The Instability of an Electrohydrodynamic Viscous Liquid Micro-Cylinder Buried in a Porous Medium: Effect of Thermosolutal Marangoni Convection

Abstract

The electrohydrodynamic (EHD) thermosolutal Marangoni convection of viscous liquid, in the presence of an axial electric field through a micro cylindrical porous flow, is considered. It is assumed that the surface tension varies linearly with both temperature and concentration. The instability of the interface is investigated for the free surface of the fluid. The expression of the free surface function is derived taking into account the independence of the surface tension of the heat and mass transfer. The transcendental dispersion relation is obtained considering the dependence of the surface tension on the heat and mass transfer. Numerical estimations for the roots of the transcendental dispersion relation are obtained indicating the relation between the disturbance growth rate and the variation of the wave number. It is found that increasing both the temperature and concentration at the axial microcylinder has a destabilizing effect on the interface, according to the reduction of the surface tension. The existence of the porous structure restricts the flow and hence has a stabilizing effect. Also, the axial electric field has a stabilizing effect. Some of previous analytical and experimental results are recovered upon appropriate data choices.

1. Introduction

The fluid flow in microjet has various applications in medical, biomedical, computer chips and chemical separations. Advent of microelectrical mechanical systems is one of the major advances in industrial technologies. Also, microjets are fundamental to connecting different devices, utilized in biochemical reaction chamber, in physical particle separation, in inkjet print heads, in infrared detectors, in diode lasers, in miniature gas chromatographs, and in heat exchanger for cooling computer chips. Understanding the flow characteristics through a microjet is very important in determining the free surface instability of the fluid flow. The augmentation of heat transfer from a solid cylinder wrapped with a porous layer was considered by Bhattacharyya and Singh [1]. An experimental analysis of unsteady heat and moisture transfer around a heated cylinder buried into a porous medium was performed by Moya et al. [2]. Furlani [3] studied the temporal instability of an infinite Newtonian cylindrical microjet (as a cylindrical fluid surface) that is subjected to a sinusoidal variation of surface tension along its length. Furlani and Hanchak [4] developed the same problem to study numerically the nonlinear analysis of the deformation and breakup of viscous microjets. The instability of non-Newtonian liquid jets (viscoelastic and power law) was investigated by Gao [5] and Gao and Ng [6]. The dispersion relation between the growth rate and the wave number for a non-Newtonian cylindrical liquid jets was derived and the instability behavior of viscoelastic jets was investigated.

The interaction of electric fields with moving fluids makes EHD a very complicated phenomenon. The process of EHD is dependent on many parameters and properties of the liquid and environment of the flow. On the other hand, EHD can be considered as the branch of fluid mechanics concerned with the electrical force effects or as the part that is involved in the influence of moving media on electric fields. It is thus concerned with the interaction between electrical fields and free or polarized charges in fluids. In the EHD stability, the electric field plays an important role in many practical problems of chemical engineering, biophysics and many other fields. Mohamed and Nayyar [7] have investigated the stability of a cylindrical jet of incompressible inviscid liquid in the presence of an axial electric field. They showed that the uniform axial electric field has a strong stabilizing influence on the cylindrical interface for short and long wavelengths in all symmetric and asymmetric modes of perturbation.

Marangoni boundary layers are dissipative layers which may occur along liquid-liquid or liquid-gas interfaces. When a free liquid surface is present, the surface tension variation is resulting from the temperature gradients along the surface. It can also induce a motion within the fluid which is called thermocapillary flow (thermal Marangoni convection) [8]. The surface tension gradients that are responsible for Marangoni convection depend on both temperature and/or concentration gradients. Earliest work in this field was first investigated by Napolitano [9, 10]. Surfactants or surface contaminants are used to control the stability of the free surface by lowering surface tension. Variations in surface tension that are introduced by surfactant can substantially alter interfacial evolution and flow. The effect of surfactants on liquid has been studied by many authors [11–15]. McGough and Basaran [16] studied the breakup mechanisms of the fluid threads containing surfactant by solving the Navier-Stokes equations. The effect of the Prandtl number on Marangoni convection over a flat plate was studied by Christopher and Wang [17]. Physically, the heat and mass transfer across the interface play an essential role in determining the flow field in some cases. For instance, when the fluid is boiling, whether it is a film or bubble, the motion of the film and bubbles is mainly dependent on the effect of mass and heat transfer. In chemical industry, the interfacial stability problems are important in the presence of mass and heat transfer. The mechanism of heat and mass transfer across an interface is of a great importance in numerous industrial and environmental problem processes. These include the design of many types of contacting equipment, for example, boilers, condensers, evaporators, gas absorbers, pipelines, chemical reactors, and nuclear reactors. The effect of heat and mass transfer in the absence of gravity on the surface tension was studied by Straub [18].

The flow through porous media is usually described by the Darcy’s law that relates the movement of the fluid to the pressure gradient acting on a parcel of the fluid. Meanwhile, the Darcy’s equations are not the fundamental equations for the flow through porous media. It is an approximation for the balance of linear momentum for the fluid flow through a porous solid within the context of mixtures. For example, the interactive force between the fluid and the porous medium, the frictional effects due to viscosity, and the inertial nonlinearities are not included in the Darcy’s model [19]. These effects can be included in more generality model like Brinkman-Darcy’s model. The basic assumptions that lead to the Brinkman-Darcy equation were illustrated by Rajagopal [19]. Elcoot and Moatimid [20] studied the instability of finitely conducting cylindrical flows through porous media, under the influence of an axial electrostatic field. They found that, under certain conditions, the field may have a stabilizing or destabilizing effect. An experimental study of the steam injection into a porous media and the stability of the interface between the steam and the water was carried out by Catton and Chung [21].

The aim of this work is to discuss the effect of axial electric field with the existence of the porous structure on the stability of the microcylindrical flow of a viscous liquid past a microsolid cylindrical surface. Also, the effect of the Marangoni thermosolutal convection (according to the dependence of the surface tension on the heat and mass transfer) is discussed. The main construction of the problem is as follows. Firstly, we studied the problem according to spatial surface tension. So, the expression of the free surface function is obtained. Secondly, the problem is studied in the presence of the heat and mass transfer rates with thermosolutal surface tension. The dispersion relation, in the latter case, is obtained as a transcendental function of the growth rate and the wave number. Finally, we studied the effect of the porous medium and the axial electric field on the stability of viscous liquid interface and tabulated results are obtained.

To clarify the problem, in Section 2, the physical description of the problem including the basic equations that govern the motion is presented. Section 3 is devoted to introducing the interfacial conditions for both the fluid and the electric field together with the boundary conditions. The method of solution according to the normal modes technique is presented in Section 4. In Section 5, the instability analysis in the absence of the heat and mass transfer is studied and the expression of the free surface function is derived. The transcendental dispersion relation in the presence of the heat and mass transfer is obtained in Section 6. Throughout Section 7, we introduced numerical estimations and gave a discussion for the stability picture according to the relation between the growth rate and the wave number. Finally, in Section 8, we give concluding remarks for this study based on the obtained results of the stability analysis.

