Heat-Flux Effect on Fluid Flow and Convective Heat Transfer Through a Rotating Curved Micro-Channel

The present paper investigates heat-flux effect and the dissemination of energy in a rotating bent square micro-channel (MC) subject to a temperature gradient between the vertical sidewalls. The flow structure prevailing the problem is solved by applying a highly accurate spectral-based numerical scheme. The flow controlling parameters are the Dean number (0< Dn ≤5000) and the Taylor number (-500 ≤ Tr ≤ 2000) for curvature 0.01 and the Grashof number, Gr =1000. After applying the arc-length path continuation technique to obtain steady solution (SS) curves, two branches of SS consisting of 2-to 8-vortex solutions are prevailed for the non-rotating case while a single branch with a symmetric 2-vortex solution is for positive rotation of the channel. Unsteady flow (UF) properties are simulated by the time-average of the solutions, and the transitional behavior is well predicted by contemplating the power spectrum and phase spaces of the solutions. Results manifest that the UF experiences a consequence ‘steady-state  multi-periodic  steady-state’ for no rotation of the channel as Dn is increased. For the rotating case, on the other hand, the flow advances in the scenario ‘steady-state  multi-periodic  steady-state’ for negative rotation and only a steady-state solution for rotation in the positive direction. Streamlines and isotherms of SS and UF for various values of the flow-controlling parameters are obtained. Centrifugal force impacts the fluid mixer, which then assists to turn the flow into chaos and prompts to intensify the convective heat transfer (CHT).


Introduction
Flow characteristics especially rotating flows and heat transfer (HT) in a rotating system of different geometrical shapes has attracted a lot of attention from prodigious researchers due to plentiful applications in the engineering fields like heating and cooling system, rocket engine, electric generators, solar energy, and many more (Kockmann, 2005;Yi and Bau, 2003;Schonfeld and Hardt, 2004). The qualitative approach in micro-biochips, micro-fuel cells, micro-coolers, and micro-reactors has been increasing at a rapid cognition due to the quick advancements in nanoelectronics and bioengineering. Since the last couple of decades, research conducted in straight or curved MC has revealed that only a few scientific or numerical evaluations of the analysis of flow behavior in curved MC have been done. (Morini, 2004;Yang et al., 2005;Wang and Liu, 2007). Microfluidic systems typically feature laminar flows, as opposed to the conventional flow processes found in macroscopic devices. Schönfeld and Hardt's (2004) work on curved MC showed that the induced secondary flows generate a mixing effect in micro-devices. The simulated results on helical flows by Jiang et al. (2004) verify the predicted mixing results by a series of experimental evaluations.
Rotating flow features are analyzed in three topics which are the steady solution (SS), unsteady solution (US), and CHT. Curved channel with rotation produces two forces; Centrifugal force and Coriolis force, and these forces directly impact the fluid flow and produce spiraling flow which is also called secondary flow (SF). The first evidence was provided by Dean (1927) where the existence of such kind of problem was analyzed and discussed in detail for motion in a curved tube with a circular cross-section. The main result of this analysis was to introduce the SF which was characterized by the Dean vortices, initiated by Dean in 1927. Since then, several enthusiastic researchers have given deep attention to investigating such types of fluids through different geometric shapes for rotation and non-rotation. Several landmark articles on curved duct flows are mentioned here, for example, Selmi et al. (1994), Alfredsson et al. (1989), Barua et al. (1954), Chengand Wu, (1970, Yamamoto et al. (1999), Yanase et al. (2002), Hasan et al. (2022aHasan et al. ( , 2022b. Analyzing the bifurcation structure, which includes the SS and its linear stability, is one of the most important characteristics of fluid flow in a curved MC. Selmi et al. (1994) investigated the structure of bifurcation for an incompressible fluid in a rotating curved channel (CC). A 6-cell revolving flow phenomenon was discovered by Yamamoto et al. (1999) in their examination. Later, Mondal et al. (2007) and Adhikari et al. (2021) evaluated how the bifurcation diagram altered as the curvature of a non-rotating CC expanded or reduced and demonstrated the precise affinity to transient behavior and the bifurcation pattern among steady branches. Wang and Liu (2007) performed a computational analysis on a completely branching structure and stability in a bending MC with a square bend to inscribe the ramifications of curvature, initial circumstances, and disturbances. They showed thatregardless of duct curvature size, the spiraling flow is always produced in the channel. They observed multiple limiting points and asymmetric branching points in their ten branches of solutions. Other pertinent studies on fluid flow through different geometric circumstances include Chanda et al. (2020Chanda et al. ( , 2021, Yamamoto et al. (2006), Hasan et al. (2019b).
Examining the nonlinear characteristics is important to figure out the unsteady behavior of the flow. Chandratilleke and Nursubyakto (2003) proposed a mathematical method for elucidating the flow characteristics using a variety of geometrical shapes with various aspect ratios. Later, Wang and Yang (2005) investigated the regular oscillations of an incompressible viscous fluid in a curved square channel (CSC) using both numerical and experimental methods, and a comparison of their findings revealed that they were completely consistent. Mondal et al. (2013 demonstrated the relationship between the UF and the bifurcation structure through linear stability. Through analyzing the oscillating behavior computationally by Mondal et al. (2017), regular and irregular oscillating flows were observed under conditions of symmetry and without symmetry, respectively. The majority of the important publications solely focused on weak rotational speed. Because of these questions to be answered, the current work focuses on how gyration and curvature affect the flow of unsteady fluid in a CRC. Chandratilleke and Nursubyakto (2003) studied the role of Dean vortices in fostering HT and demonstrated that CC is more capable than that in a straight duct to gain a better knowledge of thermo-fluid dynamics and CHT through CC. The published simulation results of Yanase et al. (2002) and  show that SFs increase HT in the flow. The numerical method of flow characteristics and HT via a rotating CSC for different curvatures was recently carried out by Hasan et al. (2019aHasan et al. ( , 2019b. Recently, Roy et al. (2020) applied a method that used Hasan et al. (2019a) to prognosticate hydrodynamic unsteadiness and CHT in a rotating CRC of medium curvature. In this study, we present the numerical analysis of the puzzling flow behavior and energy distribution of fluid flow in a tightly coiled square-shaped micro-channel in a rotating system. Flow pattern as well as US of the flow features with CHT with strong rotational speed is investigated for both co-rotating and counterrotating cases of the micro-channel.

