Study of Two-Phase Newtonian Nanoﬂuid Flow Hybrid with Hafnium Particles under the E ﬀ ects of Slip

: This paper investigates the role of slip in a two-phase ﬂow of Newtonian ﬂuid. The nano-size Hafnium particles are used in the base ﬂuid. The ﬂuid under consideration is studied for two cases namely (i) ﬂuid phase (ii) phase of particles. Both cases are examined for three types of geometries. The governing equations are simpliﬁed in nondimensional form for each phase along with boundary conditions. The resulting equations have been analytically solved to get exact solutions for both ﬂuid and particle phases. Di ﬀ erent features of signiﬁcant physical factors are discussed graphically. The ﬂow patterns have been examined through streamlines.


Introduction
The multiphase flows contribute a great role to shaping human lives for centuries, such as water flowing down from top of the hills, high waves produced in the oceans with different speed, air in the windmills, drilling of crude oil under earth surface, blood flowing in the circulatory system that deal with single, double, or multiphase flows. More precisely, the air and water flows are the examples of single-phase flow whereas flows such as solid-liquid flow, bubbly flows, particulate flows, droplets (multi-directional) flow, and water and stream flow, are the most common examples of two-phase flow. Moreover, from a medical point of view, the best example of two-phase flow is plasma-platelet in the blood flow. The multiphase flow involves such materials which has chemically different features, for example, a mixture of liquids and different types of gases, or two types of liquids that do not mix with each other. The multiphase flow term is used for any kind of fluid flow which consists of more than one component or phase. Multiphase flow was particularly studied widely in the context of the oil industry due to the increasing dependence of petroleum. In the industry, multiphase flows technology are used at a large scale nowadays, few examples among several are suspension of fibers In view of the above-mentioned studies, it is perceived that hafnium nanoparticles are widely used in biosciences to target cancer cells such as "Ionizing radiation", a new approach to cancer therapy. The aqueous solution of hafnium nano-crystallites go through different shapes and forms of arteries and veins, in the process to reach damaged tissues. The theoretically studied lubrication effects on the nanoflow will help expedite the healing process which is yet not available in existing literature. Motivated by these facts, we aim to devote our efforts to fill this gap.

Mathematical Model
If V = u ξ, η v ξ, η w ξ, η , the velocity of Newtonian fluid has features, i.e., viscous, incompressible, steady, and electrically conducting containing 40% of Hafnium particles. Investigating the multiphase flow through various geometries, paved the way on how one can face new challenges in the days to come. Here, we have considered the following three diverse geometries as shown in Figures 1-3

for illustration:
Inventions 2020, 5, x 3 of 22 In view of the above-mentioned studies, it is perceived that hafnium nanoparticles are widely used in biosciences to target cancer cells such as "Ionizing radiation", a new approach to cancer therapy. The aqueous solution of hafnium nano-crystallites go through different shapes and forms of arteries and veins, in the process to reach damaged tissues. The theoretically studied lubrication effects on the nanoflow will help expedite the healing process which is yet not available in existing literature. Motivated by these facts, we aim to devote our efforts to fill this gap.     In view of the above-mentioned studies, it is perceived that hafnium nanoparticles are widely used in biosciences to target cancer cells such as "Ionizing radiation", a new approach to cancer therapy. The aqueous solution of hafnium nano-crystallites go through different shapes and forms of arteries and veins, in the process to reach damaged tissues. The theoretically studied lubrication effects on the nanoflow will help expedite the healing process which is yet not available in existing literature. Motivated by these facts, we aim to devote our efforts to fill this gap.     In view of the above-mentioned studies, it is perceived that hafnium nanoparticles are widely used in biosciences to target cancer cells such as "Ionizing radiation", a new approach to cancer therapy. The aqueous solution of hafnium nano-crystallites go through different shapes and forms of arteries and veins, in the process to reach damaged tissues. The theoretically studied lubrication effects on the nanoflow will help expedite the healing process which is yet not available in existing literature. Motivated by these facts, we aim to devote our efforts to fill this gap.

Geometries
First geometry: otherwise. (1) Second geometry: otherwise. (2) Third geometry: otherwise. ( The associated velocities of fluid and particle phases are given: If flow is in an axial direction, then Equations (4) and (5) can be written:

Mathematical Model
In view of Equations (6) and (7), the resulting mathematical model for fluid and particle phase in components from [48] are given as: (i) For the case of phase fluid: Here, C is concentration, µ is viscosity, and "S" denotes the drag coefficient of interaction for the force exerted by the particles on the base liquid [49].
(ii) For the case of particle phase:

Boundary Conditions
Associated boundary conditions for fluid phase are: Inventions 2020, 5, 6 5 of 22 Invoking the following transformation The dimensionless form of Equations (1) to (3) and (8) to (13) are obtained as: µ s aδλ dp dξ = S u f − u p .

