# Linear Stability of Magneto Viscoelastic Walter’sB′ Type with Heat and Mass Transfer

Authors: D. M. Mostafa
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

In this paper, the couple influence of axial magnetic field and the heat and mass transfer on two rotating cylinders of Walter’sB′ 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.


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## Body

## 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’sB′ 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 B′ fluids is not investigated yet


## 2. Theoretical Framework and Governing Equations

Here we consider the two Walter’sB′ viscoelastic fluids and thermally conducting between two concentric cylinders of radius r1 and r2r2>r1 separated by an interface r=R in the presence of magnetic field H0 in the axial direction, as demonstrated in Figure 1. The inner and outer cylinders are revolving anticlockwise with velocity r1Ω1 and r2Ω2, respectively, where Ωjj=1,2. denotes the angular velocity of the cylinders. The fluid lying in the inner and outer cylinders has viscoelastic ν1′ and ν2′, viscosities ν1 and ν2, densities ρ1 and ρ2, magnetic permeabilities εm1 and εm2, temperatures T1 and T2, and thermal conductivities ϰ1 and ϰ2. The influence of gravity is ignored at the interface, while the interface is experiencing a surface tension influence σ.Figure 1
Schematic diagram of the model.The governing equations of momentum conservation and continuity for incompressible fluids are given by [1, 3]
(1)DVDt+2Ω×V=−1ρ∇P+∇12Ω×r2+ν−ν′∂∂t∇2V,∇·V=0.where ν=μ/ρ, ν=μ′/ρ, and D=∂/∂t+V·∇.A slight distortion is applied to the interface; hence, the disturbed interface may be written as follows:(2)Sr,θ,z,t=r−R−ξz,θ,t=0,where ξz,θ,t is the distortion of the interface. The unit normal to the disturbed interface is given by
(3)N=∇S∇S=1+1r∂ξ∂θ2+∂ξ∂z2−1/2e¯r−1r∂ξ∂θe¯θ−∂ξ∂ze¯z.In this analysis, the velocity potential functionsΦjj=1,2 satisfy the Laplace equations as a consequence of both fluids are assumed to be irrotational and incompressible, i.e.,
(4)∇2Φ1=0,∇2Φ2=0,where∇2=∂2/∂r2+1/r∂/∂r+1/r2∂2/∂θ2+∂2/∂z2.In this problem, it is assumed that the quasi-static approximation is valid; hence, we can drive the magnetic field from magnetic scalar potential functionΨmr,θ,z,t [17].
(5)Hj=H0e¯z−∇Ψmj=−∂Ψmj∂re¯r−1r∂Ψmj∂θe¯θ−∂Ψmj∂z−H0e¯z,j=1,2.Gauss’s law requires that the magnetic potential functions satisfy the Laplace’s equation, i.e.,(6)∇2Ψm1=0,∇2Ψm2=0.


