Developments of Electro-Osmotic Two-Phase Flows of Fourth-Grade Fluid through Convergent and Divergent Channels

: This paper discusses the development of two different bi-phase ﬂows. Fourth-grade ﬂuid exhibiting the non-Newtonian ﬂuid nature is taken as the base liquid. Two-phase suspension is obtained by using the spherically homogeneous metallic particle. Owing to the intense application of mechanical and chemical multiphase ﬂows through curved and bent conﬁgurations effectively transforms the ﬂow dynamics of the ﬂuid. Differential equations for electro-osmotically driven ﬂuid are modeled and solved with the help of the regular perturbation method. The obtained theoretical solution is further compared with the ones obtained by using two different numerical techniques and found to be in full agreement.


Introduction
In various applications, the flow of non-Newtonian fluids (such as blood, greases, drilling muds, and suspension, etc.) cannot be expressed by the classical Navier-Stokes theory, and these fluids are categorized as tangent hyperbolic fluids, power-law fluids, generalized Newtonian fluids, Ellis fluids, Williamson fluids, Burgers fluids, Johnson-Segalman fluids, Sisko fluid model, Eyring-Powell fluid, third grade fluid, etc. Due to complex rheological properties and behavior, the fourth-grade fluid [1] is a special type of non-Newtonian fluid that describe the shear thinning and shear thickening phenomena which cannot be expressed by the classical Navier-Stokes equations.The applications of fourth-grade fluids in industry, petroleum and food manufacturing, etc., have significant involvement of diffusion reaction [2] and thermal transports in parallel flows.The constitutive relation [3] of fourth-grade fluid is more complex as compared to secondand third-grade fluids due to more material parameters.So, the study of such highly viscous fluids is hard to model and predict the flow properties, due to scores of parameters.Salawu et al. [4] reported important results on fourth-grade fluid.The investigation is carried out for a parallel flow that obeys the fundamentals of Couette flow mechanism.The numerical results are reached via finite semi-discretization difference method.
Fourth-grade fluid is treated as biological flow in [5] in the curved artery channel by Khan et al. with the help of numerical technique.An approximate study of circular flows with temperature-dependent viscosity of fourth-grade fluid through is the focal point of different authors in [6,7].Aziz and Mahomed [8] present a theoretical analysis of fourth-grade fluid over a porous plate.
The flow of bulk fluids through the membrane, porous channel, capillary tube, microchannel, or any other fluid channel under the action of the electric field applied at the end of the conduit is termed electro-osmosis flow.The electroosmotic flow getting the attention of researchers and authors due to its wider applications in medical science, natural chemistry, industrial processes [9], etc.The electro-osmotic flow in non-Newtonian fluids, namely, colloidal suspension, blood, polymeric and protein arrangements, have significant usages.Currently, various studies on the electro-osmotic flow of non-Newtonian fluids have been reported by researchers by considering different constitutive models such as Eyring-Powell fluid [10], Williamson fluid [11], Casson fluid [12], Sutterby fluid [13], generalized Newtonian fluid [14], fractional Maxwell fluid [15], Walters'-B fluid [16], Phan Thien Tanner fluid [17], Power-law fluid [18], Oldroyd-B fluid [19] and third-grade fluid [20], etc.
In addition to the above literature, close analysis of some recent studies on the multiphase flow of fourth-grade fluid under the action of the electric field in two complex configurations, namely, convergent and divergent channels, is worthful investigation.The analysis of this study is a significant contribution to understanding the behavior of the multiphase flow of fourth-grade fluid in terms of physical and mathematical point of view.The modeled highly nonlinear differential equations are dealt with "Perturbation technique" to achieve an approximate solution.