Mathematical Model
The geometric scheme that is taken into consideration in this present study is shown in Figure 1. We have into account a fully formed 2D viscous incompressible, and thermally active fluid that passes through CSC. The fixed revolving speed of the system is Ω T . The applied condition to the outside bend of geometry is at high temperature, on the other hand, the inner bend wall is kept at standard room temperature whereas both the upper surface and lower portion of the wall remain adiabatically isolated. It is presumed that a constant pressure gradient controls the fluid flow as shown in Fig. 1. The working equations such as the mass conservation, momentum equation, and energy equation which describe the physical situation of the flow through a revolving duct can be referred to as Chanda et al. (2020Chanda et al. ( , 2021. Those equations are non-dimensionalized by using the following variables, where, is the uniform velocity. Here, d and υ are the representative length and the kinematic viscosity respectively. Since our study is a two-dimensional case, so we assumed . Thus, the transform equations are as follows.

Navier-Stokes equation:
Introducing sectional stream functionψ ψ as follows The processing equations for axial flow, secondary flow, and energy equations are as follows Where, Dn, Gr, Tr, and Pr are the important dimensionless variables that appear in equations (6) to (8), can be defined as; ; and Pr υ κ = .
The no-slip boundary conditions are and the conditions for thermally heated walls are 1 at 1, 1, at 1, The following symmetry criterion is fulfilled by a class of solutions with respect to y = 0.

Numerical calculation
The spectral approach is employed as a fundamental tool since the current study is entirely focused on numerical computation. The expansion of the polynomial functions ) (x n φ and ) (x n ψ , entails the variables in a group of functions comprised of Chebyshev polynomials is represented as

Resistance coefficient
The represented quantities of the resistance coefficient (λ) and the mean axial velocity * w are defined as

Nusselt number
The Nusselt number (Nu) The heat transmission rate from the inner walls to the fluid is not uniform due to temperature differences across the vertical sidewalls. Hence Nu varies on both heating and cooling sidewalls. We define the Nusselt number for the cooled ( c Nu τ ) and heated ( h Nu τ ) sidewalls for the unstable solutions as follows.
1 1 where is the time-average over a period.