## 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](7)∂Φ1∂r=0and∂Ψm1∂r=0atr=r1,∂Φ2∂r=0and∂Ψm2∂r=0atr=r2.(2)
The interfacial condition, which expresses the conservation of mass across the interface, can be written as [17](8)ρ2∂S∂t+∇Φ2·∇S=ρ1∂S∂t+∇Φ1·∇S.Then, by using Equation (2), Equation (8) in the linear form becomes
(9)ρ2∂Φ2∂r−∂ξ∂t=ρ1∂Φ1∂r−∂ξ∂tatr=R+ξ.The heat transport can be governed by [4]
(10)Lρ1∂S∂t+∇Φ1·∇S=Eξatr=R+ξ,(11)Eξ=ϰ2T0−T2rlnr2/r−ϰ1T1−T0rlnr/r1,where L denotes the latent heat released during phase transformation and Eξ is the net heat flux from the interface. In the equilibrium state, there will no mass transport through the interface. Also, at the interface r=R, ER=0 because the heat fluxes in both the phases are the same initially. Hence,
(12)ϰ2T0−T2Rlnr2/R=ϰ1T1−T0RlnR/r1=CSay,whereC is constant.Using Equations (2) and (11), then Equation (10) in the linear form can be written in the form
(13)ρ1∂Φ1∂r−∂ξ∂t=βξ,in which
(14)β=Clnr2/r1LRlnR/r1lnr2/r.(3)
The stress balance condition at the interface on adding the rotational influence, is given by(15)P2+12ρ2Ω22r2−2μ2−μ2′∂∂t∂2Φ2∂r2−P1+12ρ1Ω12r2−2μ1−μ1′∂∂t∂2Φ1∂r2+12εmHn2−Ht2=σξR2+1R2∂2ξ∂θ2+∂2ξ∂z2atr=R+ξ,whereγ=γ1−γ2 denotes the jump of the functions across the interface. Here, Ht, Hn, and P 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
(16)ρ2∂Φ2∂t−ρ2Ω22Rξ+2μ2−μ2′∂∂t∂2Φ2∂r2+H0εm2∂Ψm2∂z−εm1∂Ψm1∂z−ρ1∂Φ1∂t−ρ1Ω12Rξ+2μ1−μ1′∂∂t∂2Φ1∂r2=−σξR2+1R2∂2ξ∂θ2+∂2ξ∂z2.(4)
The continuity of tangential components of magnetic field across the interface requires [17](17)N×H1=N×H2atr=R+ξ.The unit normal vector to the interface defined to the first-order terms is as follows:(18)N=e¯r−1r∂ξ∂θe¯θ−∂ξ∂ze¯z.Equation (17), on using Equations (5) and (18), reduces to
(19)∂Ψm1∂z=∂Ψm2∂z.(5)
The normal electric displacement component is continuous at the interface; hence [17](20)εm2N·H2=εm1N·H1atr=R+ξ.Using Equations (5) and (18), then Equation (20) can be expressed as follows:
(21)εm1∂Ψm1∂r−εm2∂Ψm2∂r=−H0∂ξ∂zεm1−εm2.Our study will be based on the normal mode method. So, all disturbed quantity may be written as follows:(22)Φj=φjrexpikz+nθ−ωt+c.cΨmj=ψmjrexpikz+nθ−ωt+c.cj=1,2,(23)ξ=γexpikz+nθ−ωt+c.c,in whichγ, k, ω, and n denote the amplitude of the disturbance of the interface, wave number, growth rate, and azimuthal wave number, respectively. φjr and ψmjr are arbitrary functions of r. 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:
(24)Φ1=γkβρ1−iωA1krexpikz+nθ−ωt+c.c,Φ2=γkβρ2−iωA2krexpikz+nθ−ωt+c.c,Ψm1=iγH0εm2−εm1b2kεm1b2kB1k−εm2b1kB2kKn′kr1Inkr−In′kr1Knkrexpikz+nθ−ωt+c.c,(25)Ψm2=iγH0εm2−εm1b1kεm1b2kB1k−εm2b1kB2kKn′kr2Inkr−In′kr2Knkrexpikz+nθ−ωt+c.c,whereA1kr, A2kr, b1k, b2k, B1k, and B2k are given by
(26)A1kr=InkrKn′kr1−KnkrIn′kr1In′kRKn′kr1−In′kr1Kn′kR,A2kr=InkrKn′kr2−KnkrIn′kr2In′kRKn′kr2−In′kr2Kn′kR,bjk=In′krjKnkR−Kn′krjInkR,Bjk=Kn′krjIn′kR−In′krjKn′kR,j=1,2,in whichIn and Kn denote the modified Bessel functions of first and second kind n-order.