Development of a Mathematical Model of Multiphase Flow of Non-Newtonian Fluid with Electro-Osmotic Phenomena
Consider a two-phase flow of fourth-grade fluid through channels as shown in Figures 1 and 2, respectively.The configuration of convergent [21] and divergent [22] channels can be defined as: Geometry 1: Geometry 2: If V v f = u v f (x, y), 0, 0 and V vp = u vp (x, y), 0, 0 denote the velocity of fluid and particle phase, respectively.The governing equations for this dissemination of fluid and particle phases are:

Flow Equations for Fluid Phase
The equation of continuity which governs the conservation of mass of the flow is .0, similarly, the conservation of momentum [23,24] for the fluid phase of the considered problem is given as The mathematical expression of "" is defined as [2]

Flow Equations for Fluid Phase
The equation of continuity which governs the conservation of mass of the flow is similarly, the conservation of momentum [23,24] for the fluid phase of the considered problem is given as The mathematical expression of "T" is defined as [2] In the above one can identify The equation of continuity and momentum equations are defined in the following manner It is presumed that the velocity of the bi-phase fluid is zero and the particle concentration remains the same during the study, so the Equations ( 3), (4), ( 13) and (15) in the component's forms can be written as The momentum of the fluid phase can be obtained as The overhead expression has a lot of significance because if β 3 is zero, the Equation ( 17) turns into a momentum equation for third-grade fluid and if both are equal to zero, then the resulting equation is also a momentum equation of second-grade Newtonian fluids and if both are not turned into zero the result will be momentum equation of four grade.

Governing Equations (Particle Phase)
The Equations ( 14) and ( 15) can be expressed in the following form as For steady flow Equations ( 17), ( 18), ( 20) and ( 21) gained the shape Equation ( 23), is solved for modified pressure, which gives The boundary conditions are given as

Dimensionalization of the Problem
To predict the contribution of the most significant variables and parameters, it is mandatory to reduce or accumulate certain quantities which are of the least importance.Therefore, the following dimensionless transformation is of effective use.
The dimensionless form of Equations ( 20)-( 27) is achieved by using the expression defined in Equation (28) in the following form as (bars are omitted) Similarly, the dimensionless form of the relations described in Equations ( 1) and ( 2) are narrated as We assume that dp dx = P.

Perturbation Solution
To find the approximate analytical solution to Equation (36) can easily be achieved due to the nonlinear term.Therefore, the most effective and reliable solution with the least margin of error can be obtained if the perturbation technique is applied.For this purpose, we assume that: and more suppose that, = λε. (39) The above equation is known as the perturbation parameter.In view of Equations ( 38) and (39), Equation (36) becomes Equating and determining the equation of each order of ε 0 , ε 1 and ε 2 : Similarly, The solution to Equation (41) is given below The solution to Equation ( 43) is given below The solution of Equation ( 45) is not presented here due to lengthy expressions that appeared after solving it.The final expression of the velocity can be obtained from Equation (48), i.e., Similarly, we can get the expression for the velocity of the particulate phase u vp .
The volumetric flow rates (fluid and particle phases) can be determined from the following expressions: The mathematical expression for the total volumetric flow rate is defined as The expression for pressure P can be obtained by solving the above Equation (55).

Comparative Analysis
The comparison between numerical and perturbation solutions is displayed in Table 1.The perturbation solution is obtained in second order while the numerical solution is obtained through the spectral collocation method.In this method, we discretize the derivatives by using the Jacobi orthogonal polynomials or Chebyshev.The nonlinearity is handled through the Newton-Raphson method and finite difference approximation of the Jacobean (discrete Jacobean).Both solutions are obtained in convergent geometry.For this comparison, we obtained the numerical values of fluid velocity and particle velocity against the variation of the Hartmann number.From Table 1, it can be observed that both solutions are well-matched with each other.To validate the numerical results, we used another scheme, namely, the shooting method, and noted that both numerical results are accurate, as listed in Table 2.

