Unsteady and Incompressible Magneto-Hydrodynamics Blood Flow in an Inclined Cylindrical Channel

In the current study, the blood ﬂ ow through an inclined cylindrical tube subjected to an external magnetic ﬁ eld is evaluated. The blood ﬂ ow has been considered under the consequence of a transverse magnetic ﬁ eld. Previously the mathematical model was solved by using Caputo-Fabrizio (CF) fractional order derivative with a non-singular kernel which has the limitations like it fails to satisfy the fundamental theorem of fractional calculus. Whereas, in the present study Adomian Decomposition Method (ADM) which is suitable for all types of linear and non-linear diﬀ erential equations is used. The ﬂ ow of magnetized blood in an inclined cylindrical tube has been studied by using the ADM. An external magnetic ﬁ eld and an oscillating pressure gradient drove the blood ﬂ ow. ADM algorithm has been developed and used to ﬁ nd the Adomian solution. Computer software MATHEMATICA has been used to visualize the inﬂ uence of various ﬂ ow characteristics such as Hartmann number ( Ha ), diﬀ erent radial locations and angle of inclination on the Adomian velocity. Due to the Lorentz eﬀ ect and central radial location, the results show that the magnetic ﬁ eld diminishes the velocities of blood. Meanwhile, progressive inclination angle enhanced the blood ﬂ ow.


