Abstract
In this paper, the couple influence of axial magnetic field and the heat and mass transfer on two rotating cylinders of Walter’s viscoelastic fluids by using the potential flow theory of viscoelastic fluid has been investigated. The normal mode method is used to compute the growth rate of disturbance. The influence of gravity is ignored at the interface, while the effect of surface tension is present in the analysis. The effect of the parameters on the stability is studied. It is found that the rotation of the inner cylinder induces stability. Furthermore, the viscoelastic ratio of fluids has a stabilizing influence. The heat and mass transfer, magnetic permeability ratio, and Ohnesorge number have a stabilized influence, while viscosity ratio of fluids, density ratio of fluids, and axial magnetic field have destabilizing influence on the considered system.
1. Introduction
The flow in an annulus formed between two concentric cylinders where one or both of the cylindrical surfaces are rotating is important in a wide number of applications from shafts and axles to spinning projectile, liquid rocket fuel injectors, and cylindrical cyclone separators [1]. Fu et al. [2] studied the swirl influence on confined swirling annular layers with heat/mass transfer at the gas/liquid interface. They observed that the swirl of the liquid layer has a stabilizing influence. The nonlinear instability of Rayleigh-Taylor of two rotating superposed fluid in the presence of a vertical magnetic field has been investigated by El-Dib and Mady [3]. Awasthi and Agrawal [4] have performed an instability analysis between two concentric rotating cylindrical, with heat/mass transfer at the interface. They observed that the rotating of the inner cylinder has destabilizing effect while the rotating of the outer cylinder has stabilizing behavior. Also, they noted that heat/mass transfer has a dual influence on the stability of the considered system. El-Dib [5] studied the nonlinear azimuthal instability of two rotating fluids through porous medium in the presence of a uniform azimuthal magnetic field. The linear electrohydrodynamic instability of two incompressible rotating fluid through porous media in the presence of a uniform azimuthal electric field was investigated by Moatimid and Amer [6]. They observed that the rotation of the liquid has stabilizing nature.
In the last decades, the phenomenon of heat/mass transfer in multiphase flows has received attention because it is important in a wide number of applications such as geophysical problems and boiling heat transfer in chemical engineering. Hsieh [7] studied the influence of heat/mass transfer on Rayleigh Taylor (R-T) stability problem of liquid/vapor system. It is noted that heat/mass transfer enhances the stability of the considered system when vapor is hotter than liquid. Nayak and Chakraborty [8] observed that heat/mass transfer had a destabilizing effect on the system of the cylindrical interface. Allah [9] studied Kelvin Helmholtz instability with heat/mass in the presence of magnetic field. It is noted that heat/mass transfer and magnetic field have stabilizing influence. Moatimid et al. [10] studied the influence of mass/heat transfer on the stability of a horizontal magnetic fluid sheet. They observed that mass/heat transfer plays important role in the stability of the considered system.
The effect of the magnetic field has important role in cylindrical geometry, as many applications of engineering use cylindrical geometry. Magnetic fluid study has important role in the industrial applications. Elhefnawy and Adwan [11] studied the outcome of the magnetic field on the interface stability of flowing fluids in cylindrical geometry with mass/heat transfer. Moatimid et al. [12] studied the KHI for flow through porous medium under the effect of oblique magnetic fields by employing the viscous potential flow analysis. Recently, the combined effect of axial magnetic field and heat/mass transfer on vapor/liquid interface in cylindrical geometry was considered by Dharamendra and Awasthi [13]. They concluded that the swirl of the annular layer has a stabilizing behavior. Moreover, the magnetic field and mass/heat transfer move the interface towards stability. For recent works on this topic, see refs. [14–16].
In this paper, we attempted to study the linear instability of two concentric rotating cylinders of Walter’s with heat and mass transfer in the presence of axial magnetic field. The influence of gravity is ignored at the interface, while the effect of surface tension is present in the analysis. The normal mode procedure is applied, and equation for the growth rate of perturbation is achieved. The instability is studied through plotted figures of growth rate curves for various physical parameters which are included in our system. To our knowledge, the combined influence of heat/mass transfer and magnetic field on two concentric rotating cylinders of two viscoelastic Walter’s fluids is not investigated yet
2. Theoretical Framework and Governing Equations
Here we consider the two Walter’s viscoelastic fluids and thermally conducting between two concentric cylinders of radius and separated by an interface in the presence of magnetic field in the axial direction, as demonstrated in Figure 1. The inner and outer cylinders are revolving anticlockwise with velocity and , respectively, where denotes the angular velocity of the cylinders. The fluid lying in the inner and outer cylinders has viscoelastic and , viscosities and , densities and , magnetic permeabilities and , temperatures and , and thermal conductivities and The influence of gravity is ignored at the interface, while the interface is experiencing a surface tension influence .