2. Formulation of the Problem

Consider a microcylindrical layer of viscous fluid with density 𝜌, viscosity 𝜇, and radius 𝑟₀ flowing past an axial solid cylinder of infinitely small radius. The flow is surrounded by a vacuum (which has zero density and zero viscosity) with hypothetical sufficiently large radius 𝑅₂. The axial microsolid cylinder is elongated along the 𝑧-axis. It is earthed and has a radius 𝑅₁ (where 𝑅₁≪𝑅₂). The axial microcylindrical solid rod, which induces temperature and concentration gradients, is considered to be at a constant temperature 𝑇₁ and constant concentration 𝐶₁. The subscripts (1) and (2) refer to the liquid and the surrounding vacuum, respectively. The fluid and the vacuum are influenced by an external uniform electric field of intensity 𝐸₀ which acts along the positive 𝑧-axis. The dielectric constants for the liquid and the vacuum are 𝜀₁ and 𝜀₂, respectively. The interface between the fluid and the surrounding vacuum is considered to be free of charges. The flow jet is considered as a microcylindrical porous structure with Darcy’s coefficient 𝜈, while its porosity is assumed to be unity (this will be explained physically later). Also, we neglect the effects of gravity. Considering the previous hypotheses, the model may be illustrated graphically as in Figure 1.

Figure 1 

Physical model and flow cylindrical geometry.

After making a small disturbance, the interface function becomes 𝐹(𝑟,𝑧,𝑡)=𝑟−ℎ(𝑧,𝑡), (1) and the unit outward normal vector 𝑛 and the unit tangential one 𝑡̂ are given by 1𝑛=1+ℎ²_𝑧̂ℎ𝑟−^𝑧1+ℎ²_𝑧̂ℎ𝑧,𝑡=^𝑧1+ℎ²_𝑧̂1𝑟+1+ℎ²_𝑧𝑧, (2) where 𝑟̂and 𝑧 are the unit vectors along the radial and axial cylindrical coordinate directions and ℎ_𝑧=𝜕ℎ/𝜕𝑧.

According to the assumption of the viscosity of the fluid, considering the existence of the porous structure and the heat and mass transfer, the basic equations of the problem may take the following form [3, 19].

The continuity equation (incompressibility condition) requires ∇⋅𝑣=0. (3) The balance of linear momentum (Navier-Stokes) gives 𝜌1𝜁𝜕𝑣+1𝜕𝑡𝜁²𝑣⋅∇𝑣=−∇𝑃+𝜇_eff∇²𝑣−𝜈𝑣. (4) The balance of energy is 𝐷𝑇𝐷𝑡=𝛼∇²𝑇. (5) And finally the concentration equation becomes 𝐷𝐶𝐷𝑡=𝛽∇²𝐶. (6) In the previous equations (3)–(6), 𝜁 is the porosity of the porous medium, 𝜇_eff is the effective viscosity of the fluid (𝜇_eff=𝜇/𝜁 and if 𝜁=1 then 𝜇_eff=𝜇 as we mention in the next paragraph), and (𝐷/𝐷𝑡)=(𝜕/𝜕𝑡)+(𝑣⋅∇)𝑣. Also, 𝛼 is the thermal diffusivity, 𝛽 is the mass diffusivity, 𝑇 is the temperature, 𝐶 is the concentration, 𝑃 is the pressure, 𝑣 is the velocity vector, 𝜈 is the Darcy’s coefficient, and 𝜌 is the density.

The basic assumptions that lead to the Brinkman-Darcy equation and the form of the basic assumptions that lead to the Brinkman-Darcy equation were illustrated by Rajagopal [19], and can be summarized in the following points.(1)The porous medium is solid and thus the balance of linear momentum of the porous medium can be ignored.(2)The interactive force between the fluid and the porous medium is due to the frictional forces only and this force is proportional to the flow velocity which is represented by the term 𝜈𝑣, where 𝜈=𝜇/𝜆, 𝜇 is the fluid viscosity and 𝜆 is the permeability of the porous medium.(3)The frictional effects due to viscosity were taken into account by the term 𝜇_eff∇²𝑣, where 𝜇_eff=𝜇/𝜁.(4)The flow is unsteady and sufficiently fast, so that the inertial nonlinearities cannot be ignored; thus the term (1/𝜁²)(𝑣⋅∇)𝑣 needs to be retained.According to the previous assumptions, the balance of linear momentum can be written as in (4). Also, we want to confirm the following points.(1)The velocity of the flow through the porous medium is the average of the fluid velocity. This quantity has been given various names by different authors, such as seepage velocity, filtration velocity, superficial velocity, or Darcy velocity [22]. The seepage velocity 𝑣 is related by the velocity of the fluid 𝑉 (where 𝑉 is the flow velocity in the absence of the porous medium) by 𝑣=𝜁𝑉.(2)The porosity 𝜁 for natural media does not normally exceed 0.6. For beds of solid spheres of uniform diameter 𝜁 can vary between the limits 0.2595 (rhombohedral packing) and 0.4764 (cubic packing). Nonuniformity of grain size tends to lead to smaller porosities than those for uniform grains, because smaller grains fill the pores formed by larger grains. For man-made materials such as metallic foams 𝜁 can approach the value 1 (see the tabulated values for the porosity and permeability in [22, page 5]).(3)The term (1/𝜁²)(𝑣⋅∇)𝑣 is inappropriate if the flow is sufficiently slow. Then the inertial nonlinearities can be neglected [19]. However, this term needs to be retained in the case of highly porous media where the flow is sufficiently fast and at least the irrotational part of this term needs to be retained [22] and for incompressible fluids this term reduces to 𝜁⁻¹∇[𝜁⁻¹𝑣⋅𝑣], and for highly porous media we can assume 𝜁=1 (as we assume in the present problem).(4)Brinkman sets the viscosity 𝜇 and the effective viscosity 𝜇_eff equal to each other but in general that is not true [22]. The difference between 𝜇 and 𝜇_eff is due to the momentum dispersion and 𝜇_eff is dependent on the type of the porous medium as well as the strength of flow, and it is common practice to take 𝜇_eff equal to 𝜇 for high porosity media [23].(5)Experimental checks of Brinkman’s theory have been indirect and few in number. The Brinkman’s equation reduces to Darcy’s equation when the spatial length scale is much greater than (𝜇_eff𝜆/𝜇)^1/2. Therefore the term ∇²𝑣 is negligible in comparison with the term proportional to 𝑣 [22]. But, the Brinkman’s model holds only for particles whose size is of order 𝜗³, where 𝜗≪1 is the distance between neighboring particles, and for large particles the fluid filtration is governed by Darcy’s law [22].Because of the quasi-static approximation and in poorly conducting fluid [24, 25], the induced magnetic field is negligible. Also, we deal with very small electric conductivity and there is no applied magnetic field. So, the electric field is conservative. Since, the electric field is curl free vector the Maxwell’s equations are ∇×𝐸=0 and ∇⋅𝜀𝐸=0. So, the electric field has an electric scalar potential 𝜙 such that 𝐸=𝐸₀𝑧−∇𝜙. Therefore the electric potential 𝜙_𝑗 satisfies the Laplace’s equation ∇²𝜙_𝑗=0𝑗=1,2. (7) The boundary and interfacial conditions for the problem may be displayed in the following section.

3. Boundary Conditions

The boundary conditions adopted here must be classified into three categories. The first relates the hydrodynamic part at the boundaries. The second is concerned with the electric part at the boundaries. Meanwhile, the third relates the combination condition of the electric and hydrodynamic balance of the stress tensor components at the interface between the liquid and the surrounding vacuum.

(1) For the hydrodynamic part, because 𝐹 (in (1)) is a scalar function that is always equal to zero at any point on the fluid interface, its time derivative following any material point on the interface is obviously equal to zero, which means that there is no phase transformation occurring [26, Chapter 2, page 75]. So, the continuity of the normal velocity at the interface requires 𝐷[(𝐷𝑡𝑟−ℎ𝑧,𝑡)]=0,at𝑟=ℎ, (8) where 𝑟 is the radial distance and ℎ is the radius for the disturbed fluid surface.