Results
After basic manipulation, the electro-osmotic potential function Φ(η), given in Equation (19), can be attained as In view of Equation (24), Equation (19) can be written as The velocity of the particles in view of Equation (23) can be expressed as Inventions 2020, 5, 6 6 of 22 Using Equation (26) in Equation (25), yields By using routine calculation, the exact solution of the second order linear but nonhomogeneous ordinary differential Equation (27) by means of boundary conditions in Equations (20) and (21) yield Further simplification of Equation (28) yields: Which is the required velocity of the base fluid. By using the base fluid values given above in Equation (29) in Equation (26) one gets

Simplification of Equation (30) further leads to
Now, for fluid and particle phases, the volumetric flow rates are obtained as: However, the total fluid particle volumetric flow rate can be obtained as: Inventions 2020, 5, 6 Pressure gradient is the most important feature of the considered fluid. By simplifying the expression for pressure gradient from Equation (35). dp dξ =  Figure 5a-c, the velocity of particles phase also declines due to variation in magnetic field. This phenomenon basically is caused in the response of Lorentz force which resists the flow. Moreover, the magnetic effects are intense for the case of diverging type of geometry (i.e., geometry III) as observed in Figures 4c and 5c, respectively. This also confirms that the magnetic induction is much stronger for diverging-converging nozzles, as compared to geometry I and II. The variation of electro-osmotic parameter m, is displayed in Figures 6  and 7. One sees that the motion of both fluid and Hafnium particles is supported by the concerned parameter, by considering the flow in any form of geometry. However, the influences of m are so prominent in the convergent type of geometry. Therefore, the velocity of fluid and particles inclines rapidly, as shown in Figures 6a-c and 7a-c, respectively. This changing trend is the contribution of the electrokinetic term in the governing equation.

Discussion
An opposite behavior in the flow is observed for Helmholtz-Smoluchowski velocity U HS . By looking at each diagram shown in Figures 8a-c and 9a-c, the velocity of both phases reduces in strength for the higher values of U HS . The most significant parameter of this survey, is the lubrication parameter β 1 , which is shown in Figures 10a-c and 11a-c. The effects of lubrication at both walls of each geometry resist the performance of fluid and particle. Therefore, by increasing the slip parameter velocity of the fluid, the particle also declines.
The pattern of fluid flow through the given geometries, can be described by stream lines. In this regard, Figures 12-15 have been sketched. Deterioration in velocities is observed in both phases for the increasing values of M which indicates hindrance in the course of flow. Thus, escalation in M causes the emergence in terms of extra stream lines as observed in Figure 12a,b,e,f, for geometries I and III whereas for geometry II, the fluid's velocity gets the highest momentum, thus the lines of magnetic fields can only manage to compress the streamlines inwards and the same is seen in Figure 12c,d. Consequently, the fall in velocity of the fluid-particle suspension is perceived. The role of m on streamlines varies in all cases. The streamlines that shrink against higher m is detected in Figure 13a,b. One can detect that fluid flow gets stress-free having no hurdle in its way in geometry I, whereas streamlines remain unchanged in the rest of two geometries II and III, as displayed in Figure 13c on streamlines varies in all cases. The streamlines that shrink against higher m is detected in Figure  13a,b. One can detect that fluid flow gets stress-free having no hurdle in its way in geometry I, whereas streamlines remain unchanged in the rest of two geometries II and III, as displayed in Figure  13c

Conclusions
The expressions of the velocities for fluid and particle phase are analytically obtained in the multiphase flow of water and Hafnium particles ( − ). The flow is examined through three

Conclusions
The expressions of the velocities for fluid and particle phase are analytically obtained in the multiphase flow of water and Hafnium particles (HF − H 2 O). The flow is examined through three different geometries for various parameters. The effects of slip is studied on multiphase flow synthesis of nano-sized particles. Some important observations are listed as: • The velocity decreases against Hartmann number M and Helmholtz-Smoluchowski U HS in all cases. • The velocity of particle phase decreases against the slip parameter, β 1 in geometry I and geometry II whereas it increases in geometry III.

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The velocity increases against m, in the three geometries for both particles and fluid phases.

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Inclined behavior of velocity is observed for both phases in three geometries against m.

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In the case of Hartmann number M, Helmholtz-Smoluchowski U HS , and slip parameter β 1 , the graphs of streamlines are the same for all three geometries. However, the reduction of the streamlines for electro-osmotic parameter m, is observed. This behavior is due to the curve-like structure of the channel.