Effects of Viscous Dissipation and Joule Heating on Micropolar Hybrid Nanouid in a Stretching/Shrinking Channel including Thermal/Solar Radiation

: This study aims to investigate the behavior of the micropolar hybrid nanofluid (Cu-TiO 2 ) in a stretching and shrinking channel while maintaining a continuous laminar flow. External effects from magnetic field, thermal and solar radiation are also considered in the model. The governing differential (momentum, micro-rotation, and energy) equations are transformed into dimensionless forms via similarity transformations based on the boundary layer theory. The resulting ordinary differential equations are solved numerically using the fourth-order Runge-Kutta-Gill technique with the shooting method. The effects of nanoparticles transport on fluid flow and heat transfer are discussed and compared to the pure water case. The results are presented in tables or graphs such as velocity, temperature, angular-velocity profiles, and concentration.


Introduction
On the other hand, micropolar fluids have a microstructure, including fluids having nonsymmetrical stress sensors. They are essential in hydrodynamic-fluid problems because they are made up of randomly oriented particles suspended in a viscous medium. Micropolar fluids have a wide range of applications, such as extrusion of plastic sheets, paper production, and glass blowing [3,4,5]. Eringen [3] was the first to formulate the theory of micro-polar fluids. Srinivasacharya et al. [6] reviewed the entropy production and heat transfer of micro-polar fluid flow in an annulus with a magnetic field. A computational study into the heat and mass transfer of a micro-polar fluid flow over a stretching surface was conducted by Mabood et al. [7]. Many other researchers have studied micropolar fluid flow researchers [8][9][10][11][12][13]. Ellahi et al. [14] investigated mixed convection nanofluid flow over a wedge in a porous medium.
Akbarzadeh et al. [15] have studied laminar nanofluid flow and force convective heat transfer within a wavy tube. Ellahi et al. [16] produced a two-dimensional heat transfer flow of a nanofluid in the vicinity of a stagnation stage. Sheikholeslami and Ganji [17] studied the hydrothermal activity of nanofluid flow between two parallel plates in the presence of a magnetic field. The magneto-hydrodynamic nanofluid flow between parallel sheets was studied by Sheikholeslami and Ganji [18]. Rahbari et al. [19] conducted an analytical analysis of blood flow, including nanoparticles, through porous vessels. EndoScopic study of peristaltic blood flow of Sisko fluid with Titanium magneto-nanoparticles through a uniform tube was performed by Bhatti et al. [20].
Bhatti et al. [21,22] looked at the effects of coagulation (blood clot) and a variable magnetic field on three-dimensional unsteady MHD boundary layer flow of viscous nanofluid containing gyrotactic microorganisms. The combined impact of thermodiffusion and thermal radiation on Williamson fluid was studied by Bhatti and Rashidi [23]. Rashidi et al. [24] investigated heat transfer over a stretching sheet in a nanofluid flow. Sheremet et al. [25] investigated the unsteady natural convection of a water-based nanofluid inside a wavy-walled cavity under the influence of a uniform inclined magnetic field. Free convection was studied by Sheremet et al. [26] in a square differentially heated porous cavity filled with a nanofluid. Sheremet et al. [27] investigated steady-state natural convection heat transfer between non-parallel walls in a three-dimensional MHD nanofluid flow. Flow and heat transfer between two flat plates of magneto-hydrodynamic (MHD) nanofluid by Dogonchi et al. [28]. Dogonchi and Ganji [29] investigated the Brownian motion effect of nanofluid flow and heat transfer between non-parallel stretching walls. Dogonchi et al. [30] studied the unsteady squeezing flow and heat transfer in MHD. The flow of nanofluids near a stagnation point was studied by El-dawy et al. [31]. El-dawy and Gorla [32] examined the turbulent flow of a nanofluid over a wedge. As cooling and heating fluids, nanofluids in minichannels of different configurations are used [33]. In the presence of injection and suction, heat transfer and fluid flow of a pseudo-plastic non-Newtonian nanofluid over a permeable surface have been overcome. Maleki et al. [34] studied nanofluid flow in microchannel heat sinks of similar length and hydraulic diameter by Goodarzi et al. [35].
The effects of hybrid nanoparticles (Cu-TiO2) on micropolar nanofluid flow, heat transfer and concentration in stretching and shrinking channel with radiations and heat source are studied in this paper. The paper is organized as follows. In Sec. 2, the problem is described and the mathematical modeling is presented. Then, the results are presented and discussed in Sec. 3. Finally, conclusion is drawn in Sec. 4.