According to the microscale of the cylindrical fluid radius, the standard no-slip boundary condition in the classical fluid mechanics does not apply, so that the shear stress at the boundary vanishes (i.e., there is no considerable surface fraction between the fluid and the axial bar) and hence 𝜕𝑣_𝑧/𝜕𝑟=0 at the axial bar. Meanwhile, the normal velocity must be vanishing at the axial bar. So the boundary conditions for the velocity at the surface of the microcylindrical axis may be written as follows: 𝑣_𝑟=0,𝑟=𝑅₁,𝜕𝑣_𝑧𝜕𝑟=0,𝑟=𝑅₁, (9) where 𝑅₁ is the microradius of the axial microcylindrical bar such that 𝑅₁ is sufficiently small (𝑅₁≪1) and 𝑣_𝑟,𝑣_𝑧 are the radial and axial velocity components, respectively.

(2) For the electric part, because there is no surface charges accumulated at the interface, the normal electric displacement must be continuous across the interface. So that 𝑛⋅‖𝜀𝐸‖=0 and the interfacial condition for the normal electric field displacement, in the linear form, yields 𝐸₀ℎ_𝑧𝜀_𝑗+𝜀_𝑗𝜕𝜙_𝑗𝜕𝑟=0,at𝑟=ℎ, (10) where the notation ‖‖ denotes the subtract at the fluid and surrounding vacuum. In other words ‖𝑓‖=𝑓₁−𝑓₂. The subscript 𝑗, in the electric potential 𝜙_𝑗 and the dielectric constant 𝜀_𝑗, takes the values 𝑗=1,2 at the fluid and the surrounding vacuum, respectively.

The continuity of the tangential electric filed components at the interface requires 𝑛×‖𝐸‖=0 and in the linear form (where the linear terms only are retained and the nonlinear terms are omitted) becomes 𝜕𝜙_𝑗𝜕𝑧=0,at𝑟=ℎ. (11)

Since the axial microsolid bar and the vacuum at infinity have no electric field, the electric potential at the boundaries satisfies ∇𝜙₁=0,𝑟=𝑅₁,∇𝜙₂=0,𝑟=𝑅₂. (12)

(3) The balance of the stress components at the interface may be presented as follows.

At the interface between the liquid and vacuum, the fluid and the electrical stresses must be balanced. The components of these stresses consist of the electric hydrodynamic stresses together with the surface tension force [27]. The combination of the electric stress 𝜏^electro_𝑖𝑗 and hydrodynamic stress 𝜏^hydro_𝑖𝑗 parts may be presented in the total stress as 𝜏_𝑖𝑗=𝜏^hydro_𝑖𝑗+𝜏^electro_𝑖𝑗. (13) The electric force density 𝐹_𝑒 can be written in terms of the electric field as 𝐹_𝑒=(∇⋅𝜀𝐸)𝐸. Since ∇×𝐸=0, the electric force can be written as 𝐹_𝑒=(∇⋅𝜀𝐸)𝐸+(∇×𝐸)×𝜀𝐸 and the Maxwell stress tensor 𝜏^electro_𝑖𝑗, in the index notation, may be written as [28] 𝜏^electro_𝑖𝑗=𝜀𝐸_𝑖𝐸_𝑗−12𝜀𝐸²Δ_𝑖𝑗, (14) meanwhile, the hydrodynamic stress tensor may be described by the constitutive relation 𝜏^hydro_𝑖𝑗=−𝑃Δ_𝑖𝑗+𝜇𝜕𝑣_𝑖𝜕𝑥_𝑗+𝜕𝑣_𝑗𝜕𝑥_𝑖, (15) where Δ_𝑖𝑗 is the usual kronecker delta and 𝑣_𝑖, 𝑣_𝑗, 𝑥_𝑖, 𝑥_𝑗 are a general notations for the velocity components and the coordinates, respectively.

According to the simple interface description, which involves only interfacial tension, the forces acting on any segment of an interface are of two kinds. First, there are the bulk pressure and stresses that act on the faces of the interface element and produce a net effect that is proportional to the surface area; second, there is a tensile force that is due to surface or interfacial tension that acts in the plane of the interface at the edges of the surface element and is specified by means of the magnitude of the surface or interfacial tension as a force per unit length [26, Chapter 2, page 76]. So, the stress balance can be written as [3] 𝑛⋅𝜏=−2𝐻𝜎(𝑧)𝑛+∇_𝑆𝜎(𝑧), (16) where 𝐻 is the curvature of the interface, ∇_𝑆 is the surface gradient at the interface, and 𝜎(𝑧) is the surface tension.

The previous condition can be decomposed into the normal and tangential stress tensor components as follows.(i) The normal stress component requires (𝑛⋅𝜏)⋅𝑛=−2𝐻𝜎(𝑧). (17)(ii) The tangential stress components yield (𝜏⋅𝑛)⋅𝑡̂=𝑡̂⋅∇_𝑆𝜎(𝑧), (18) where the surface curvature and the gradient at the interface may be given as follows: 1𝐻=21∇⋅𝑛=21ℎ1+ℎ²_𝑧−ℎ_𝑧𝑧1+ℎ²_𝑧^3/2,∇_𝑆≡12∇−𝑛𝑛=𝑟̂ℎ⋅∇^𝑧1+ℎ²_𝑧𝜕+𝑧1𝜕𝑧1+ℎ²_𝑧𝜕.𝜕𝑧 (19)Finally, the normal and tangential stress tensor conditions as given in (17) and (18) (at 𝑟=ℎ) may be written as follows: −𝑃+2𝜇1+ℎ²_𝑧𝜕𝑣_𝑟𝜕𝑟+ℎ²_𝑧𝜕𝑣_𝑧𝜕𝑧−ℎ_𝑧𝜕𝑣_𝑟+𝜕𝑧𝜕𝑣_𝑧+1𝜕𝑟21+ℎ²_𝑧𝜀_𝑗1−ℎ²_𝑧𝜕𝜙_𝑗𝜕𝑟²−1−ℎ²_𝑧𝐸₀−𝜕𝜙_𝑗𝜕𝑧²+4ℎ_𝑧𝜕𝜙_𝑗𝐸𝜕𝑟⁰−𝜕𝜙_𝑗(𝑧)𝜕𝑧=−2𝜎𝐻,𝜇1+ℎ²_𝑧2ℎ_𝑧𝜕𝑣_𝑟−𝜕𝑟𝜕𝑣_𝑧+𝜕𝑧1−ℎ²_𝑧𝜕𝑣_𝑟+𝜕𝑧𝜕𝑣_𝑧+1𝜕𝑟1+ℎ²_𝑧𝜀_𝑗ℎ_𝑧𝜕𝜙_𝑗𝜕𝑟²−ℎ_𝑧𝐸₀−𝜕𝜙_𝑗𝜕𝑧²−1−ℎ²_𝑧𝜕𝜙_𝑗𝐸𝜕𝑟⁰−𝜕𝜙_𝑗=̂𝜕𝑧𝑡⋅∇_𝑆𝜎(𝑧). (20)(21)

We study the solution of the equations of motion (4) according to the boundary and interfacial conditions in two cases. The first case is obtained in the absence of the heat and mass transfer where the spatial surface tension is independent of the rate of heat and mass transfer. In this case, we follow the same analysis of Furlani [3] to gain the function of the free surface of the microcylindrical fluid flow. In the second one, taking into account the dependence of the surface tension on the heat and mass transfer, the surface tension becomes a function of the time and the axial distance. So, in this case, the dispersion relation may be obtained as a transcendental function and its roots can be computed numerically hence the stability picture may be obtained graphically. Before studying the previous cases, we obtain the solution of the electric field according to the boundary and interfacial conditions.