The governing equations of momentum conservation and continuity for incompressible fluids are given by [1, 3] where , , and .
A slight distortion is applied to the interface; hence, the disturbed interface may be written as follows: where is the distortion of the interface. The unit normal to the disturbed interface is given by
In this analysis, the velocity potential functions satisfy the Laplace equations as a consequence of both fluids are assumed to be irrotational and incompressible, i.e.,
where
In this problem, it is assumed that the quasi-static approximation is valid; hence, we can drive the magnetic field from magnetic scalar potential function [17].
Gauss’s law requires that the magnetic potential functions satisfy the Laplace’s equation, i.e.,
3. Boundary Conditions and Linear Stability Analysis
The solutions of magnetic and velocity scalar potential functions must satisfy the following boundary conditions: (1)Conditions at rigid cylindrical surfaces are given by [10](2)The interfacial condition, which expresses the conservation of mass across the interface, can be written as [17]Then, by using Equation (2), Equation (8) in the linear form becomes
The heat transport can be governed by [4] where denotes the latent heat released during phase transformation and is the net heat flux from the interface. In the equilibrium state, there will no mass transport through the interface. Also, at the interface , because the heat fluxes in both the phases are the same initially. Hence,
where is constant.
Using Equations (2) and (11), then Equation (10) in the linear form can be written in the form in which (3)The stress balance condition at the interface on adding the rotational influence, is given by
where denotes the jump of the functions across the interface. Here, , , and denote tangential component, normal component of magnetic field, and pressure of the fluid, respectively.
Using Bernoulli’s equation to obtain the pressure, hence the linear perturbed form of Equation (15) can be written as (4)The continuity of tangential components of magnetic field across the interface requires [17]
The unit normal vector to the interface defined to the first-order terms is as follows:
Equation (17), on using Equations (5) and (18), reduces to (5)The normal electric displacement component is continuous at the interface; hence [17]
Using Equations (5) and (18), then Equation (20) can be expressed as follows:
Our study will be based on the normal mode method. So, all disturbed quantity may be written as follows:
in which , , , and denote the amplitude of the disturbance of the interface, wave number, growth rate, and azimuthal wave number, respectively. and are arbitrary functions of On using Equation (22) to solve Equations (4) and (6) according to the previous boundary conditions, hence the solution of potential functions can be written in the following form:
where , , , , , and are given by
in which and denote the modified Bessel functions of first and second kind -order.
4. Dispersion Relation
By substituting , , , , and from Equations (23)–(25) into the boundary condition (16), we get the following:
where
From the dispersion relation Equation (27), we get the following conclusions: (1)In the absence of viscoelastic, that is, , and axial magnetic field, that is, , Equation (27) reduces to the same expression obtained previously by Awasthi and Agarwal [4](2)If inner cylinder is stationary and outer cylinder is rotating, that is, , and viscoelastic is absent , we get the same dispersion relation equation obtained by Dharamendra and Awasthi [13].
The parameter is supposed to be complex. Hence, putting into Equation (27) and separating the real and imaginary parts of Equation (27), we have
Equation (30) has two roots, and we will plot the maximum of those two values. Now from Equation (30), we have
For marginal stability analysis, ; therefore, Equation (30) changes to
Equation (32) shows that viscoelasticity does not affect the marginal stability criterion.
The dimensionless form of parameters including in our analysis is written as [3]
Here, represents the Ohnesorge number, and indicates the centrifuge number. The nondimensional form of dispersion relation (30) can be written as follows:
Here is the viscoelastic parameter. The solution of Equation (34) takes the form where
5. Stability Discussion
In this section, the variation of growth rate with wave number is plotted using the Mathematica software for physical parameters included in our considered system. The system is stable when , while the system is said to be unstable if . The case is the neutrally stable. Figures 2–11 are depicted in asymmetric case (). The growth rate of disturbance with different values of Ohnesorge number has been plotted in Figure 2. It demonstrates in Figure 2 that the increasing of leads to decrease of growth rate, which indicates that the system goes toward stability by increasing Ohnesorge number. This result is consistent with the result obtained by Amer and Moatimid [18]. The Ohnesorge number depended on the density of the outside fluid; hence, has stabilizing effect on the considered system. Also, as surface tension increases, the Ohnesorge number increases, and consequently, the growth rate decreases. Therefore, surface tension has a stabilizing effect. The viscosity of outside fluid decreases by increasing Ohnesorge number; hence, viscosity of outside fluid has destabilizing nature.
