The Wall Properties Effect on Peristaltic Transport of Micropolar Non-Newtonian Fluid with Heat and Mass Transfer

Abstract

The problem of the unsteady peristaltic mechanism with heat and mass transfer of an incompressible micropolar non-Newtonian fluid in a two-dimensional channel. The flow includes the viscoelastic wall properties and micropolar fluid parameters using the equations of the fluid as well as of the deformable boundaries. A perturbation solution is obtained, which satisfies the momentum, angular momentum, energy, and concentration equations for case of free pumping (original stationary fluid). Numerical results for the stream function, temperature, and concentration distributions are obtained. Several graphs of physical interest are displayed and discussed.

1. Introduction

The peristaltic transport is traveling contraction wave along a tube-like structure, and it results physiologically from neuromuscular properties of any tubular smooth muscle. Peristalsis is now well known to physiologists to be one of the major mechanisms for fluid transport in many biological systems. In particular, a peristaltic mechanism may be involved in swallowing food through the esophagus, in urine transport from the kidney to the bladder through the ureter, movement of chyme in the gastrointestinal tract, in the transport of spermatozoa in the ducts efferentes of the male reproductive tracts and in the cervical canal, movement of ovum in the female fallopian tubes, the transport of lymph in the lymphatic vessels, and the vasomotion in small blood vessels such as arterioles, venues, and capillaries. In addition, peristaltic pumping occurs in many practical applications involving biomedical systems. It has now been accepted that most of the physiological fluids behave in general like suspensions of deformable or rigid particles in a Newtonian fluid. Blood, for example, is a suspension of red cells, white cells, and platelets in plasma. Another example is cervical mucus, which is a suspension of macromolecules in a water-like liquid [1]. However only a few studies have considered this aspect since the initial investigation by Raju and Devanathan [2, 3]. The biviscosity fluid can represent the behavior of blood in vessels of small diameter as the mean shear rate is about 20–150 s⁻¹ [4].

Several studies have been made to analyze both theoretical and experimental aspects of the peristaltic motion of non-Newtonian fluids in different situations. Böhme and Friedrich [5] studied the mechanism of peristaltic transport of an incompressible viscoelastic fluid by means of an infinite train of sinusoidal waves traveling along the wall of the duct in the case of plane flow. The effects of an Oldroyd-B fluid on the peristaltic mechanism are examined by Hayat et al. [6], under the long wavelength assumption. They noticed that in the narrow part of the channel, the behavior of an Oldroyd-B fluid is much more different from that of a Newtonian fluid than in the wide part of the channel. El-Shehawey et al. [7] studied the peristaltic motion of an incompressible non-Newtonian fluid through a porous medium. They showed that the pressure rise increases as the permeability decreases and noted that both pressure rise and friction force do not depend on permeability parameter at a certain value of flow rate. Haroun [8] investigated the peristaltic flow of a third-order fluid in an asymmetric channel under the assumption of long wavelength approximation. He expanded the velocity components and pressure in a regular perturbation series in a small parameter Deborah number that contained the non-Newtonian coefficients appropriate to shear-thinning. The effects of slip boundary conditions on the dynamics of fluids in porous media is investigated by El-Shehawey et al. [9]. They studied the flow of non-Newtonian Maxwell fluids in an axisymmetric cylindrical tube (pore), in which the flow is induced by traveling transversal waves on the tube wall (peristaltic transport). Hakeem et al. [10] discussed the effect of an endoscope on the peristaltic mechanism of a generalized Newtonian fluid. Mekheimer [11] studied the effect of a uniform magnetic field on peristaltic transport of a blood in nonuniform two-dimensional channels, when blood is represented by a couple-stress fluid. Tsiklauri and Beresnev [12] analyzed the effect of viscoelasticity on the dynamics of fluids in porous media by studying the peristaltic flow of a Maxwell fluid in circular tube, in which the flow is induced by a wave traveling on the tube wall.

The micropolar fluid represents fluids, which consist of rigid, randomly oriented (or spherical) particles suspended in a viscous medium where the deformation of the particles is ignored. The theory of thermo-micropolar fluid was developed by Eringen [13]. Agrawal and Dhanapal [14] considered the micropolar fluid to be the model of blood flow in small arteries, and the calculation of theoretical velocity profiles is observed, in good agreement with experimental data [15]. They have many applications in physiological and chemical engineering [16]. Srinivasachrya et al. [17] studied the problem of peristaltic transport of a micropolar fluid in a circular tube. Girija Devi and Devanathan [18] studied the peristaltic flow of a micropolar fluid in a cylindrical tube with a sinusoidal deformation of small amplitude travelling down its flexible wall for the case of low Reynolds number flow devoid of wall properties like tension and damping. However, the wall properties are essential to be taken into consideration in various real situations. The peristaltic motion of a simple microfluid which accounts for microrotation and microstretching of the particles contained in a small volume element is studied by Philip and Chandra [19], using long wavelength approximation.

