Rotation and Magnetic Force Effects on Peristaltic Transport of Non -Newtonian Fluid in a Symmetric Channel

In this paper, the impact of magnetic force, rotation, and nonlinear heat radiation on the peristaltic flow of a hybrid bio -nanofluids through a symmetric channel are investigated. Under the assumption of a low Reynolds number and a long wavelength, the exact solution of the expression for stream function, velocity, heat transfer coefficient, induced magnetic field, magnetic force, and temperature are obtained by using the Adomian decomposition method. The findings show that the magnetic force contours improve when the magnitude of the Hartmann number M is high and decreases when rotation increases. Lastly, the effects of essential parameters that appear in the problem are analyzed through a graph. Plotting all figures is done using the MATHEMATICA software.

of tiny blood vessels, blood pump in the heart-lung machine, and the transport of urine from the renal to the bladder [4]. There are some academic research has investigated the peristaltic flow process of nanofluids, and some of these researches, including . [5] discussed the peristaltic transport affects the magnetic field and thermal properties of copper-water nanofluids. [6] discussed Zinc Oxide nanoparticles moving via tapering arteries while suspended in blood and subject to magnetic effects. [7] examined the impacts of various nanoparticle types on the peristaltic transport of nanofluid. [8] studied the impact of the apply magnetic field on heat radiation and the magnetic force on gold and copper nanoparticles in peristaltic flow. [9] discussed the effect of nanoparticles in the motion of blood in a vertical channel. [10] discussed the peristaltic of couple stress nanofluid organized by the presence the electrical field and magnetic field into micro channel.
The rotation phenomenon has several uses in cosmic and geophysical processes .It can be used to understand when galaxies arise and the oceans circulate. The orientation of nanoparticles in fluids is explained by rotational diffusion.The peristalsis of magnetic field fluid in the presence of rotation is significant in some flow situations involving the motion of physiological fluids, such as saline water and blood. The magnetic field and rotation are beneficial for the movement of bio fluids through the intestines, ureters, and arterioles. Numerous scholars have been interested in the impact of rotation and the peristaltic transport mechanism since they were examined by [11][12][13][14][15][16][17].
This study intends to investigate how both magnetic force and rotation affect the peristaltic transport of hybrid bio-nanofluids through a symmetric channel. The precise solutions for magnetic force, stream lines, heat transfer coefficient, temperature, and velocity have been obtained utilizing the Adomian decomposition technique. Graphs are used to illustrate physical characteristics that affect the flow.

Adomain Decomposition Method (ADM)
This method can be utilized to solve linear and non-linear differential equations as well as integral equations, and it produces better results than other methods. George Adomian first proposed this method, and it has since been used to solve a variety of problems [18 and19]. To illustrate a general overview of the Adomain Decomposition Method, let's consider a form equation [20][21][22][23].
Where L is invertible linear operator, N represent the nonlinear terms, R is reminder of the linear operator. Applying the L −1 on both sides in equation (1), yield: Now L −1 represent the n-fold integration for nth order L,by using the given conditions all are assumed to be prescribed ,the ADM define the solution: Where the 0 , 1 , 3 , … recursively determined by using the relation : The last term of the Eq. (3) can be computed by substituting : Where represent generated Adomain polynomials for the specified nonlinearity, they depend only on the 0 , … , components and form rapidly converge series, the are given: And can be found from the formula :