4. Solution of the Eclectic Field

Our analysis will be based on the normal modes technique. Therefore, the electric potential may take the following form: 𝜙_𝑗(𝑟,𝑧,𝑡)=𝜑_𝑗(𝑟)𝑒^{𝜔𝑡+𝑖𝑘𝑧}, (22) where 𝜔 is the frequency of the surface wave and 𝑘 is its wave number which is assumed to be real and positive. Using (7) with the boundary conditions (10)–(12), we may obtain the solution for the electric potential functions as follows: 𝜙₁()=𝑟,𝑧,𝑡𝑖𝑟₀𝐸₀𝜀₂−𝜀₁𝑔₂(𝑘)𝜀₂𝑔₁(𝑘)𝐺₂(𝑘)−𝜀₁𝑔₂(𝑘)𝐺₁(𝑘)×𝐾₀(𝑘𝑟)𝐼^′₀𝑘𝑅₁−𝐾^′₀𝑘𝑅₁𝐼₀(𝑘𝑟)𝑒^{𝜔𝑡+𝑖𝑘𝑧},𝜙₂()=𝑟,𝑧,𝑡𝑖𝑟₀𝐸₀𝜀₂−𝜀₁𝑔₁(𝑘)𝜀₂𝑔₁(𝑘)𝐺₂(𝑘)−𝜀₁𝑔₂(𝑘)𝐺₁(𝑘)×𝐾₀(𝑘𝑟)𝐼^′₀𝑘𝑅₂−𝐾^′₀𝑘𝑅₂𝐼₀(𝑘𝑟)𝑒^{𝜔𝑡+𝑖𝑘𝑧}, (23) where 𝐼₀ and 𝐾₀ are the modified Bessel’s functions of first and second kinds of order zero, respectively. The dashes denote the differentiation with respect to 𝑟 and 𝑖=√−1. Also, the functions 𝑔₁(𝑘), 𝑔₂(𝑘), 𝐺₁(𝑘), and 𝐺₂(𝑘) are given in the appendix.

5. Instability in the Absence of Heat and Mass Transfer

This section is devoted to solving the equations of motion and the continuity equation (see (3) and (4)) according to the interfacial and boundary conditions (see (8) and (9)) with employing the normal and tangential stress tensor interfacial conditions. We follow the same procedure given by Furlani [3] to get the function that describes the free surface. So, we expand the velocity components and the pressure as a power of 𝑟 (depending on the microradius of the flow jet), as follows: 𝑣_𝑧(𝑟,𝑧,𝑡)=𝑣₀(𝑧,𝑡)+𝑣₂(𝑧,𝑡)𝑟²𝑃()+⋯,𝑟,𝑧,𝑡=𝑃₀(𝑧,𝑡)+𝑃₂(𝑧,𝑡)𝑟²+⋯. (24)

From the continuity equation (3), one gets 𝑣_𝑟(𝑟,𝑧,𝑡)=−𝜕𝑣₀(𝑧,𝑡)𝑟𝜕𝑧2−𝜕𝑣₂(𝑧,𝑡)𝑟𝜕𝑧4³+⋯, (25) where 𝑣₀(𝑧,𝑡), 𝑣₂(𝑧,𝑡) are perturbed velocity functions and 𝑃₀(𝑧,𝑡), 𝑃₂(𝑧,𝑡) are the perturbed pressure functions. It is important to notice here that the expansions in (24)-(25) are compatible with the boundary conditions in (9) and with the continuity equation (3). Where 𝑅₁ and 𝑟 are a microradii of the axial microcylinder and the microporous cylinder that contains the fluid, respectively. So, all terms containing 𝑂(𝑟) and 𝑂(𝑅₁) or higher orders can be neglected according to their tendency to zero.

According to the previous expansions, the equation of motion (4) for 𝑣_𝑟 can be neglected because it is identical to the lowest order of 𝑟. Therefore the equation of motion for 𝑣_𝑧 can be written as 𝜌𝜕𝑣₀𝜕𝑡+𝑣₀𝜕𝑣₀𝜕𝑧=−𝜕𝑃₀𝜕𝑧+𝜇4𝑣₂+𝜕²𝑣₀𝜕𝑧²−𝜈𝑣₀. (26) The second order term of the velocity 𝑣₂ can be determined from the tangential stress condition as given in (21); on neglecting the terms of 𝑂(𝑟) and all higher orders, it can be written as follows: 𝑣₂=12𝜇ℎ𝜕𝜎+𝜕𝑧3ℎ_𝑧2ℎ𝜕𝑣₀+1𝜕𝑧4𝜕²𝑣₀𝜕𝑧²+𝜀_𝑗𝐸₀2𝜇ℎ𝜕𝜙_𝑗−𝜀𝜕𝑟^𝑗𝐸₀ℎ_𝑧𝜇ℎ𝜕𝜙_𝑗.𝜕𝑧 (27) The zero order term of the pressure 𝑃₀ can be determined from the normal stress condition, as given in (20), in the form 𝑃₀=−𝜇𝜕𝑣₀𝜀𝜕𝑧+2𝜎𝐻+_𝑗𝐸₀𝜕𝜙_𝑗.𝜕𝑧 (28) Now, substituting from (27) and (28) into (26), we obtain 𝜌𝜕𝑣₀𝜕𝑡+𝑣₀𝜕𝑣₀𝜕𝜕𝑧=−()+𝜕𝑧2𝜎𝐻2𝜀_𝑗𝐸₀ℎ𝜕𝜙_𝑗−𝜕𝑟4𝜀_𝑗𝐸₀ℎ_𝑧ℎ𝜕𝜙_𝑗𝜕𝑧−𝜀_𝑗𝐸₀𝜕²𝜙_𝑗𝜕𝑧²+3𝜇ℎ²𝜕ℎ𝜕𝑧²𝜕𝑣₀+2𝜕𝑧ℎ𝜕𝜎𝜕𝑧−𝜈𝑣₀. (29) The kinematic condition equation (8) gives 𝜕ℎ𝜕𝑡=−𝑣₀𝜕ℎ−ℎ𝜕𝑧2𝜕𝑣₀.𝜕𝑧 (30) Also, the axial velocity 𝑣₀(𝑧,𝑡), the radial position ℎ(𝑧,𝑡), and the surface tension 𝜎(𝑧) can be represented in a perturbed and unperturbed parts as follows: 𝑣₀(𝑧,𝑡)=𝑢₀+𝑢(𝑧,𝑡),ℎ(𝑧,𝑡)=𝑟₀(1+𝛿(𝑧,𝑡)),𝜎(𝑧)=𝜎₀+𝜎₁(𝑧), (31)(32)(33) where 𝑢₀, 𝑟₀, and 𝜎₀ are the constant unperturbed axial streaming velocity, radius, and surface tension, respectively. The analysis may be simplified by using the transformation 𝜂=𝑧−𝑢₀𝑡. (34) By using (31)–(34) into (29) and (30), after linearizing the resulting equations, we can obtain 𝜌𝜕𝑢=1𝜕𝑡𝑟₀𝜕𝜎₁+𝜕𝜂2𝜀_𝑗𝐸₀𝑟₀𝜕𝜙_𝑗𝜕𝑟−𝜀_𝑗𝐸₀𝜕²𝜙_𝑗𝜕𝜂²𝜕+3𝜇²𝑢𝜕𝜂²+𝜎₀𝑟₀𝜕𝛿𝜕𝜂+𝜎₀𝑟₀𝜕³𝛿𝜕𝜂³𝜕−𝜈𝑢,₂𝑢𝜕𝜂²𝜕=−2²𝛿.𝜕𝜂𝜕𝑡 (35) Equations (35) can be combined into one equation that determines the free surface function 𝛿(𝜂,𝑧) as follows: 𝜕²𝛿𝜕𝑡²+𝜕𝜈𝜕𝑡𝜌𝛿−3𝜇𝜌𝜕²𝛿𝜕𝜂²+𝜎₀2𝜌𝑟₀𝜕²𝛿𝜕𝜂²+𝑟²₀𝜕⁴𝛿𝜕𝜂⁴+𝜀_𝑗𝐸₀2𝜌𝑟₀2𝜕²𝜙_𝑗𝜕𝑟𝜕𝜂−𝑟₀𝜕³𝜙_𝑗𝜕𝜂³1=−2𝜌𝑟₀𝜕²𝜎₁𝜕𝜂². (36) Initially, the disturbed fluid radius 𝛿(𝜂,𝑡) is independent of the time (also its partial derivative with respect to the time). So, the disturbed fluid radius 𝛿(𝜂,𝑡) has the following initial conditions: 𝛿𝜕𝜂,0=0,𝛿𝜕𝑡𝜂,0=0. (37) Assuming that the surface tension 𝜎₁(𝜂) has a spatially periodic form, which is independent of the heat and mass transfer. Then, the surface tension function may be written in the following form: 𝜎₁𝜂=−Δ𝜎21+𝑒^𝑖𝑘𝜂, (38) where Δ𝜎 is a constant variation of the surface tension.

