The Effect of Rotation on the Heat Transfer of a Couple Stress Fluid in A Non-uniform Inclined Asymmetrical Channel with Inclined MHD

The purpose of this research is to investigate the effects of rotation on heat transfer using inclination magnetohydrodynamics for a couple-stress fluid in a non-uniform canal. When the Reynolds number is low and the wavelength is long, math formulas are used to describe the stream function, as well as the gradient of pressure, temperature, pressure rise and axial velocity per wavelength, which have been calculated analytically. The many parameters in the current model are assigned a definite set of values. It has been noticed that both the pressure rise and the pressure gradient decrease with the rise of the rotation and couple stress, while they increase with an increase in viscosity and Hartmann number. The explanation of parameters is shown graphically by a series of figures using "MATHEMATICA".

easy to see how the walls slide. In technology, boundary slip condition fluids can be used to polish artificial hearts. See [8][9][10][11][12][13] for a list of studies that use this condition.
Researchers have recently looked into how a magnetic field and rotation affect how fluid moves through an asymmetric channel, Abd-Alla and Abo-Dahab [14]. The effects of rotation and MHD on an asymmetric channel through a porous medium where a Jeffrey fluid flows nonlinearly have been looked at by Abdulhadi and Al-Hadad [15]. In porous media with non-symmetric canals, rotating waveform motion in two-dimensional channels of non-Newtonian fluid was explored by Alshareef [16]. For viscoplastic fluid and variable viscosity on peristalsis, they discussed the influence of rotation of the mixture on convection heat transfer analysis by [17,18].
This study's objective is to examine the effects of rotation on heat transfer with inclination magnetohydrodynamics in a non-uniform channel containing a couple stress fluid. The equation for the governing equation is examined using low Reynolds approximations and long-wavelength assumptions, respectively. Graphs are used to show the impact of various factors on fluid flow.

mathematical Formulation
Consider that the effect of a magnetic field and rotation on heat transfer can be explained by a porous medium with an inclined asymmetry of the couple stress fluid.
Let's say * ̅ =ℎ 1 (X ̃ , t ̃ ) and * ̅ =ℎ 2 (X ̃ , t ̃ ) here is a representation of the channel's top and bottom walls. (1) Where 1 * 2 * are the amplitudes of waves λ is wavelength, ̃ is the time c is wave speed , (0 ≤°≤ ) the phase difference °b etween the walls of the channel the rectangular Cartesian coordinates are used ̃a nd ̃. The channel's axis is measured by ̃, and the transverse axis is measured by ̃, which is perpendicular to ̃. The constant heights of the upper and lower walls of the channel from the central line are denoted by 1 2 , respectively. It's worth noting that °= 0 corresponds to an asymmetry with out-of-phase waves, whereas °= relates to waves that are in phase. Furthermore, 1 , 2 , 2 * , 1 * and ° satisfy the condition; 1 * 2 + 2 * 2 + 2 1 * 2 * °≤ ( 1 + 2 ) 2 .
The governing equations.
Where V ′ ⃑⃑⃑ =(̃, Ṽ, 0) be the velocity vector, Ὼ is rotation, ̇ the Coefficient of thermal expansion , ̃ is the fluid pressure , is density of fluid , Φ is the inclination of the magnetic field angle, μ the dynamic viscosity , η is a constant linked to the couple stress, is the specific heat at constant pressure. impact B * ⃑⃑⃑⃑ =( * Φ , * Φ ,0) the magnetic field vector, ° is the permeability parameter ,T is the temperature , is the acceleration by gravity , ̃ is the fluids electrical conductivity ,K is the thermal conductivity . The induced electric field is not taken into consideration at all, because assuming a low magnetic Reynolds number The fixed frame's(̃ , ̃) flow field and the wave frame's(̀ , y) wave field are also considered the motions of an unsteady and steady-state . It's important to think about the relationship between the wave (̀ , y)and the fixed frames (̃ , ̃) with a velocity of c as they move apart from one another as a result of the transformations below . v̀=̃ , y= Ỹ , T = T , x̀ = ̃−, ù =̃− , ̀=̃ In which (u, v) and (Ũ, Ṽ)are the waves' and laboratories' velocities, respectively. the governing equations (3) ,(4) , (5), and (6) be expressed in Wave frame as using the transformations mentioned above.
In terms of stream function ° , the dimensionless variables are specified in equations. (9), (10), and (11) were translated into the following equations.
. 3 Cross differentiation from dimensionless equations is used to remove the pressure term. (17), (18), and (19) may be formulated in the context of stream function differential equation and temperature under low Reynolds approximations and long-wavelength assumptions(δ<< 1) .
The dimensionless boundary conditions in the wave frame are [5]: As a result of solving equations (18)   The axial pressure gradient can be calculated once we've identified the stream function.
In non-dimensional form, the pressure rise per wavelength Δ * is defined as

