitates to discuss the exact results evaluated for velocity, pressure gradient and wall shear stress of flow through stenosed artery graphically by comparison of Cu-CuO/blood, Cu-blood and pure blood in a channel with electroosmosis effect.This analysis comprises experimental results of Cu-CuO/blood, Cu-blood nano particles cases which are listed in Tables 1 and 2. The current phenomena are discussed by using Helmholtz-Smoluchowski velocity approximately equal to the 2 cm −1 ionic meditation ranging from 1m-1Mm.Electric field of strength up to 1kV cm −1 is applied by knowing about the relative permittivity of base fluid 22,23 .Graphical configurations are plotted by considering laminar flow with hybrid nanofluid, and pure blood as base fluid with electroosmotic effect with different parameters of interest.The solutions evaluated above are plotted against the radial coordinate with enhanced values of dimensionless parameters of interest.

Velocity profile
The velocity profile depicts flow behavior in stenosed artery.The velocity of a viscous fluid flow in a channel with a steonsis at the outer walls is discussed using a graphical configuration.Figures 2, 3, 4 and 5 are planned to discuss the velocity significance for Cu-CuO/blood, Cu-blood nanoparticles and pure blood cases for different enhanced values of parameter η (velocity slip) m, n(stenosis shape) and U.In Fig. 2, We find that increasing η results in an increase in fluid velocity near the walls with stenosis, but a decrease in fluid velocity in the middle of the channel.To find out the visible effect of the parameter m on the fluid velocity, Fig. 3 is sketched and noted that by enhancing its value, the corresponding result is a deceleration at the center of the flow channel.Graphical (38) Table 1.Different shape features of nanoparticles 9 .

Geometrical appearance
Shape Bricks Cylinder Platelets Shape factor 3.7 4.9 5.7 Table 2. Thermophysical significances of Cu, CuO and base fluid blood 9 .result displayed in Fig. 4 against Helmholtz-Smoluchowski (HS) velocity parameter which shows minimum flow velocity output at the center of fluid flow channel but opposite behavior is noted near the walls having multiple stenosis.Velocity profile for symmetric and non-symmetric shape parameter n is designed in Fig. 5, it shows parabolic trajectory for both symmetric and non-symmetric multiple stenosis and profile of velocity decreases in the center of the channel more prominently as compared to the walls of the artery.It has been observed that the velocity of hybrid nanofluid near the arterial wall is much higher than the nanofluid and the base fluid.

Temperature significance
This section facilitates to discuss the heat transfer significance through diseased artery having multiple stenosis at outer walls with applied electric field phenomena.The heat transfer rate is discussed through significant   Influence of parameter m on temperature profile is found in Fig. 9.It is observed from this phenomenon that an increase in this parameter appears to lead to a decline in the temperature profile.Figure 10 is designed to discuss the temperature profile for symmetric and non-symmetric stenosis which shows decline in temperature profile for both the cases.In all these graphs, it is found that temperature is higher at the middle of channel as compared to the stenosed walls having electroosmosis effect.Thus, the rate of temperature flow can be reduced by the application of electric field.These graphs show that the temperature profile of a hybrid nanofluid (Cu-Cu/

Shear stress graph
To analyze the behavior of stress on the stenotic wall, shear stress graphs are planned and displayed in Fig. 11, 12, 13 and 14 for different significant dynamical and thermodynamically feature based parameters of interest.
In Fig. 11 it is noted that there is decline in wall shear stress by enhancing values of velocity slip parameter.

Figure 2 .
Figure 2. Velocity profile in a channel for velocity slip parameter.

Figure 3 .
Figure 3. Velocity profile in a channel for parameter m.

Figure 5 .
Figure 5. Velocity profile in a channel for multiple stenosis shape parameter.

Figure 7 .
Figure 7. Temperature profile in a channel for parameter Brinkman number.
Figures 12 and 13 are designed against enhancing parameters Helmholtz-Smoluchowski (HS) velocity parameter and m.But opposite behavior is found for these parameters.All these graphs depict that the shear stress profile of hybrid nanofluid (CuO-Cu/blood) is less than the nano fluid (Cu-blood) and much more than base fluid.

Figure 8 .
Figure 8. Temperature profile in a channel for velocity slip parameter.

Figure 9 .
Figure 9. Temperature profile in a channel for parameter m.

Figure 10 .
Figure 10.Temperature profile in a channel for multiple stenosis parameters.

Figure 11 .
Figure 11.Shear stress on stenotic walls for velocity slip parameter.

Figure 14 .
Figure 14.Shear stress for multiple stenosis shape parameter.