The initial value problem given by (36) and the initial conditions equation (37) can be solved by decomposing the solution into a homogenous and particular parts, 𝛿_ℎ(𝜂,𝑡) and 𝛿_𝑝(𝜂,𝑡), respectively. Based on the form of the surface tension 𝜎₁(𝜂), which is given in (38), the homogenous solution of the wave function 𝛿_ℎ(𝜂,𝑡) may be written in the form 𝛿_ℎ𝜂,𝑡=𝑒^{𝜔𝑡+𝑖𝑘𝜂}. (39) The final form of the general solution of the free surface function, 𝛿(𝜂,𝑡)=𝛿_ℎ(𝜂,𝑡)+𝛿_𝑝(𝜂,𝑡), can be written as follows: 𝛿𝜂,𝑡=−Δ𝜎𝑘²4𝜌𝑟₀𝑏₀𝜔₋𝜔₊−𝜔₋𝑒^𝜔+𝑡−𝜔₊𝜔₊−𝜔₋𝑒^𝜔−𝑡+1𝑒^𝑖𝑘𝜂, (40) where, the roots 𝜔_± and 𝑏₀ are defined in the appendix.

It is worthwhile to mention here that the final solution in (40) reduces to the same solution obtained earlier by Furlani [3] for the flow of cylindrical microjet surface in the absence of the electric field (𝐸₀=0). It is worthwhile to mention here that the final solution in (40) reduces to the same solution that was obtained earlier by Furlani [3] for the flow of cylindrical microjet surface in the absence of the electric field (𝐸₀=0) with neglecting the porous structure (𝜈=0) and with taking the real part only of the free surface function.

6. Instability of Thermosolutal Marangoni Convection

If the two-fluid system is a multicomponent system, it is often the case that there may be a preferential concentration of one or more of the components at the interface (e.g., if we consider a system of pure 𝐴 and pure 𝐵, which are immiscible, with a third solute component 𝑆 that is soluble in 𝐴 and/or 𝐵 but that is preferentially attracted to the interface), and then the interfacial tension will also be a function of the (surface-excess) concentration of these solute components. Both the temperature and the concentrations of adsorbed species can be functions of position on the interface, thus leading to spatial gradients of the surface tension [26]. So, to study the effect of the heat and mass transfer on the interfacial instability of the microcylindrical surface, the surface tension is assumed to be a function of the temperature and the solute concentration and the surface tension will depend on temperature and concentration linearly [29], as follows: 𝜎𝜂,𝑡=𝜎₀+𝜎₁𝜂,𝑡=𝜎₀1−𝛾_𝑇𝑇−𝑇₀−𝛾_𝐶𝐶−𝐶₀, (41) where 𝛾_𝑇 and 𝛾_𝐶 denote the temperature and concentration coefficients of the surface tension. Also, 𝑇₀ and 𝐶₀ are the reference temperature and concentration, respectively. Note that (41) is the same as (33) with 𝜎₁(𝑧) being 𝜎₁(𝜂,𝑡) that depends on the heat and mass transfer as in (41). 𝜎₀ is the unperturbed constant value of the surface tension.

To determine the form of the surface tension, we need firstly to obtain the solution of the heat and concentration equations. Again, the analysis will be based on the normal modes technique as given by 𝑇𝑟,𝜂,𝑡=𝑇_𝑠(𝑟)𝑒^{𝜔𝑡+𝑖𝑘𝜂},𝐶𝑟,𝜂,𝑡=𝐶_𝑠(𝑟)𝑒^{𝜔𝑡+𝑖𝑘𝜂}. (42) With the help of the coordinate transformation in (34), the energy and concentration equations (see (5) and (6)) may be written in the following forms: 𝑑²𝑇_𝑠𝑑𝑟²+1𝑟𝑑𝑇_𝑠𝑑𝑟−𝑚²𝑇_𝑠=0,𝑑²𝐶_𝑠𝑑𝑟²+1𝑟𝑑𝐶_𝑠𝑑𝑟−𝑛²𝐶_𝑠=0, (43)(44) where 𝑚²=𝑘²+(𝜔/𝛼) and 𝑛²=𝑘²+(𝜔/𝛽).

If we consider the diffusion of the thermal energy and the concentration in an infinite isolated column of stationary flow of radius 𝑟₀ then there are no temperature and concentration gradients at 𝑟=𝑟₀. So, the temperature and concentration boundary conditions are as follows [3]: 𝑇_𝑠=𝑇₁,𝐶_𝑠=𝐶₁,at𝑟=𝑅₁,𝜕𝑇_𝑠𝜕𝑟=0,𝜕𝐶_𝑠𝜕𝑟=0,at𝑟=𝑟₀. (45) Equations (43) and (44) represent the standard Bessel’s differential equation. Therefore, the general solution of the temperature and concentration function may be written as follows: 𝑇=𝑇𝑟,𝜂,𝑡₁𝐾^′₀𝑚𝑟₀𝐼₀(𝑚𝑟)−𝐼^′₀𝑚𝑟₀𝐾₀(𝑚𝑟)𝐾^′₀𝑚𝑟₀𝐼₀𝑚𝑅₁−𝐾₀𝑚𝑅₁𝐼^′₀𝑚𝑟₀𝑒^{𝜔𝑡+𝑖𝑘𝜂},𝐶=𝐶𝑟,𝜂,𝑡₁𝐾^′₀𝑛𝑟₀𝐼₀(𝑛𝑟)−𝐼^′₀𝑛𝑟₀𝐾₀(𝑛𝑟)𝐾^′₀𝑛𝑟₀𝐼₀𝑛𝑅₁−𝐾₀𝑛𝑅₁𝐼^′₀𝑛𝑟₀𝑒^{𝜔𝑡+𝑖𝑘𝜂}. (46) On combining the following equations:(1)the general solution of the heat and mass transfer (see (46)),(2)the surface tension expression (see (41)),(3)the expression of the free surface function, which may be given based on the normal modes technique as 𝛿(𝜂,𝑡)=𝑒^{𝜔𝑡+𝑖𝑘𝜂}, into the free surface deferential equation given by (36).