Combined heat and mass transfer problems are of importance in many processes and have, therefore, received a considerable amount of attention in recent years. In processes such as drying, evaporation at the surface of a water body, energy transfer in a wet cooling tower and the flow in a desert cooler, heat and mass transfer occurs simultaneously. Possible applications of this type of flow can be found in many industries. For example, in the power industry, among the methods of generating electric power is one in which electrical energy is extracted directly from a moving conducting fluid. Many practical diffusive operations involve the molecular diffusion of a species [20]. Radhakrishnamacharya and Radhakrishna [21] investigated the peristaltic flow with heat transfer in a nonuniform channel. The effect of elasticity of the flexible walls on peristaltic transport of an incompressible viscous fluid, with heat transfer, in a two-dimensional uniform channel under long wavelength approximation is explained by Radhakrishnamacharya and Srinivasulu [22]. El Dabe et al. [23] studied heat and mass transfer of a steady slow motion of a Rivlin-Ericksen fluid in tube of varying cross-section with suction. Agrawal et al. [24] obtained numerical solutions of flow and heat transfer of a micropolar fluid at a stagnation point on a porous stationary wall. They observed that the heat sources increase the velocity and temperature in the pipe while the heat sinks decrease them.

Muthu et al. [25] carried out a study of the peristaltic motion of an incompressible micropolar fluid in two-dimensional channel. They investigated the effects of viscoelastic wall properties and micropolar fluid parameters on the flow using the equations of the fluid as well as of the deformable boundaries. The non-Newtonian property and equations of heat and mass transfer were not taken into their consideration. Because of the wide range of practical importance of the heat and mass transfer, the present study considered the heat and mass transfer of an unsteady peristaltic motion of a micropolar non-Newtonian biviscosity fluid [4]. The following analysis includes the dynamic boundary condition. Analytical approximate solutions for the stream function, microrotation velocity, temperature, and concentration equations are obtained as a power series in terms of the small amplitude ratio. We have shown the relation between the different parameters of motion in order to investigate how to control the motion of the fluid by changing these parameters.

2. Formulation of the Problem

Consider a two-dimensional symmetric unsteady flow of an incompressible micropolar biviscosity fluid in an infinite channel of uniform thickness 2d, with heat and mass transfer. The walls of the channel are flexible membranes on which they are imposed traveling sinusoidal waves of moderate amplitude (see Figure 1).

Figure 1 

Geometry of two-dimensional peristaltic channel.

We choose a rectangular coordinate system such that the axes 𝑥 and 𝑦 are in the directions of wave propagation and normal to the mean position of the membranes, respectively. The origin is located at the center line of the channel. It was shown by Wilson and Taylor [26] that the limiting process by which the biviscosity model is approached with the limiting process implicit in lubrication theory. Models of this type are much easier to handle mathematically than many models such as Oldroyd’s and represent the experimental facts just as well, at least in many cases.

The biviscosity model (which has been used in similar contexts by Nakayama and Sawada [4]) can be written as𝜏_𝑖𝑗=2(𝜇_𝐵+𝑝_𝑦√2𝜋)𝑒_𝑖𝑗,𝜋≥𝜋_𝑐,2(𝜇_𝐵+𝑝_𝑦2𝜋_𝑐)𝑒_𝑖𝑗,𝜋<𝜋_𝑐. (2.1) The following quantity is introduced as a nondimensional parameter including 𝜋_𝑐𝛾=𝜇_𝐵2𝜋_𝑐𝑝_𝑦, (2.2) where 𝜇_𝐵 is the plastic viscosity, 𝑝_𝑦 is the yielding stress, 𝜋=𝑒_𝑖𝑗𝑒_𝑖𝑗, where 𝑒_𝑖𝑗 is the (𝑖,𝑗) component of the deformation rate and the value of 𝛾 denotes the upper limit of apparent viscosity coefficient. The biviscosity model is approached in the limit 𝛾→∞ (see Nakayama and Sawada [4] for a few further details).