Validation of numerical result
This study's underlying algorithm's precision is guaranteed by grid accuracy. In light of this, it is crucial to evaluate the algorithm's effectiveness. We have shown the validation of numerical exactness by determining heat flux (Q) and axial flow (w). The following relationship is used to estimate the error percentage. In this study, we consider five grid sizes,16 16,18 18, 20 20, 22 22, 24 24 × × × × × i.e. N = M. The grid size of 20×20 was good enough to guarantee the accuracy of the solution as represented in Table 1.

Results and Discussion
Here, the completely formed 2D viscous incompressible fluid through a curved micro-channel (CMC) is computationally evaluated with rotating and non-rotating cases for curvature β= 0.01. For the non-rotating case, the Dean number, 0<Dn≤5000, is the flow-controlling parameter while for the rotating case, the Taylor number, -500 ≤ Tr ≤ 2000, controls the flow. Two cases arise in the present study: Case I: Non-Rotating channel Case II: Rotating channel In the following, SS, time advancement, and HT are discussed, in detail, for the above-mentioned two cases.

Structure for Steady Solutions
As investigated by Mondal et al. (2007Mondal et al. ( , 2013 for finding the steady solutions, our study prevailed in two branches of SS for 0<Dn≤5000, Gr = 1000, β= 0.01 (see Fig. 3). These two branches are called Branch 1 (solid red line) and Branch 2 (solid black line). According to Mondal et al. (2006) discussion, the path continuation approach is used to acquire the SS branches using a variety of initial hypotheses. The velocity of the non-isothermal flow and isotherm is shown in Fig. 4. It is clear that Branch 1 is made up of a sym- metric 2-to 8-vortex solution, whereas Branch 2 is made up of a 4-to 8-vortex solution. Finally, the variation of velocity concerning grid points in the primary flow is depicted in Figure 5 which gives us the idea of CMC flows heating wall and cooling wall as shown in Figure 2.

Unsteady solutions
By investigating the time-advancement solution for CMC with the heating wall and cooling wall for 0<Dn ≤3840, as visualized in Fig. 6, the unsteady flow is discovered to have a steady-state solution. Steady-state flow in this region always possesses a 2-vortex solution.The nature of the flow quickly changes to regular oscillation i.e., periodic as Dn increases. For 3845 ≤ Dn ≤ 4700, the unsteady flow (UF) remains in regular oscillation and the pattern of the secondary flow fluctuate in a certain range. The irregularities of the flow arise as seen in Fig. 6 for Dn = 5000. Contours of PF, SF, and thermoprofile are shown in Figure 7 for various Dn at the mentioned time. A 2-vortex asymmetric solution is remained up to the periodic stage and in the chaotic state, a new vortex is produced from the channel's outer wall. This new vortex is called Dean-vortex which is due to a high-pressure gradient. As seen from Figure 7, the velocity increases in the axial direction as Dn increases, and the contours of the isotherms distribute from the center of the outer lateral wall to the inner lateral wall because of the buoyancy force and curvature effect.

Convective heat transfer (CHT)
To explore the CHT from the walls to fluid, we study temperature gradients (TG) which simply can explain the rate of heat at a particular location. The TGs are executed for some specific Dn values and presented for the cooled wall (CW) in Figure 8(a) and for the heated wall (HW) in Figure 8(b). It is noticed from Figure   8(a) that on the CW the temperature difference is dropped down in the middle region of the duct as Dn increases which are taken place due to the advection of outward SF as well as the influence of centrifugal force. On the contrary, outside the central region, is increased monotonically, which is resulted from the advection of inward SF and due to the fluid mixing. Besides, is prolonged at the central portion of the duct on the HW monotonically by the opposite SF in the inward direction.  Case II: Rotating channel

Steady solutions for positive rotation
We have searched the SS branches with the help of the path continuation technique and arc-length method and obtained a single branch of SS which exists in the entire ranges for β = 0.01. Fig. 9 shows a unique SS branch, where we see that the branch is very simple. In Fig. 10 Due to the small curvature, Coriolis, and centrifugal forces, the primary velocity and isotherms only little altered as the rotation is increased. The velocity of the flow in the axial direction over various grid points is also shown in Figure 11 for Tr = 500 and Tr = 2000 which provides us the scenario of velocity increase in a rotating square channel flow for the heating wall and cooling wall. The unsteady solution (US) of the entire domain of the investigated Tr (0<Tr≤2000) gives us the same result as the steady-state with a 2-vortex solution (see Fig. 12 and 13). There is no notable change occurring in the primary flow and temperature distribution and implicit with the secondary vortices.