We finally get the transcendental dispersion relation of the instability of the interfacial free surface as 𝑓(𝜔,𝑘)=𝜔²+𝑎₁(𝑘)𝜔+𝑏₁(𝜔,𝑘)=0, (47) where the functions 𝑎₁(𝑘) and 𝑏₁(𝜔,𝑘) are defined in the appendix.

It is convenient to write the stability criterion in an appropriate dimensionless form. This can be done in a number of ways depending primarily on the choice of the characteristic length, time, temperature, and mass. Consider the following dimensionless forms depending on the characteristic length = 𝑟₀, the characteristic time = 1/𝜔, the characteristic temperature = 𝑇₁, and the characteristic mass = 𝜎₀/𝜔², where 𝜔 is a characteristic value of 𝜔. The other dimensionless quantities are given by 𝑘𝑘=^∗𝑟₀,𝜌=𝜌^∗𝜎₀𝑟³₀𝜔²,𝐸²₀=𝐸²₀^∗𝜎₀𝑟₀,𝜇=𝜇^∗𝜎₀𝜔𝑟₀,𝑇=𝑇₁𝑇^∗,𝛾_𝑇=𝛾^∗_𝑇𝑇₁,𝜈=𝜈^∗𝜎₀𝑟³₀𝜔,𝑟=𝑟^∗𝑟₀,𝜔=𝜔𝜔^∗,𝛼=𝛼^∗𝑟²₀𝜔,𝛽=𝛽^∗𝑟²₀𝜔. (48) The superscript asterisks refer to the dimensionless quantities. From now on, these will be omitted for simplicity. The dispersion relation in (47) and its coefficients 𝑎₁(𝑘) and 𝑏₁(𝜔,𝑘) may be written as follows: 𝑓(𝜔,𝑘)=𝜔²+𝑎(𝑘)𝜔+𝑏(𝜔,𝑘)=0, (49) where the functions 𝑎(𝑘) and 𝑏(𝜔,𝑘) are defined in the appendix.

The dimensionless transcendental dispersion relation in (49) describes the instability of the interfacial free surface of the microfluid jet. We will discuss the instability criteria numerically and graphically in the next section.

7. Numerical Estimations

In the case of the thermosolutal Marangoni convection, the surface tension depends on the heat and mass transfer and the dispersion relation is the transcendental equation (49). The stability criteria of the system may be discussed by getting the values of the growth rate 𝜔 from the roots of the transcendental equation (49). The positive values of 𝜔 indicate a disturbance that grows with time (instability) and the negative values indicate decays with time (stability). So, to study the instability of the microcylindrical surface, we indicate the relation between the growth rate 𝜔 and the wave number 𝑘 graphically. In all coming curves, the positive values region of 𝜔 indicates the instability region, denoted by letter 𝑈. Meanwhile, the negative values region of 𝜔 indicates the stability region, denoted by letter 𝑆.

It is well known that increasing the wave number enhances the stability of the system. As seen in Figure 2, the stability occurs at 𝑘≅0.7, when the intensity of the electric field 𝐸₀=1000. Meanwhile, at 𝐸₀=1 the stability occurs at 𝑘≅1. This means that the stability occurs at less value of 𝑘 and at the higher value of 𝐸₀, which means that the system is more stable at highly electric field intensity and the electric field has a stabilizing effect. This is an early result obtained by many researchers [7, 20, 24, 25]. Also, we can notice that the maximum instability in all figures, except Figure 2, according to the peak of the curve occurs at 𝑘≅0.6. But at high values of the electric field, the peak of the curve approaches the stability region and the maximum instability occurs at 𝑘≅0.4. This implies that existence of the electric field causes a high stability of the interface.

Figure 2 

Stability diagram for a system having the particulars 𝜌=1, 𝜇=0.5, 𝑅1=10−5, 𝑅2=10, 𝜈=0.5, 𝜀1=5×10−5, 𝜀2=10−4, 𝑟0=5×10−3, 𝛾𝑇=0.01, 𝛾𝐶=0.2, 𝛼=0.01, 𝑇1=0.05, 𝐶1=0.03, and 𝛽=0.2.

Figure 3 illustrates the electric field and the growth rate relation. It is clear from this figure that the maximum value of the electric field 𝐸₀ is congruent to the zero value of the growth rate 𝜔. The zero value the growth rate 𝜔 corresponds to the steady state of the interface. This result confirms the same result of Figure 2; that is, the electric field has a stabilizing effect. Figure 4 illustrates the effect of the dielectric constants difference 𝜀̂=𝜀₁−𝜀₂ on the stability picture. The dielectric difference 𝜀̂ has a stabilizing effect, where the higher value of 𝜀̂ is identical to the minimum peak value of the curve.

Figure 3 

The electric field plotted versus the growth rate at 𝜌=1, 𝜇=0.5, 𝑅1=10−5, 𝑅2=10, 𝐸0=10, 𝛾𝑇=0.01, 𝛾𝐶=0.03, 𝑟0=5×10−3, 𝛼=0.01, 𝐶1=10, 𝑇1=10, 𝛽=0.2,  𝜀1=5×10−5, and 𝜀2=10−4.

Figure 4 

Stability diagram for a system having the particulars 𝜌=1, 𝜇=0.5, 𝑅1=10−5, 𝑅2=10, 𝐸0=10, 𝛾𝑇=0.01, 𝛾𝐶=0.03, 𝑟0=5×10−3, 𝛼=0.01, 𝐶1=0.04, 𝑇1=1, and 𝛽=0.2.

Figures 5–7 illustrate the effect of Darcy’s coefficient, the viscosity, and the density of fluid on the stability behavior. The of Darcy’s coefficient, the viscosity, and the density of fluid have stabilizing effect for 𝑘≤1 according to reducing the peak of the curve and decreasing the number of the points in the instability region. Figure 5 illustrates the effect of the existence of the porous structure, according to the different values of the Darcy’s coefficient, on the stability picture. It is clear that the microcylindrical surface is more stable, in case of the flow through a porous medium for majority values of the wave number, due to an increase in the values of the Darcy’s coefficients in comparison with the case of the pure flow (𝜈=0). This is due to the fact that the greatest peak value of the curve in the instability region is at 𝜈=0. Hence the Darcy’s coefficient 𝜈 is equal to 𝜇/𝜆, where 𝜆 is the permeability of the porous medium, so an increase in the values of the Darcy’s coefficients corresponds to a decrease in the permeability of the porous medium. This in turn restricts the streaming velocity of the fluid flow. It is known that when the streaming velocity decreases, the stability of the system increases. In other words, when the permeability of the medium increases (𝜆 increases and 𝜈 decreases) the holes of the porous medium are very large and the resistance of the medium may be neglected so that the streaming velocity increases and causes instability of the system. The same result was illustrated experimentally by Catton and Chung [21], where the authors studied the interfacial stability between the water and the steam. Two porous media were studied experimentally: a bead of glass with diameter 0.6, 2, and 6 and with permeability 0.14, 1.57, and 18.0 (×10⁻⁹ m²), respectively, and a bead of steel with diameter 4 and with permeability 9.02 (×10⁻⁹ m²). It was found that in the more permeable cylinder the steam bubble is chaotic just after it has been formed. This means that the interface becomes unstable and the chaotic turbulence becomes more and more profound as the water flow rate increases (according to increasing the permeability of the porous medium).

