Flow Induced By A Rotating Microchannel-A Numerical Study

In this paper, a mathematical foundation has been developed for the primary understanding of complex interaction of the wall slip with the Coriolis and Lorentz forces acting orthogonally on the Electromagnetohydrodynamic (EMHD) flow of a power-law fluid in a microchannel. Modified Navier Stokes equations are solved numerically by incorporating the fully implicit computational scheme with suitable initial and boundary conditions, which generates numerical results in excellent comparison with the literature for a certain limiting case. An extensive effort has been made to understand how the Hartmann number, fluid behavior index, rotating Reynolds number, and slip parameter affects the flow. Results show the velocity of the power-law fluid depends strongly on flow parameters. Critical Hartmann number can be obtained for the power-law fluid in presence of uniform electric and magnetic fields. As a promising phenomenon, existence of a cross over point (which depends upon the fluid behavior index) for the centerline flow velocity, has also been predicted. Reduction in the shear stress and fluid viscosity can be controlled effectively by incorporating a slippery film of lubricant on the periphery of the microchannel. This work is useful to meet the upcoming challenges of future generation, like improvement in bio-magnetic-sensor technologies as well as electrical and mechanical mechanisms.


Introduction
Microfluidics plays a fundamental role in various technological processes and their applications, concerning the examination of biological and chemical samples, detection, separation, and the professional design for transfer of mass and heat. In recent years, microfluidic transport based lab-on-achip has fascinated the researchers and attained extraordinary awareness due to its applications (Holger et  investigation has yet been made for evaluating the results of hydrodynamic slip on the EMHD power-law fluid model for Newtonian fluid in microfluidic systems ( Yves 1971;Brunn 1975;Prakash et al. 2011).
In recent years, rotating EMHD flow has been a scorching topic as it plays a vital role in improving the equality of mixing liquids of the microfluidic system with a substantial reduction in the time of mixing. The investigation has a significant impact on the improvement of an electromagnetic gadget in a rotary system, which can be beneficial in the area of drug delivery and species control separation.

Mathematical formulation and Power-law fluid model
Physical demonstration of the rotating electromagnetohydrodynamic flow of a power-law fluid model can be seen in Fig. 1(a). At the center of the microchannel, Cartesian coordinates   , , x y z are developed, since the rotary motion of the entire system is about z-axis so, the system is assumed in non-inertial frame.
Height of the microchannel is 2H . Microchannel is filled with fluid and produces a Lorentz force in the xdirection, induces uniform electric field 0 E  in y-direction, and uniform magnetic field 0 B  in z-direction.
We assume that the microchannel is rectangular having length L and the breadth W in x and y-directions, respectively. It is considered that the microchannel aspect ratio is too small, i.e. , 2 . W L H W     Cross section outlook of a microchannel can be noticed in Fig. 1(b). With the constant angular velocity  , the entire microchannel spins about z-axis. In a rotating frame of reference, and flow velocity is called the relative velocity. An incompressible fluid is packed in the microchannel and continuity equation is Here, t is time,  is fluid density, dynamic pressure is P , i.e. 2 5 10  , so, it is implies that magnetic field is independent of the flow velocities. Lorentz force is expressed as (Zhi-Yong et al. 2017) where , , and e E B     are the electrical conductivity, electrical field strength, and magnetic field strength respectively. Dynamic viscosity  in a power-law fluid model is: where  and n are represented as consistency index and fluid behavior index. Shear-thinning fluid viscosity ( 1 n  also known as pseudo-plastic fluid) reduces with an enhancement in shear rate. Pseudoplasticity can be verified by shaking a ketchup bottle, causes an irregular alteration in viscosity. The shearthinning fluid viscosity ( 1 n  also known as dilatants fluid) boost up with the shear rate. The result of dilatants fluid can be observed with a cornstarch and water combination when thrown adjacent to a surface. When 1 n  fluid behaves like a Newtonian fluid and this is a reason, we recognize the model of a powerlaw fluid as the Newtonian model. Distinction between Newtonian and non-Newtonian fluids is that the viscous stress of non-Newtonian fluids does not have a linear relationship with strain rate tensor. Strain rate tensor in magnitude is denoted by  and can be demonstrated as Here,  is strain rate tensor. Using the relationship of the strain rate tensor and viscous stress rate tensor, the equation for power-law fluid model is signifying as Here, the velocity gradient tensor is V   and its transpose is .
Regarding the above theoretical arguments, we summarize that the microchannel aspect ratio is very small and the electromagnetohydrodynamic velocity components are functions of z and t only i.e.
. For incompressible fluid, the continuity equation can be easily satisfied. In power-law fluid model, dynamic viscosity is represented as The shear stress in component form can be illustrated as Therefore, in the rotating coordinate system, the Navier-Stokes momentum equations are represented as Concerning initial and boundary conditions are where  is the slip length. Electromagnetohydrodynamic flow determined by the Lorentz force has already In the power-law fluid, the maximum velocity is denoted by p u . It is obtained by pure pressure between the micro parallel plates in the flow problem.
Here, the pressure gradient characteristic is c  and the viscosity characteristic is c u , which are derived from momentum and viscous shear stress equations. Using the non-dimensionless variables from eq.(12) into eqs.
(7), (9), and (10) along with boundary conditions (11). Dynamic viscosity and Navier-Stokes equations in non-dimensionless form are represented as where the dimensionless pressure gradients are x  and y  in x and y directions, respectively. Converted initial and boundary conditions are given by: It is note that for our own convenience in eqs. (14)- (17), all dimensionless variables are shown by ignoring the star sign "*". Many dimensionless factors related to governing problems are: Here, Re  recognized as rotating Reynolds number, that has an inverse relation with the Ekman number, which provides a frequency scale. The estimate electric field strength is  and the ratio of magnetic to the viscous forces is called Hartman number and denoted by Ha and the slip parameter is denoted by L S .
According to the literature survey (Zhi-Yong et al. 2017), the analysis of the rotating flow proves that the pressure gradient must be constant in x and y directions. By using this assumption in present study, the pressure gradients are neglected. Reinstating the dynamic viscosity in eqs. (15), (16) with eq. (14), we get , , , and K z K z K z K z are functions in non-dimensional form and written as