Results and Discussion
A comprehensive parametric study is carried out in this section.The momentum of the particulate flow is predicted via the change in the numerical values of fourth-grade parameter " ", electro-osmotic "m", particle concentration "C" and volumetric flow rate "Q".Because of the diverse shapes and layout, the velocity acts entirely differently.In Figures 3 and 4, the graphs of the most significant parameter , the fourth-grade parameter are drawn against the different values in both channels.It is of great interest that the velocity of both phases inclines with the respect to the variation in the dimensionless quantity.However, both geometries affect the flow quite differently.This opposite impression of the geometry of the multiphase flow can easily be apprehended due to Bernoulli's principle of fluid dynamics.
The variation of the electro-osmotic parameter "m" on fluid and particle phases is shown in Figures 5 and 6.It can be viewed from the plotted graphs that the electro-osmotic parameter inversely impacts the velocity profile of the fluid and particle phases, respectively.This inverse relationship introduces a force of hindrance across the flow.Therefore, the momentum of the fluid and particles diminishes gradually.Variation in the concentration of metallic particles is depicted in Figures 7 and 8. Unlike previous graphs, the impact of C is quite different, all depending on the configuration of the geometry through which the bi-phase suspension is transported.The momentum of both phases declines by opting for the convergent channel.On the other hand, there is a tremendous enhancement in the velocity of fluid and particle phases when the channel is considered to be the divergent one.
The volumetric flow rate is also a pivotal parameter of this analytic study.In the most recent decade, when every appliance has reduced in size, the need of the hour is to conduct such research where micro-size geometries and channels are considered; so, in this regard, the volumetric flow rate is especially important to measure.The volumetric flow rate is also known as the rate of fluid flow or volume velocity.This is the volume of the fluid which is passing through the considered geometry per unit of time.Its units in system international (SI) are cubic meter/second; however, cubic centimeters per minute is also in practice.The volumetric flow rate is mathematically defined as and this is the scalar quantity.In Figures 9 and 10, the graphs of the volumetric flow rate are plotted against the different values of the parameters Q for both the phases in convergent and divergent channels.As the values of enhanced, the velocity profile of fluidic and particulate phases increased in convergent and divergent channels.The same behavior of the graph has been seen in both phases and the simultaneous effect is observed for diversely shaped convergent and divergent channels.This is because when the volumetric flow rate is increased, the velocity in the geometries experienced pressure and the velocity is enhanced due to extra pressure of the flowing fluid, the fluid entering the channel and gaining high velocity.

Concluding Remarks
A closed-form pronouncement for the velocity dispersal of utterly evolve flow of hafnium particles and fourth-grade base fluid adjournment via two different geometries diverse in shape are dispensed.The impact of germane parameters such as fourth-grade parameters, electro-osmotic parameter, the concentration of nanoparticles and volumetric flow rate in a couple of channels such as convergent and divergent flow has been exhibited and inspected graphically.The most noteworthy remarks itemized are:

O
An increase in the behavior of both particle and fluid phase velocities is viewed in convergent and divergent geometries when enhancement is made in the fourth-grade parameter; O A remarkable decrease in the velocity profiles of fluid and particle phases in both channels is noted when the value of the electro-osmotic parameter is enhanced; O The credible incline is measured in the velocity profile of both phases in the divergent channel when the value of particle concentration is increased, and a very dubious decline has been seen in the velocities of both phases in the convergent channel; O When the volumetric flow rate upraised in both channels the velocity profile of fluid and particle phases improved as the volumetric flow rate more in velocities.

Figure 4 .Figure 4 .
Figure 4. Impact of the four-grade parameter on particle velocity.

Figure 5 .
Figure 5. Impact of electro-osmotic parameters on fluid velocity.

Figure 5 .
Figure 5. Impact of electro-osmotic parameters on fluid velocity.

Figure 6 .
Figure 6.Impact of electro-osmotic parameters on particle velocity.

Figure 6 .
Figure 6.Impact of electro-osmotic parameters on particle velocity.

Figure 9 .
Figure 9. Impact of volumetric flow rate on fluid velocity.

Figure 9 .
Figure 9. Impact of volumetric flow rate on fluid velocity.

Figure 10 .
Figure 10.Impact of volumetric flow rate on particle velocity.

Figure 10 .
Figure 10.Impact of volumetric flow rate on particle velocity.

Table 1 .
Absolute error between perturbation and numerical solutions.

Table 2 .
Absolute error between shooting method and pseudo-spectral collocation method.