## 4. Dispersion Relation

By substitutingξ, Ψm1, Ψm2, Φ1, and Φ2 from Equations (23)–(25) into the boundary condition (16), we get the following:
(27)Dω,k=a1ω2+a2iω+a3=0,where(28)a1=ρ1A1kR−ρ2A2kR+2k2μ2′G2kR−μ1′G1kR,a2=βA1kR−A2kR−2k2μ2G2kR−μ1G1kR+2k2βμ2′ρ2G2kR−μ1′ρ1G1kR,a3=2k2βμ2ρ2G2kR−μ1ρ1G1kR+Rkρ1Ω12−ρ2Ω22+σkR21−m2−k2R2+k2H02εm2−εm12b1kb2kεm1B1kb2k−εm2B2kb1k,(29)GjkR=1+mkR2AjkR−1kR,j=1,2.From the dispersion relation Equation (27), we get the following conclusions:
(1)
In the absence of viscoelastic, that is,μ1′=μ2′=0, and axial magnetic field, that is, H0=0, 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,Ω1=0,Ω2=Ω, and viscoelastic is absent μ1′=μ2′=0, we get the same dispersion relation equation obtained by Dharamendra and Awasthi [13].The parameterω is supposed to be complex. Hence, putting ω=ωR+ωI into Equation (27) and separating the real and imaginary parts of Equation (27), we have
(30)a1ωI2+a2ωI−a3=0.Equation (30) has two roots, and we will plot the maximum of those two values. Now from Equation (30), we have
(31)ωI=−a2±a22+4a1a32a1.For marginal stability analysis,ωI=0; therefore, Equation (30) changes to
(32)2kβRμ1ρ1G1kR−μ2ρ2G2kR+ρ2Ω22−ρ1Ω12+σR3k2R2+m2−1+k2H02εm2−εm12b1kb2kRεm2B2kb1k−εm1B1kb2k=0.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]
(33)r^1=r1h,r^2=r2h,R^=Rh,k^=kh,h^1=h1h=h^,h^2=h2h=1−h^1,ω^I=ωIτ,τ=ρ2h3σ,ρ^=ρ1ρ2,t^=tτ,μ^=μ1μ2,μ∧′=μ1′μ2′,ε^m=εm1εm2,H∧2=εm2H02hσ,β^=τβρ2,Oh=ρ2σhμ2,Ce1=ρ1Ω12h3σ,Ce2=ρ2Ω22h3σ.Here,Oh represents the Ohnesorge number, and Ce indicates the centrifuge number. The nondimensional form of dispersion relation (30) can be written as follows:
(34)ρ^A1k^R^−A2k^R^+2k∧2LG2k^R^−μ∧′G1k^R^ω^I2+β^A1k^R^−A2k^R^+2k∧2Ohμ^G1k^R^−G2k^R^+2k∧2β^LG2k^R^−μ∧′ρ^G1k^R^ω^I−k^R^Ce1−Ce2+2k∧2β^OhG2k^R^−μ^ρ^G1k^R^+k^R21−m2−k∧2R∧2+k∧2H∧21−ε∧m2b1k^b2k^ε^mB1k^b2k^−B2k^b1k^=0.HereL=ν2′/h2 is the viscoelastic parameter. The solution of Equation (34) takes the form
(35)ω^I=−a^2±a^22+4a^1a^32a^1,where
(36)a^1=ρ^A1k^R^−A2k^R^+2k∧2LG2k^R^−μ∧′G1k^R^,a^2=β^A1k^R^−A2k^R^+2k∧2Ohμ^G1k^R^−G2k^R^+2k∧2β^LG2k^R^−μ∧′ρ^G1k^R^,a^3=k^R^Ce1−Ce2+2k∧2β^OhG2k^R^−μ^ρ^G1k^R^+k^R21−m2−k∧2R∧2+k∧2H∧21−ε∧m2b1k^b2k^ε^mB1k^b2k^−B2k^b1k^.