Problem Formulation
Let's consider the peristaltic flow of an electrically conducted hybrid bio-nanofluid in the presence of rotation and magnetic field, blood is considered in this research as a basic non-Newtonian fluid, while copper and gold are considered nanoparticles in a 2-D channel of width 2 1 . A (X ̅ , Y ̅ ) Cartesian coordinate system. We selected X ̅ in the direction of wave propagation and Y ̅ transverse to it, a constant magnetic field of strength 0 acting in the transverse direction results in an induced magnetic field ( ̅ (X ̅ , Y ̅ , t ̅ ), ̅ (X ̅ , Y ̅ , t ̅ ) + 0 , 0). H * refers to the total magnetic field composed of ( ̅ (X ̅ , Y ̅ , t ̅ ), ̅ (X ̅ , Y ̅ , t ̅ ) + 0 , 0). The channel wall's geometry is depicted by: The flow of the fluid is induced by the infinite sinusoidal wave that travels along the channel walls at a wave speed of C and a wavelength of . Where 2 is the wave amplitude, 1 is the half channel width, and t ̅ is the time. The governing for the problem are: Where V is velocity vector, is electrical conductivity, J is current density; E is electric field ,and is magnetic permeability. The governing equations for an incompressible, unsteady, hydromagnetic, viscous bio-nanofluid (Au-Cu Nanoparticles) include momentum, induction, and heat [24and 25]: Where P is fluid pressure, = , represent the magntic diffusivity, is the electrical conductivity,T represent the temperature distribution, = − 4 * 3 * 4 is the radiative heat flux, * and * are mean absorption coeffient and Stefan Boltzmann constants, Q 0 is the heat source, experimental formulation for the physical characteristics of hybrid nanofluid [25] are shown in Table 1, which are given by: Table 1 Physical properties of hybrid nanofluid .
property Nanofluids Thermal conductivity k nf = ( Where ∅ 1 is the volume fraction of gold nanoparticles, ∅ 2 is the volume fraction of copper nanoparticles, and m represents the shape of a factor of the indicated nanoparticles, respectively. The physical characteristics of nanoparticles are classified in the Table 2. The velocity Choose a wave frame( ̅̅̅̅ , ̅) that moves at a speed of C away from the fixed frame by the following transformation given below.
The governing equations were simplified to take the following form: Using the non-dimensional quantities listed below: In the preceding expressions is temperature at upper wall , is temperature at lower wall , ̅ is stream function, is strength of the electric field, is wavenumber, Reynolds number, Ω is rotation, is magnetic Reynolds number, isEckert number, is Prandtl number, ̅ magnetic force function, M is Hartmann number, is temperature in the nondimensional form , is internal heat generation, and S is Strommer number respectively, the nondimensional form of the peristaltic wave can be expressed by : Introduction of dimensionless magnetic force function and stream function by using the relations : Substituting Eq.(16) into Eqs. (12 -15), giving us the equations:   ) 2 ) + ( 4 ) + (20) ℎ is the sum magnetic and ordinary pressure, which is the total pressure. The corresponding stream function, temperature function, and magnetic force function boundaries for non-conductive elastic walls in the wave frame are as follows: Where q represent mean of flow rate. Using the long wave length approximation and neglect the wavenumber along the low Reynolds number, one can find from Equations (17)- (20) That: By using cross derivation to eliminate the pressure from Eq. (22) and Eq.(23), giving us the equation: Join Eq.s (24) and (26), become :
The equation for the coefficient of heat transfer at the wall is: So,the electric field E is obtain from : . .

Discussion and Graphs of the Results
The purpose of this section is to examine the graphic results of a variety of important parameter utilized in the specified modeling. In particular, streamlines, velocity, induced magnetic field, magnetic force contours, heat transfer coefficient, and temperature distribution are depicted in the figures below. We examined numerous cases, including the Hartmann number M, the magnetic Reynolds number Rm,and rotation Ω. The numerical variables were chosen based on previous literature [26][27][28], and flow trapping is discussed graphically. All figures are plotted using from the MATHEMATICA program.

Velocity Profile
Figure1 depicts the velocity changes with respect to the axial y for various Hartmann number M, rotation, and dynamic viscosity values µ. The effect of Hartmann number M on velocity is shown in Fig.1a. As M increases, the velocity u near the channel's middle rises, whereas the opposite occurs near the peristaltic wall. This fact is related to the Lorentz force, which occurs when an external magnetic field is used and, in turn, leads the fluid motion to slow down. It demonstrates that the Lorentz force is much stronger near the wall than in the channel's middle. Display the effect of dynamic viscosity on velocity in Fig.1b. It is noticed that when µ increases, the velocity along the channel walls slowly decreases, whereas it increases at the channel center. Fig.1c displays the rotational effect. It can be seen that as rotation increases, the velocity u decreases near the channel's middle, whereas the reverse behavior can be seen near the peristaltic wall.