3.Volumetric flow rate
In the laboratory frame, the volumetric flow rate is equal to whereℎ 1 andℎ 2 are functions of ̃ and ̃ .
In the wave frame, the volumetric flow rate is calculated: The relationship between and can is calculated as follows: The time-mean flow over a span of time * fixed in place ̅ as =°+ 1 + 2 + 2 tañ+ 1 cos(2 ) + 2 cos(2 +°) (31) The non-dimensional of equation (28)

Results and Discussion
This section focuses on " velocity"W, "temperature" θ°, " gradient of pressure " δP * , "pressure rise" Δ * , and "stream function" ° . To get the numerical values corresponding to the abovementioned analytical formulas, we utilized "MATHEMATICA " program.

The distribution of velocity
Now, we'll look at the effect of axial velocity (W) variation across the canal on several parameters. Note that the velocity is parabolic. In the center, the axial velocity rises with increasing rotation parameter " ̀" , density "ρ", the width of the channel " 1 ", Darcy number " " and the magnetic field inclination angle "Φ", whereas the axial velocity reduces near the channel wall's edge, as seen in Figure 1 (a-e). According to Figure 1(f) "°", is shown to increase axial velocity in the middle of the canal while axial velocity lowers at one wall boundary and rises at the other.

Pumping characteristics
In this subsection, we will analyze the pressure through the channel.

pressure gradient"
* " Each of the figures in this section illustrates the pressure gradient and the axial axis of fluctuation along the canal at one wavelength ∈[0,1]. Figures like this show that flow is restricted in the narrowing portion of channel ∈[0.2,0.6]. As a result, a larger pressure gradient is needed to achieve a normal flow. Because of the smaller pressure gradient in the larger area of the channel ∈ [0,0.2]∪ [0.6,1] fluid may pass readily. Figure 2(a-b) shows that when viscosity " " and the Hartmann number" ℋ" raises, the pressure gradient also increases in size. This graphic shows that greater pressure is required to move some volume of fluid through the narrower region of the channel. When looking at channel parameters such as the magnetic field inclination angle "Φ" , the inclination" ̃" of the channel, rotation parameter " ̀" ,density " " ,width of the channel " 1 " , Grashof numbers "G " ,slip parameter" ", Darcy number " " , Reynolds number" ",couple stress " " , temperature "θ° ", the phase difference " °" and Non-uniformity " " as depicted in Figures 2( c -o), the pattern is the opposite. Because of the inclination, less pressure is needed to move liquids.