Figure 5 

Stability diagram for a system having the particulars 𝜌=1, 𝜇=0.5, 𝑅1=10−5, 𝑅2=10, 𝐸0=10, 𝜀1=5×10−5, 𝜀2=10−4, 𝑟0=5×10−3, 𝛾𝑇=0.01, 𝛾𝐶=0.2, 𝛼=0.01, 𝑇1=0.05, 𝐶1=0.03, and 𝛽=0.2.

Figure 6 

Stability diagram for a system having the particulars 𝜌=1, 𝜈=0.3, 𝑅1=10−5, 𝑅2=10, 𝐸0=10, 𝜀1=5×10−5, 𝜀2=10−4, 𝑟0=5×10−3, 𝛾𝑇=0.01, 𝛾𝐶=0.2, 𝛼=0.01, 𝑇1=0.05, 𝐶1=0.03, and 𝛽=0.2.

Figure 7 

Stability diagram for a system having the particulars 𝜈=0.3, 𝜇=0.5, 𝑅1=10−5, 𝑅2=10, 𝐸0=10, 𝜀1=5×10−5, 𝜀2=10−4, 𝑟0=5×10−3, 𝛾𝑇=0.01, 𝛾𝐶=0.2, 𝛼=0.01, 𝑇1=0.05, 𝐶1=0.03, and 𝛽=0.2.

The viscosity 𝜇 and the density 𝜌 have a stabilizing effect, for larger values of the wave number, as shown in Figures 6 and 7. This influence occurs due to an increase in the viscosity damping and in the inertia. So, the interface of the microcylindrical surface becomes stable with an increase in the viscosity and the density.

The effect of heat transfer on the free surface is illustrated in Figure 8. To study the instability influence, according to the heat transfer, we notice that the temperature coefficient is 𝛾_𝑇=±(1/𝜎₀)(𝑑𝜎/𝑑𝑇) [30], where 𝛾_𝑇 has a positive or negative sign according to the increasing or decreasing of the surface tension. When 𝑇<𝑇₁, the axial microsolid cylinder, which induces the heat and mass gradients, is heated and then the temperate transfers from it to the free surface by convection currents. This in turn tends to evaporate the fluid easily and provokes higher instabilities at the interface. So, in this case, 𝛾_𝑇 has a negative value according to the decreasing of the surface tension. Conversely, 𝛾_𝑇 has a positive value when 𝑇>𝑇₁. In this case the free surface loses its temperature that is transferred to the core region of the flow jet and hence the free surface becomes stable. This phenomenon was illustrated in Figure 8 where the temperature coefficient 𝛾_𝑇 has a stabilizing effect when it is positive and the inverse occurs at 𝛾_𝑇<0. In other words, heating the axial microsolid surface 𝑇₁ has destabilizing effect at the interface and conversely for cooling. Figure 9 confirms this result. We notice from this figure that the unstable state, for the positive values of the growth rate 𝜔, occurs at the temperature values corresponding to 𝑇<𝑇₁. Meanwhile, the temperature 𝑇>𝑇₁ is for the negative growth rate in the stability region. This result was obtained experimentally by D’Aubeterre et al. [31]. The authors stated that the temperature has a destabilizing effect due to the fact that a substance is more volatile for increasing the temperate; this means that when a temperature gradient occurs, the substance tends to evaporate easily and provokes higher instabilities at the interface. Also, depending on the alcohol studied [31], turbulence begins at the interface at different temperature gradients. Instabilities begin near the interface, showing a sinuous movement. At higher temperatures and temperature gradients, instabilities increase and movements become continuous. The most evident effect occurs when temperature gradients are 15--17^∘C; at these values a sinuous wave was observed [31].

Figure 8 

Stability diagram for a system having the particulars 𝜌=1, 𝜇=0.5, 𝑅1=10−5, 𝑅2=10, 𝐸0=10, 𝜀1=5×10−5, 𝜀2=10−4, 𝑟0=5×10−3, 𝛾𝐶=−0.03, 𝛼=0.01, 𝜈=0.3, 𝐶1=0.03, 𝛽=0.2, and 𝑇1=10.

Figure 9 

The temperature plotted versus the growth rate at 𝜌=1, 𝜇=0.5, 𝑅1=10−5, 𝑅2=10, 𝐸0=10, 𝛾𝑇=0.01, 𝛾𝐶=0.03, 𝑟0=5×10−3, 𝛼=0.01, 𝐶1=10, 𝑇1=0.5, 𝛽=0.2, 𝜀1=5×10−5, and 𝜀2=10−4.

Similar behavior can be noticed for the concentration as seen in Figure 10. In this figure we use 𝛾_𝐶=±(1/𝜎₀)(𝑑𝜎/𝑑𝐶) [30]. The concentration coefficients of the surface tension 𝛾_𝐶 become negative if 𝐶<𝐶₁. This implies that the concentration at the interface is less than the concentration at the core region. Because the surface tension is proportional with the concentration, the surface tension at the free surface accordingly reduces. So, the free surface loses its interfacial rigidity and its ability to movement increases. Therefore the instability of the free surface occurs. On the other hand, when the concentration of the interface is higher than that of the core region, the stability in the interface occurs. This means an increase of the interfacial rigidity, which in turn restricts any surface movements or Marangoni convection. Therefore, the surface becomes stable when 𝛾_𝐶>0 and the instability occurs for 𝛾_𝐶<0, as shown in Figure 10. Figure 11 confirms this result, where the concentration is plotted versus the growth rate. The region of the positive values of the growth rate 𝜔, that is, the unstable region, occurs at 𝐶<𝐶₁. Meanwhile, the stable region, that is, the negative growth rate values, occurs at 𝐶>𝐶₁. Similar results were obtained by Agble and Mendes [32] and compared with experimental results.

Figure 10 

Stability diagram for a system having the particulars 𝜌=1, 𝜇=0.5, 𝑅1=10−5, 𝑅2=10, 𝐸0=10, 𝜀1=5×10−5, 𝜀2=10−4, 𝑟0=5×10−3, 𝛾𝑇=−0.01, 𝛼=0.01, 𝜈=0.3, 𝑇1=1,  𝛽=0.2, and 𝐶1=10.

Figure 11 

The concentration plotted versus the growth rate at 𝜌=1, 𝜇=0.5, 𝑅1=10−5, 𝑅2=10, 𝐸0=10, 𝛾𝑇=0.01, 𝛾𝐶=0.03, 𝑟0=5×10−3, 𝛼=0.01, 𝐶1=0.1, 𝑇1=0.5, 𝛽=0.2, 𝜀1=5×10−5, and 𝜀2=10−4.

Finally, the instability criterion of the system can be discussed by studying the roots of the dispersion relation (49) to get the values of the growth rate 𝜔, as shown in Table 1. The negative values of the roots of 𝜔 indicate stability of the microcylindrical surface and the inverse occurs of the positive root values. It can be noticed from the roots of the growth rate 𝜔 that the interface becomes stable when the microradius 𝑟₀ increases which means an increase in the inertia with 𝑟₀.