Introduction
The term "magnetohydrodynamics" (MHD) is formed from the words "magneto" (magnetism) "hydro" (water), and "dynamics" (mobility).Hannes Alfven pioneered the ield of MHD [1], for which he received the Nobel Prize in Physics.The MHD nanoliquid low is evaluated by taking it with the magnetic and aluminum oxide nanoparticles.The variable wall temperature was considered.The velocity of the ferro-luid is strongly dependent on the viscosity and thermal conductivity along with the Lorentz's forces.Fluids that are impacted by magnetic ields are known as magnetic luids.Blood is the most common biomagnetic luid that may be handled as a magnetic luid since erythrocytes contain hemoglobin particles, which are iron oxides that are only found in adult blood.Furthermore, the condition of oxygenation has an impact on its magnetic property [2,3].Blood is a viscous crimson liquid that circulates via blood arteries.It is one of the fundamental components that make up the human body and has a powerful nourishing in luence on it.Blood is a mixture of white, red and platelets cells suspended in consistent solution known as plasma.Plasma includes 7% of the major proteins and 90% water [4].
The luid low through an inclined cylinder has tremendous applications in industry and engineering.This type phenomena are very important in cooling electronic devices, in a gas turbine during cooling the wall of combustion chamber, lows in the atmosphere and oceans, automobile demister, during an emergency shutdown of atomic reactor, boilers, defroster system, in the system of solar energy and in a low velocity situation heat exchangers set [5].It is clear that in biological mechanisms, not all capillaries are horizontal, and only a few are slanted.The blood artery is represented as an inclined cylindrical tube exposed to an external magnetism in the current study, which was inspired by the previous investigations.The luctuating pressure differential in the z-direction drove blood low.
Previously the blood low model was solved in the presence of magnetic particles by using CF fractional order derivative with non-singular kernel which has limitation like it fails to satisfy the fundamental theorem of fractional calculus [6][7][8][9].Since they not do allow the reality of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which assess an abnormal regulation on differential https://doi.org/10.29328/journal.ijpra.1001065equations and ideal where these derivative can be used.Where as in the present study ADM is used which is suitable for all types of differential equations [11,12] linear, non-linear homogenous or inhomogeneous.The Adomain decomposition point of view was used to manage collections of linear and non-linear problems.
The research of MHD is extremely bene icial in biotechnology, particularly in the areas of Magnetic Resonance Imaging (MRI), cancer therapy and heart attack [10].A mathematical formalism of blood low in a microscopic artery exposed to a magnetic intensity has earlier been explored [11].Their outcomes showed that as the magnetic pitch grew, the low and velocity rate decreased.The blood low in a catheterized stenosis artery exposed to slip velocity while a transverse magnetic ield was present was investigated [12].It was discovered that when the intensity of the transverse magnetic ield rose, the wall shear stress increased.Gold nanoparticles were used to study MHD blood low [13].In a hollow blood artery, the blood was administered as a nano luid.The least square technique (LSM) and the RK4 method were used to tackle this issue.The velocity was observed to decrease due to the transverse action of the magnetic ield.The in luence of magnetic ield on a blood low inside oscillatory arteries was studied using a fractional derivative-based mathematical model [14].Tassaddiq [15] analyzed the upshot of nanomaterials on the thermal ef iciency of nanomaterials using heat low model (micropolar luid).Vaidya, et al. [16] inspected the effects of various features on Bingham luid in MHD peristaltic transmission.Dynamical forces acting on the conducting luid cause the luid stream to shift.Since this idea focuses on medicine transfer and blood low management during procedures, the impact of MHD low with radiative and permeable boundary conditions is addressed.Alghamdi, et al. [17] used blood as a transportation luid to study the laminar low of silver and aluminum oxide hybrid nanoliquids within two impermeable channels.In this study, the medicine administration strategy, as well as the microcirculatory system's low dynamic mechanism, may be altered.Under convective conditions, Nisar, et al. [18] investigated how wall suppleness changes MHD peristaltic low through nanoliquid.
Blood is a liquid-like luid that circulates all across the human body and that of animals.Because the luid in this study is presumed the cleanest blood low under the magnetic force through an inclined cylinder, it is noticed that the blood thickness is not static according to medicinal perceptions and that it can regulate the body compression, hemoglobin percentage, temperature, and vessel/artery dimension.As a result, we may consider blood to be a Newtonian luid.In order to explore the various bloodstream con iguration, a numerical approach based on Newtonian blood low conditions and stable state was devised [19].It was discovered that different functionalities could in luence hemodynamic factors.The gold-blood principle was initiated over a longitudinal upper side of a parabolic [20].Cuo blood transportation was studied theoretically through a stenosis artery with speci ic hemodynamic properties [21].For biomedical sciences, a numerical estimate of blood in human blood vessels was replicated using Newtonian techniques with heat instability [22].The low of nanoliquids over an irregular channel was compared [23].The erythrocyte cell is a signi icant magnetic component that has the potential to in luence blood low via arteries.By supposing magnetic in luence on blood low, a theoretical conclusion was obtained [24].Conducting luids are mostly employed for cellular partition, MRI or just as a con licted agent, active substantial measurement, therapy of tumor infected cells, MHD micro pump and in the biomedical area [25].Rana, et al. [26] took Williamson's luid blood and has studied the expanding and shrinking surface.Khan, et al. [27] used entropy to analyze the brain and its abnormalities, whereas gold used nanoparticles to cure cancer.Following up on the nature of blood's thermo physical property and the use of gold nanoparticles on curved shrinking/stretched surfaces.Some arteries in the general biological system are not precisely level or vertical.As a result, while calculating low in sloping arteries, the impact of gravity should be taken into account.An angled, catheterized, and overlapping stenosis was studied for its unstable non-Newtonian blood low [28].The blood was represented as an incompressible conducting luid moving in an asymmetric inclined channel.To tackle the problem, the authors employed the analytical perturbation approach.According to indings, as the inclination angle grows, the pressure gradient enhance.Jamil, et al. [29] explored a non-Newtonian conducting bloodstream in an inclined stenotic artery using the CF fractional derivative without a singular kernel.The impact of a diagonal magnetic pitch on unstable blood low via an inclined porous media with the pulsatile pressure gradient was explored by Shah, et al. [30].