The growth rates have been plotted in Figure 3 for different values of density ratio of fluids It has been noted that the increasing of density ratio of fluids leads to increase of growth rates. This concludes that density ratio of fluids has destabilizing nature. Also, it is seen that the density ratio of fluids depends on the density of inside fluid; hence, the density of inside fluid has destabilizing effect on the considered system.
The growth rate curves for , 6, and 3 have been plotted in Figure 4. From this figure, it is noted that as the centrifuge number of inside cylinder increased, the values of the growth rate of perturbation decreased and so the increase in resists disturbances from developing at the interface of two fluids, and therefore, flow moves towards stability. Figure 5 shows the effect of on the growth rate curves. It is observed from Figure 5 that the increasing of leads to the increase of the growth rate. Hence, the rotation of the outside cylinder makes the interface unstable. Influence of rotation on growth rate is illustrated in Figure 6. It is observed that if the outer cylinder is fixed while the inner cylinder is rotating, the growth rate of disturbance is minimum; hence, the system moves toward stability, which indicates that the rotating of inner cylinder plays an important role in the stability of system. On the other hand, if the outer cylinder is rotating while the inner cylinder is stationary, the growth rate of disturbance is the highest and the flow gets destabilized.
The influence of the mass and heat transfer, during the parameter , on the growth rate versus wave number is illustrated in Figure 7. It is observed that as the heat and mass transfer parameter increases, the values of the growth rate decrease. Hence, one may conclude that the mass and heat transfer parameter has stabilizing effect on the considered system. This result is similar to a result that had been obtained by Aswasthi and Agarwal [4].
Figure 8 shows the influence of magnetic field on the growth rate curves. It has been noted that as the magnetic field increases, growth rate increases, and therefore, the axial magnetic field has destabilizing behavior. Our result is different from previous studies, and this adverse result, in the present work, probably occurs in accordance with the viscoelastic influence. In Figure 9, we have displayed the growth rate curves for various values of magnetic permeability ratio It is observed that the growth rate decreases on increasing magnetic permeability ratio Therefore, the magnetic permeability ratio has a stabilizing nature.
Figure 10 illustrates the variation of growth rate with wave number for different values of viscoelastic ratio of two fluids , 0.7, and As the viscoelastic ratio of two fluids increases, the growth of disturbance waves decreases. Hence, the viscoelastic ratio has a stabilizing effect on the considered system. The influence of viscosity ratio of fluids on the growth rate is displayed in Figure 11. It is observed that as the viscosity ratio of fluids increases, the growth rate increases, and the system gets destabilized. Therefore, has destabilizing behavior
6. Conclusions
The linear instability analysis of two concentric rotating cylinders of Walter’s viscoelastic fluids with the heat and mass transfer in the presence of magnetic field in axial direction has been investigated. The viscoelastic potential flow theory is used to solve the mathematical equations. We obtain the dispersion relation which is a quadratic equation in growth rate. The variation of imaginary part of growth rate is plotted versus wave number to study the influence of various parameters, and we conclude the following: (1)Our main result is that asymmetric perturbation is stable if the rotation of the inner cylinder is faster than the rotation of the outer cylinder(2)The viscoelastic ratio of fluids has stabilizing effect on the considered system(3)The heat and mass transfer parameter stabilizes the interface(4)The Ohnesorge number and the ratio of magnetic permeability have a stabilizing influence on the stability of the considered system, while viscosity ratio of fluids, axial magnetic field, and density ratio of fluids have destabilizing behavior
It is based on the importance of nonlinear effects on rotating concentric cylinder phenomena. So, we will discuss the nonlinear stability for this problem in a subsequence article.
Data Availability
The data used to support the finding of this study are included within the article.
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
The researcher would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project, on the material support for this research under the number 4001.