Induced Magnetic Field Profile
This subsection describes an examination of induced magnetic field . Graphics have been drawn to show the development of the induced magnetic field profile in Figure 2.
To study the impacts of magnetic Reynolds numbers Rm, Hartmann number M, and rotation Ω on the induced magnetic field with respect to axial y. We are carried out from Figure 2a-2d.The effects of magnetic Reynolds number Rm, and magnetic field M on the induced magnetic field profile can be observed in Figs.2a and 2b. It is noticed that the relationship between and y is inversely proportional to each other , with the rise in Rm and M in the region ≤ 0 ,the induced magnetic field decreases and while in the region ≥ 0, the induced magnetic field rises with the increase in Rm and M. The impact of rotation Ω on the induced magnetic field is seen in Fig.2c. At ≥ 0, the induced magnetic field increases as rotation Ω increases, but it decreases at ≤ 0.  Fig.3b, this is because heat generation form friction brought by shear in the flow is more pronounced when the fluid is highly viscous or moving quickly. Same behavior is observed for Eckert number that can be shown from Fig.3c As the rotation parameter values increase, the temperature profile is seen to decrease Fig. 3d.
Influence of heat absorption β is depicted in Fig. 3e the temperature profile decreases in the center of the channel and merges for near the wall. Fig.3f shows the impact of the radiation Rd on the temperature profile , initially, the decreases and then merges near the wall.

Heat Transfer Coefficient
The behaviors of Prandtl number pr, heat absorption β, Eckert number Ec, rotation Ω, and heat radiation Rd on heat transfer coefficient have been noticed through Fig.4. Fig.4a shows that when rotation increases, the heat transfer coefficient between the fluid and the wall of the channel decreases. In Fig.4b depicts the behavior of the heat transfer coefficient for various values of heat radiation Rd. An increase in the heat transfer is observed for rising values of the heat radiation Rd. The impact of Eckert number Ec and Prandtl number Pr on heat transfer coefficient, it is noticed from Fig.4c and 4d that heat transfer coefficient enhances for higher values of Ec and Pr.

Trapping Phenomenon
Typically, streamlines have the same shape as a boundary wall in the wave frame. However, under certain situations, some stream lines split and enclose a bolus, which moves as a whole with the waves. "Trapping" is the term for this phenomenon. In order to investigate the impacts of the trapping phenomenon at various values of Hartmann number M and rotation Figs.5 and 6 were drawn. It can be shown from Fig.5 that as Hartmann number M increases, the trapped bolus rises in size. The impact of rotation Ω on trapping can be noticed from Fig .6, it is shown that when the value of Ω increases, the size of the trapped bolus diminishes.  Fig.7 illustrates that rotation affects the magnetic force contours; it is observed that as rotation increases, the magnetic force lines diminish and the size of the bolus changes. Figs.8 and 9 illustrate how magnetic Reynolds number Rm and Hartmann number M influence the magnetic force contours. As the values of M and Rm are raised,the magnetic force contours move themselves forward and the magnetic force gradually increases, as can be seen in these figures.

Conclusion
The impact of magnetic force ,rotation,and nonlinear heat radiation on hybrid bio-nanofluids peristaltic flow in a symmetric channel under the influence of a magnetic field are discussed. The governing equations representing momentum, Maxwell, and heat equations are considered. The exact expressions for velocity, heat transfer coefficient, stream lines, temperature, induced magnetic field, and magnetic force are obtained by using the Adomain decomposition method. The major results can be summarized as follows: • The behavior of the velocity distribution u near the center part of the channel increases with a rises in Hartmann number M, while rotation parameters oppose the velocity. • Induced magnetic field exhibits dual conduct in the two zones ≤ 0, ≥ 0 against all the important parameters. • The temperature profile in the Prandtl number Pr and the Eckert number Ec is improved.
• The temperature profile decline when higher values of rotation.
• The heat transfer coefficient raises with rising values of the radiation parameter, Prandtl number, and Hartmann number M, and decreases with rotation.
• The trapped bolus rises in size at increasing of Hartmann number M and decline at increasing of rotation. • The contours of magnetic force improve because of an increase in the magnitude of Hartmann number M. • The contours of magnetic force decrease because of an increase in the magnitude of the rotation Ω. Finally, we conclude that the findings of this paper should be beneficial for researchers aiming to develop fluid mechanics as well as those in biomedical engineering and technology.