## 5. Stability Discussion

In this section, the variation of growth rateω^I with wave number k^ is plotted using the Mathematica software for physical parameters included in our considered system. The system is stable when ω^I<0, while the system is said to be unstable if ω^I>0. The case ω^I=0 is the neutrally stable. Figures 2–11 are depicted in asymmetric case (n=1). The growth rate of disturbance with different values of Ohnesorge number Oh has been plotted in Figure 2. It demonstrates in Figure 2 that the increasing of Oh 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, ρ2 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.Figure 2
Growth rate curves for various values of Ohnesorge number atρ^=0.002, μ∧′=0.6, μ^=0.4, L=0.3, β^=0.6, ε^m=0.3, Ce2=3, Ce1=6, and H^=20.Figure 3
Influence of density ratio of fluids on growth rate curves atOh=100, μ∧′=0.6, μ^=0.4, L=0.3, β^=0.6, ε^m=0.3, Ce2=3, Ce1=6, and H^=20.Figure 4
Influence of the centrifuge numberCe1 on growth rate curves at ρ^=0.002, μ∧′=0.5, μ^=0.03, L=0.3, β^=0.3, ε^m=0.2, Oh=100, Ce2=9, and H^=10.Figure 5
Influence of the centrifuge numberCe2 on growth rate curves at ρ^=0.002, μ∧′=0.5, μ^=0.03, L=0.3, β^=0.3, ε^m=0.2, Oh=100, Ce1=9, and H^=10.Figure 6
Influence of rotation on growth rate curves atρ^=0.002, μ∧′=0.5, μ^=0.03, L=0.3, β^=0.3, ε^m=0.2, Oh=100, and H^=10.Figure 7
Effect of heat and mass transfer parameterβ^ on growth rate curves at ρ^=0.002, μ∧′=0.6, μ^=0.4, L=0.3, Oh=100, ε^m=0.3, Ce2=3, Ce1=6, and H^=20.Figure 8
Effect of magnetic field on growth rate atρ^=0.002, μ∧′=0.5, μ^=0.03, L=0.3, β^=0.3, ε^m=0.3, Ce2=3, Ce1=6, and Oh=100.Figure 9
Effect of magnetic permeability ratio of two fluids on growth rate curves atρ^=0.002, μ∧′=0.5, μ^=0.03, L=0.3, β^=0.3, H^=20, Ce2=3, Ce1=6, and Oh=100.Figure 10
Effect of viscoelastic ratio of fluids on growth rate atρ^=0.002, Oh=100, μ^=0.4, L=0.3, β^=0.6, ε^m=0.3, Ce2=3, Ce1=6, and H^=20.Figure 11
Effect of viscosity ratio of fluids on growth rate curves atρ^=0.002, Oh=100, μ^=0.4, L=0.3, β^=0.6, ε^m=0.3, Ce2=3, Ce1=6, and H^=20.The growth rates have been plotted in Figure3 for different values of density ratio of fluids ρ^=0.006,0.004,and0.002. It has been noted that the increasing of density ratio of fluids ρ^ 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 forCe1=9, 6, and 3 have been plotted in Figure 4. From this figure, it is noted that as the centrifuge number Ce1 of inside cylinder increased, the values of the growth rate of perturbation decreased and so the increase in Ce1 resists disturbances from developing at the interface of two fluids, and therefore, flow moves towards stability. Figure 5 shows the effect of Ce2 on the growth rate curves. It is observed from Figure 5 that the increasing of Ce2 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 parameterβ^, on the growth rate ω^I versus wave number k^ is illustrated in Figure 7. It is observed that as the heat and mass transfer parameter β^ increases, the values of the growth rate decrease. Hence, one may conclude that the mass and heat transfer parameter β^ has stabilizing effect on the considered system. This result is similar to a result that had been obtained by Aswasthi and Agarwal [4].Figure8 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 ε^m=0.9,0.8,and0.7. It is observed that the growth rate decreases on increasing magnetic permeability ratio ε^m. Therefore, the magnetic permeability ratio ε^m has a stabilizing nature.Figure10 illustrates the variation of growth rate ω^I with wave number k^ for different values of viscoelastic ratio of two fluids μ∧′=0.8, 0.7, and 0.6. 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 μ^ on the growth rate is displayed in Figure 11. It is observed that as the viscosity ratio of fluids μ^ increases, the growth rate increases, and the system gets destabilized. Therefore, μ^ has destabilizing behavior


## 6. Conclusions

The linear instability analysis of two concentric rotating cylinders of Walter’sB′ 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 behaviorIt 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.



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