Table 1 

The root of the growth rate 𝜔 for different values of the micro radius 𝑟₀ at 𝜌=1, 𝜇=0.5, 𝜀₁=10⁻⁵, 𝜀₂=10⁻³, 𝑅₁=10⁻⁵, 𝐸₀=10, 𝑅₂=10, 𝛾_𝑇=0.5, 𝛾_𝐶=0.03, 𝛼=0.01, 𝛽=0.2, 𝑇₁=1, 𝐶₁=0.04, and 𝜈=0.3.

8. Conclusion

In this paper, we have examined the influence of the existence of the porous structure on the EHD instability of the free microcylindrical fluid surface with thermosolutal Marangoni convection. The surface tension depends linearly on the heat and mass transfer. The main results of our study can be epitomized in the following points.(1)The existence of the porous structure has a stabilizing effect for the free surface. Existence of the porous medium restricts the streaming velocity of the fluid. Similar behavior was obtained experimentally by [21] for the steam-liquid interface thorough a porous structure.(2)As it is known, the electric field has a stabilizing effect, and this result coincides with the previously published results in the literature [7, 20, 24, 25]. (3)The heating of the axial microsolid cylinder reduces the surface tension and makes the free surface volatile easily and hence the surface becomes unstable and the inverse occurs for cooling the axial microsolid cylinder. The same results were obtained experimentally by [31].(4)The surface tension is proportional to the concentration variation. So, high concentration at the free surface inhibits the movement of the surface. This means stability of the free surface. The inverse behavior was obtained for a high concentration at the core region. Similar trend was obtained experimentally by [32].

Finally, our calculation of the free surface function was compared with the previous results of [3] in the absence of the porous medium and the electric field. Also, the present results were compared with the experimental results of [21, 31, 32]. There was compatibility between the results.

Appendix

(1) The functions 𝑔₁(𝑘), 𝑔₂(𝑘), 𝐺₁(𝑘), and 𝐺₂(𝑘) are given as 𝑔₁(𝑘)=𝐾₀𝑘𝑟₀𝐼^′₀𝑘𝑅₁−𝐼₀𝑘𝑟₀𝐾^′₀𝑘𝑅₁,𝑔₂(𝑘)=𝐾₀𝑘𝑟₀𝐼^′₀𝑘𝑅₂−𝐼₀𝑘𝑟₀𝐾^′₀𝑘𝑅₂,𝐺₁(𝑘)=𝐼^′₀𝑘𝑟₀𝐾^′₀𝑘𝑅₁−𝐾^′₀𝑘𝑟₀𝐼^′₀𝑘𝑅₁,𝐺₂(𝑘)=𝐼^′₀𝑘𝑟₀𝐾^′₀𝑘𝑅₂−𝐾^′₀𝑘𝑟₀𝐼^′₀𝑘𝑅₂. (A.1)(2) The roots 𝜔_± are defined as 𝜔_±=−𝑎₀±𝑎²₀−4𝑏₀2, (A.2) where 𝑎₀=3𝜇𝜌𝑘²+𝜈𝜌,𝑏₀𝜎=−⁰𝑘²2𝜌𝑟₀+𝜎₀𝑟₀𝑘⁴+𝐸2𝜌₂₀𝑘³𝑟₀𝜀2𝜌²−𝜀₁²𝑔₁(𝑘)𝑔₂(𝑘)𝜀₂𝐺₂(𝑘)𝑔₁(𝑘)−𝜀₁𝐺₁(𝑘)𝑔₂(𝑘). (A.3)(3) The functions 𝑎₁(𝑘) and 𝑏₁(𝜔,𝑘) are defined as 𝑎₁(𝑘)=3𝜇𝜌𝑘²+𝜈𝜌,𝑏₁()𝜎𝜔,𝑘=−⁰𝑘²2𝜌𝑟₀+𝜎₀𝑟₀𝑘⁴+𝐸2𝜌₂₀𝑘³𝑟₀𝜀2𝜌²−𝜀₁²𝑔₁(𝑘)𝑔₂(𝑘)𝜀₂𝐺₂(𝑘)𝑔₁(𝑘)−𝜀₁𝐺₁(𝑘)𝑔₂(𝑘)+𝜎₀𝑘²2𝜌𝑟₀𝛾_𝑇𝑇_𝑠𝑟₀+𝛾_𝐶𝐶_𝑠𝑟₀. (A.4)

Also, the functions 𝑎(𝑘) and 𝑏(𝜔,𝑘) are defined as 𝑎(𝑘)=3𝜇𝜌𝑘²+𝜈𝜌,𝑏()=1𝜔,𝑘2𝜌−𝑘²+𝑘⁴+𝐸²₀𝑘³𝜀2𝜌²−𝜀₁²𝑔₁(𝑘)𝑔₂(𝑘)𝜀₂𝐺₂(𝑘)𝑔₁(𝑘)−𝜀₁𝐺₁(𝑘)𝑔₂(𝑘)+𝑘²𝛾2𝜌^𝑇𝑇_𝑠𝑟₀+𝛾_𝐶𝐶_𝑠𝑟₀. (A.5)

Nomenclature

𝐶:	Concentration
𝐶0:	Concentration at the interface
𝐶1:	Concentration at the microcylindrical solid axis
𝐸0:	Intensity of the electric field
𝐹:	Interface function
ℎ:	Radius for the disturbed fluid surface
𝐻:	Curvature of the interface
𝐼0,𝐾0:	Modified Bessel’s function of the first and second kinds
𝑘:	Wave number
𝑛:	Unit normal vector to the interface
𝑃:	Pressure
𝑃0(𝑧,𝑡),𝑃2(𝑧,𝑡):	Perturbed pressure functions
(𝑟,𝜃,𝑧):	Cylindrical coordinates
𝑟0:	Microradius of the fluid cylindrical surface
𝑅1:	Microradius of the cylindrical solid axis
𝑅2:	Radius of the surrounded vacuum
𝑡:	Time
𝑡̂:	Unit tangential vector to the interface
𝑇:	Temperature
𝑇0:	Temperature at the interface
𝑇1:	Temperature at the surface of microcylindrical axis
𝑢(𝑧,𝑡):	Axial streaming velocity function
𝑢0:	Constant streaming velocity
𝑣:	Velocity vector
𝑣𝑟,𝑣𝑧:	Radial and axial velocity components
𝑣0(𝑧,𝑡),𝑣2(𝑧,𝑡):	Perturbed velocity functions.

Greek Symbols

𝛼:	Thermal diffusivity
𝛽:	Mass diffusivity
𝜎(𝑧,𝑡):	Surface tension
𝜎0:	Unperturbed surface tension
𝜎1(𝑧,𝑡):	Perturbed surface tension
Δ𝜎:	A constant variation of the surface tension
𝛾𝑇=±(1/𝜎0)(𝑑𝜎/𝑑𝑇):	Temperature coefficient of the surface tension
𝛾𝐶=±(1/𝜎0)(𝑑𝜎/𝑑𝐶):	Concentration coefficient of the surface tension
𝜙:	Electric potential function
𝜏𝑖𝑗:	Stress tensor
𝜔:	Growth rate of the surface wave
𝜇:	Viscosity
𝜇eff:	Effective viscosity
𝜈:	Darcy’s coefficient
𝜀1,𝜀2:	Dielectric constants of the liquid and vacuum, respectively
𝛿(𝑧,𝑡):	Dimensionless radius of the disturbed fluid surface
𝜂=𝑧−𝑣0𝑡:	Transformation of the axial distance
𝜌:	Density.