Convective heat transfer (CHT)
To determine the CHT from the heatingwall to the fluid, the influence of positive rotation on the heatgenerating mechanism is examined. We compute TG at the CW and HW for this reason. Fig. 14(a) illustrates how the difference in temperature on the HW grows in the central region owing to a slight vibration of the heating wall and varies in other places as a result of the fluid mixing. Figure 14(b) shows that on the CW diminishes in the middle area about y = 0 and gradually increasesthe whole part of the duct except the middle area. The propagation of the SF in an outward direction is what leads to these results around y = 0 because of the centrifugal force. These results demonstrate that when Tr rises, there is a substantial augmentation in heat transmission from the heating lateral wall to the fluid.
Here, we studied the transient flow behavior for negative rotation of the curved MC for -500≤Tr<0, Dn = 2000, Gr= 1000, and β = 0.01. Fig. 15(a) and 15(b) show that the UF is a steady-state with a 2-vortex solution for Tr = -10. When Tr = -250 then we see that the UF is multi-periodic with a 2-to 4-vortex solution (see Fig. 16 and 18). Fig. 17(a) and 17(b) demonstrate the phase-space diagram and spectrum density analysis for conformation of the multi-periodicity flow. Next, we investigate the UF for Tr = -500 where it can be (a) (b) visible that the UF returned to the previous state i.e., the steady-state but the remarkable scenario is that the UF is composed of a 4-cell solution (Fig. 19). The interaction among centrifugal, Coriolis, and buoyancy forces results in the formation of these vortices.

Vortex diagram in the plane
We demonstrate the structure fluctuation of secondary vortices for various parameters of Tr to observe the structure of SF in the light of vortices. Fig. 20 depicts how many vortices can the flow be formed for a specific value of curvature at a Taylor number and a two-to eight-vortex solution is discovered in the secondary flow pattern for distinct values of Tr. It has become possible to figure out maximum 8-vortex solutions for curvature β= 0.001, compared to 4-and 2-vortex solutions for β = 0.1 and β = 0.5 correspondingly. It is revealed that when Tr rises, the vortices' number falls. In this work, the steady-state solution has been discovered to have 2-vortex solutions, the periodic solution has 2-to 4-vortex solutions, and the chaotic solution has 2-to 6-vortex solutions. Due to the fact that chaotic solutions generate a large number of vortices in the surrounding concave wall, it is suggested that they be used instead of steady-state or periodic solutions to enhance heat transmission.

Conclusions
In the present study, the heat-flux effect and the dissemination of energy through a bent MC are determined using a spectral-based computational scheme for rotating and non-rotating cases with a temperature gradient between the vertical lateral walls. From the present study, the following outcomes are listed: • For non-rotating case, two asymmetric SS are revealed with 2-to 8-vortex solutions. The nonlinearity of the time-advancement flow demonstrates as 'steady-state periodic chaotic' which consists of a 2-to 4-vortex solution. • For positive rotation, an asymmetric SS branchis found with a 2-vortex solution.The nonlinearity of the time-advancement flow is 'steady-state' only and consists of a two-vortex solution. • By analyzing non-linear behavior for negative rotation, the unsteady flow goes through-'steady-statemulti-periodic steady-state' with a 2-to 4-vortex solution. • CHT is increased significantly with the increase of Dn for non-rotating duct and imposing rotational speed i.e., raising Tr number. Higher Tr leads to the development of the extremely complicated secondary flow field. • The research demonstrates that buoyancy force interacts with centrifugal and Coriolis forces hart impact the fluid mixing, which then assists to turn the flow into chaos and prompts the intensification of the convective heat transfer (CHT).