Problem description
In a two-dimensional tube, consider the steady laminar flow of a micropolar hybrid nanofluid. The channel walls are in two cases stretching and shrinking, and the fluid enters the channel or exits by the uniform speed of v 0 . The temperature of bottom wall is defined by T 1 , while T 2 indicates the temperature of top wall. The channel walls are located at y = ±h (see Fig. 1).

Fig. 1: Schematic diagram of the problem under consideration
It is assumed that there is a thermal equilibrium between the nanoparticles and the base fluid and there is no slip velocity between the phases. The governing boundary layer equations of momentum, energy, micro-rotation and energy are: where u and v are the velocity components, P is the pressure, T is the temperature, ℎ , 0 is the electric conductivity of the fluid, Volumetric rate of heat generation/ absorption, N is the microrotation velocity, k is the vortex viscosity, ℎ = ( ℎ + 2 ) is the spin gradient viscosity, j is the micro-inertia density and g is the gravity, T, TW , T∞ and the temperature of the fluid inside the thermal boundary layer, the plate and the free-stream flow, respectively; , ℎ , ℎ , ( ) ℎ are density, thermal conductivity, dynamic viscosity and specific heat of the hybrid nanofluid respectively, .q rad is the radiative heat flux, kr (reaction rate).
We consider Rosseland approximation to express the for radiation: where σ* is the Stefan-Boltzmann constant and ℎ * is the mean absorption coefficient of the nanofluid. Further, such that 4 may be expanded in a Taylor series. So, expanding 4 about ∞ and ignoring higher order terms we get,  The boundary conditions are: When v0 > 0, suction occurs and when v0< 0, injections occur. Here, s is a boundary parameter and represents the microelements' rotational degree of freedom near the walls.
When s=0, there is no rotation for microelements which are close to walls.
This situation corresponds to concentrated particle flows. Researchers also have considered other particular cases such as s=0.5 and s=1, which represent the weak concentration and the turbulent flow, respectively. We introduce these dimensionless parameters: where T2 =T1 +Ax in which A is constant.
By substituting these expressions into the governing equations and then eliminating the pressure gradient, Eqs. (2)-(5) transform to the coupled system of nonlinear equations: where S is parameter stretching and shrinking S< 0 Shrinking case S > 0 Stretching case, Pr is the Prandtl number, Re is the Reynolds number where Re > 0 corresponds to suction n and Re < 0 corresponds to injection, R is the radiation parameter, M is the magnetic parameter, K is the micropolar parameter. These parameters are defined as below:

Result and discussion:
The governing partial differential equations were transformed into ordinary differential equations. The obtained self-similar equations were solved numerically by using the    Fig. 2, we see the effect of the magnetic parameter M on the velocity in two cases.
First, in the stretching case, the magnetic field parameter increases the velocity is decreasing but, in the shrinking case, the magnetic field parameter increases the velocity is increasing, and we see then decreasing is more than increasing. Fig. 3 describes the effect of the hybrid nanoparticles in angular momentum in two cases. In the stretching case, when the volume of hybrid increases, angular velocity decreases but near a boundary surface of the channel, the angular velocity increase, but in the Shrinking case, when the volume of hybrid increases, angular velocity decreases. Fig. 4 shows the effect of the magnetic parameter M on the angular velocity in two cases.
First, in the stretching case, the magnetic parameter increases the angular velocity, but at a certain point, inverse decreases angular velocity with increasing magnetic parameter. In the Shrinking case, the magnetic parameter increases the angular velocity decreases but, at a certain point inverse, increases angular velocity with increasing magnetic parameter. As the concentration of nanoparticles increases, the temperature is creasing because of the high thermal conductivity of the nanoparticles. Fig. 6 illustrates the effect of the magnetic parameter on the temperature. As the magnetic parameter increases, the temperature is increasing. From Fig. 7, one may observe the effect of the heat source/sink parameter on the temperature. As the heat source/sink parameter increases, the temperature is increasing. From Fig. 8, it can be seen that see the effect of the Schmidt number parameter on the concentration. As the Schmidt number parameter increases, the concentration is increasing. Finally, Fig. 9 illustrates the influence of chemical reaction parameters on the concentration. As the chemical reaction parameter increases, the concentration is decreasing.