pressure rise"
* " Figures 3(a-o) show the relationship between pressure rise and volumetric flow rate. The connection between pressure rise and the rate of volumetric flow for each wavelength is seen to be linear. The whole region is considered into five parts (1) the peristaltic pumping region where(∆ * > 0, > 0) (2) when (∆ * > 0 , < 0), then it is a retrograde pumping region (3) augmented pumping (co-pumping) region where (∆ * < 0 , > 0) (4) There is a co-pumping region where (∆ * < 0 , < 0) (5) (∆ * = 0)corresponds to the free pumping region. Figure3 (a-b) shows the impact of viscosity " " and Hartmann number " ℋ "on ∆ * on It has been noticed in the retrograde pumping region. The pumping rate increases as Hartmann number "ℋ"increases. Co-pumping reduces when in this region and increases in " ℋ "and " " . Volumetric flow rate and pressure increase ∆ * are shown to be linearly related in Figures 3 (c -f ) for a variety of Grashof numbers "G " , " ", " ̃" and "θ° ". We observed that a rise results in an increase in retrograde pumping rate and augmented pumping (co-pumping) region an increase in the pressure rise. Figures 3 (g -m ) Graph shows that with an increase in rotation parameter " ̀" ,density " " ,width of the channel "d 1 ", " ", " ", " " and" Φ " . the pumping rate decreases in the region of retrograde pumping while in the augmented pumping region found to increase. Figure 3 (n)shows an impact of channel non-uniform parameter " "on ∆ * . It is observed that in a retrograde pumping region and a free pumping region (∆ * = 0), the pumping rate decreases with an increase in " ". Figure 3 (o) depicts the effect of "°" on ∆ * . It is observed that in a retrograde pumping region and a free pumping region (∆ * = 0), the pumping rate increases with an increase in"°" .

temperature profile "° "
An examination of the fluid temperature profile for a fixed value of x = 1 yields a parabolic temperature profile, with a higher graph in the middle. Fluids that have a high viscosity are more likely to convert kinetic energy into internal energy, which causes them to get hotter. There is no doubt about that. Its flow and movement resistance diminish with increasing temperature, a phenomenon caused by the tiny distances between molecules and the cohesive forces that exist between them. Figures 4 (a-g) show that the temperature increases values " ̀" , " " , "d 1 " " " ," H " , " "and " "increase, so does the temperature in the channel's center and temperature parabolic. The increase in temperature is accompanied by " ".

The trapping phenomena
In order to explain the trapping phenomena, the formation of a circulating bolus of fluid, which is a closed streamline region, at the speed of the wave. There will be points in the wave frame where the fluid's velocity is zero due to the trapping phenomenon. The volumetric flow rate via a line linking any two places is computed by taking into account the difference in stream function values at the two sites in question, which is why studying streamline patterns is so important. Figures 5-15 indicate that bolus formation occurs from both sides of the center line in the extended region. Bigger and larger volumes of the trapped bolus can be extracted from the system by raising the strength of the magnetic field in Figures ( 5-9) (a, b, and c), as "Φ " , " ̀" , " " , "d 1 " and " ". In Figures (10 and 11) (a ,b and c), as " °" and " " changes, so does the variation in streamlines. According to our research, we've found that bolus sizes decrease" " and " °" increase. The Figure 12(a ,b and c) shows that the wall draws fluid in the widest section of the duct, but this fluid is pushed away from the wall in the narrower section and the bolus disappears in the central part as " " increases. Graphing the non-uniformity parameter " "of the asymmetric channel as shown in Figure 13(a, b, and c), when it is raised, the trapped bolus decreases in size and migrates downstream. These figures 14and15 (a, b, and c) demonstrate how In increasing the Hartman number " ℋ " ,and couple stress " " , the incidence of trapped bolus diminishes in size and vanishes in the direction of downstream, Another effect that may help protect red blood cells and other elements is the tendency to reduce bolus volume.

Conclusions
In this research, we studied the effects of rotation and incline magnetohydrodynamics to investigate the effects of heat transfer and couple stress fluid as they move via an inclined asymmetric channel and porous medium under low Reynolds approximations and longwavelength assumptions in the transport of bodily fluids by the use of non-Newtonian fluid models. Analytically, using Mathematica software. This investigation focused on studying the distribution of velocity, the pumping characteristics, the distribution of temperature, and the trapping phenomena.