Methodology
As depicted in Figure 1, blood low is in a cylindrical tube.A homogeneous magnetic ield is employed orthogonal to the low direction while blood lows in an axial direction.At t = 0, eq. ( 10) can further be simpli ied by taking, Where Appling 1 t L  on both sides of eq. ( 11), and using the initial condition stated in eq. ( 8), where we get: First inding u 1 from eq. ( 14) Now inding u 2 by putting n=1 in eq. ( 14), we get And using u 1 from eq. ( 17), we have  (20) Now inding u 3 by putting n = 2 in eq. ( 14), and using u 2 from eq. ( 20), we have the blood motion begins at rest, and the no-slip condition at the tube's wall is applied, resulting in the blood having zero velocity at the tube's wall.The Navier-stokes equations, which describe blood low, and Maxwell's relations, which describe the magnetic ield, are the governing equations.
The problem entails solving mutually the momentum equations for luid low and Maxwell's for the magnetic ield connections.
By Ohm′s law, the current density j  is stated by Where  are the electrical conductivity, electric ield intensity, magnetic lux intensity and velocity vector.Maxwell′s equations are: . 0, , , 0 Where μ 0 is the magnetic permeability and F m   is the electromagnetic force characterized as The pulsatile low in a cylindrical tube of blood with radius 'r' is thought under the effect of the uniform diagonal magnetic ield.The modeled momentum equation can be rebound as: Here ρ is the luid density, p is the pressure, u is the luid velocity.

cos , 0 1
Where A 0 and A 1 present the constant amplitude and pressure gradient, giving rise to diastolic and systolic pressure.
We present the subsequent dimensionless parameters and variables: Where Ha Ba    (Hartmann number).
Using the above parameters and dimensionless, Eqs.(4), can be inscribed as The initial condition is: ADM operator form of eq. ( 7) can be written as, Let us assume that, then we can write eq. ( 9), as (10)   Finally combining all the previous results obtained in eq. ( 14), ( 7), ( 10) and ( 22), we have,

Skin friction coeffi cient
Skin friction is friction of blood that is against the artery membrane.

 
Since, in our study we are considering external pressure gradient as given, so one of possible solution of the skin friction is,

Results & Discussions
In this segment, the properties of different constraints on Adomian based solutions of unsteady and incompressible magneto-hydrodynamics blood low in an inclined cylindrical channel are inspected, discussed and graphically explained. .
ADM solution versus different low parameters are plotted in the Figures 2-4.Moreover, to explain the effect of radial axis   Figure 2 is showing blood velocity at different radial r components against t.Cylindrical channel low at different radial positions is inspected.Initially for t ≤ 0.2 blood low velocity is almost same at all the radial locations but for t > 0.2, it is maximum at the boundary then as compared to the central location.
In Figure 3 external magnetic ield effect on the blood low is investigated.Enhanced low behavior was recorded, due to reduction of blood viscosity.But later own due to the gradual enhancement of Lorentz force the normal blood low velocity become decreases.We can conclude that by applying the suf icient amount of external magnetic ield we can control the blood low.This inding might be bene icial in some medical treatment practices.
The velocity pro ile u(r, t) at different inclination angles φ are plotted in Figure 4.It was noted that the luid is faster with the arterial inclination, as compared to the non-inclined artery.

Conclusion
The blood low through an inclined cylinder has been exposed to a transverse magnetic ield has been studied using a mathematical model.Previously the mathematical model was solved in the presence of magnetic particles by using Caputo-fabrizio (CF) fractional order derivative with nonsingular kernel.The semi analytical ADM technique was used to solve the controlling non-dimensional PDEs and to create a velocity pro ile for blood low.Initially, the velocity of blood increases due to reduced viscosity but later own decreases due to Lorentz force with Hartmann number.Nonetheless, when the inclination angle increases, the velocity increases.The numerical indings reveal that the inclination angle affects the low velocity inside the cylindrical artery signi icantly.This discovery might aid in the identi ication and treatment of certain medical conditions.Furthermore, the recent discovery may lead to improved pad and machine designs.

Figure 1 :
Figure 1: Blood fl ow through an inclined cylindrical channel.


Whereas, in Figures2-4, graphs of physical low parameters of interest contains different radial axis (0.2≤r≤0.5), rexternal magnetic ield (0.2≤Ha≤0.5) and inclination angle ( Pi Pi Pi Pi .In our model we only impose initial condition on the governing equation and there is no condition on the boundary due to which all the curves are not converging to the single point, where the boundary condition is used, due to which all the plots are converging.Other luid low constants

Figure 2 :
Figure 2: Velocity plot against t for the various values of r.

Figure 3 :
Figure 3: Velocity plot against t for the various values of Ha.

Figure 4 :
Figure 4: Velocity plot against t for the various values of φ.