Numerical Method
The governing equations involve two types of derivatives, one is the time derivative whereas the other is spatial. For discretization purpose, we use the forward difference for the time derivatives in both the equations. On the other hand, the central difference is employed for spatial derivatives because of its relatively better accuracy. Further, their spatial derivatives are treated implicitly which yield the following system of algebraic equations.
where         In the above difference eqs.  Table 1 based on the physical properties given in [7][8][9]13]. Findings of current study may facilitate one to look for an appropriate configuration for the most favorable presentation of EMHD-based micropumps.   Table 2.   Fig. 2(a) that, the center velocity is superior to the other two-fluid models. Motivation behind this observable fact is that fluid viscosity depends on location of the channel. Newtonian fluid viscosity at the midpoint of the channel is greater, compared to dilatant fluid but smaller than that of a pseudoplastic fluid. Due to negligible influence of rotating microchannel, the velocity u at the center of the channel for pseudo-plastic fluid becomes large.
As a result, under the influence of the same electrical and magnetic fields, the depression for both Newtonian and dilatant fluids becomes visible.  Fig. 2(b) and 3(b), which persuades in the -ydirection. The magnitude of the lateral velocity v depicts an increasing trend with n due to the rotational effect of the channel.
In Fig. (4-5), the velocity distribution in dimensionless form for different Hartmann numbers is illustrated.
The related constraint is stated in the caption. From the analysis of the sketch, it is found that for low Also, unlike Fig. 5(a), for further increase in Hartmann number Ha the depression in velocity profiles disappears as shown in Fig. 7(a). Due to the revolving effect of EMHD, influence of Coriolis force weakens as compared to the Lorentz force. However, from Fig. 4(a), it is concluded that due to large fluid viscosity in the pseudoplastic fluid model the depression cannot easily appear. For the high Ha , the depression vanishes as can be observed from Fig. 6(a). Coriolis force performs a significant role as compared to the Lorentz force for large values Re  . Therefore, for the dilatants fluid, the centerline velocity is smallest for the large Re  due to the smaller viscosity of a dilatant fluid. Similar for small Ha in Fig. 9(a), an identical fashion of the velocity differences in the main flow can be initiated in Fig. 10

Conclusion
In the present work, a mathematical model has been formulated for incorporating the impact of wall slip in the rotating electromagnetohydrodynamic (EMHD) flow of power-law fluid in a microchannel. Following conclusions may be drawn: 1. Wall slip may be considered as an effective tool for controlling the flow and shear stress in a rotating channel. Therefore, the slip effects can not be simply ignored.

Funding
There is no financial support to carry out this research.

Data availability statement
All the relevant material is available upon request.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.