The governing continuity, momentum, angular momentum, temperature, and concentration equations for this problem can be written as [13]𝜕𝑢+𝜕𝑥𝜕𝜐𝜕𝑦=0,𝜌[𝜕𝑢𝜕𝑡+𝑢𝜕𝑢𝜕𝑥+𝜐𝜕𝑢𝜕𝑦]=−𝜕𝑃𝜕𝑥+(2𝜇_𝐵(1+𝛾⁻¹)+𝑘2)∇²𝑢+𝑘𝜕𝑁_𝜃𝜕𝑦,𝜌[𝜕𝜐𝜕𝑡+𝑢𝜕𝜐𝜕𝑥+𝜐𝜕𝜐𝜕𝑦]=−𝜕𝑃𝜕𝑦+(2𝜇_𝐵(1+𝛾⁻¹)+𝑘2)∇²𝜐−𝑘𝜕𝑁_𝜃𝜕𝑥,𝜌𝐽[𝜕𝑁_𝜃𝜕𝑡+𝑢𝜕𝑁_𝜃𝜕𝑥+𝑣𝜕𝑁_𝜃𝜕𝑦]=−2𝑘𝑁_𝜃+𝛾∇²𝑁_𝜃+𝑘[𝜕𝜐−𝜕𝑥𝜕𝑢𝜕𝑦],𝜌𝐶_𝑝[𝜕𝑇𝜕𝑡+𝑢𝜕𝑇𝜕𝑥+𝜐𝜕𝑇𝜕𝑦]=𝑘_𝑐∇²𝑇+(2𝜇_𝐵(1+𝛾⁻¹)+𝑘2)×[2(𝜕𝑢)𝜕𝑥²+2(𝜕𝜐)𝜕𝑦²+(𝜕𝑢+𝜕𝑦𝜕𝜐)𝜕𝑥²]+2𝑘[𝑁_𝜃²−𝑁_𝜃(𝜕𝜐−𝜕𝑥𝜕𝑢𝜕𝑦)]+𝛾[(𝜕𝑁_𝜃)𝜕𝑥²+(𝜕𝑁_𝜃)𝜕𝑦²]+𝛼_𝑐[𝜕𝑇𝜕𝑥𝜕𝑁_𝜃−𝜕𝑦𝜕𝑇𝜕𝑦𝜕𝑁_𝜃𝜕𝑥],𝜕𝐶𝜕𝑡+𝑢𝜕𝐶𝜕𝑥+𝜐𝜕𝐶𝜕𝜕𝑦=𝐷(²𝐶𝜕𝑥²+𝜕²𝐶𝜕𝑦²)+𝐷𝑘_𝑇𝑇_𝑚(𝜕²𝑇𝜕𝑦²+𝜕²𝑇𝜕𝑥²), (2.3)(2.4)(2.5)(2.6)(2.7)(2.8) where 𝑢(𝑥,𝑦,𝑡) and 𝜐(𝑥,𝑦,𝑡) are the velocity components in the x and y directions, respectively, 𝑁_𝜃(x,y,t) is the microrotation velocity component in the direction normal to both the x and y axes. Here 𝜌 is the density of the fluid, P is the pressure, J is the microinertia constant, k is the vortex viscosity coefficient (also known as the coefficient of gyroviscosity), and 𝛾 is the spin-gradient viscosity.

Further, the material constants 𝜇_𝐵 and k satisfy the following inequalities [13]:2𝜇_𝐵+𝑘≥0,𝑘≥0,𝜇_𝐵≥0. (2.9)

The temperature and concentration of fluid are 𝑇(𝑥,𝑦,𝑡) and 𝐶(𝑥,𝑦,𝑡), 𝐶_𝑝 is the specific heat at constant pressure, D is the coefficient of mass diffusivity, K_T is the thermal diffusion ratio, T_m is the mean fluid temperature, k_c is the thermal conductivity and 𝛼_𝑐 is the heat conduction parameter for a micropolar fluid. The last term in (2.8) signifies the thermal-diffusion effect.

We consider a symmetric motion of the flexible walls in the boundary conditions. Let the vertical displacements of the upper and lower walls be 𝜂 and −𝜂, respectively, where𝜂(𝑥,𝑡)=𝑎cos2𝜋𝜆(𝑥−𝑐𝑡), (2.10) where 𝑎 is the amplitude, λ is the wavelength, and c is the wave speed.

We assume that the walls are inextensible so that only lateral motion takes place and the horizontal displacement of the wall is zero [25].

Thus the no-slip boundary conditions for the velocity and microrotation are 𝑢=0,𝑁_𝜃=0at𝑦=±(𝑑+𝜂(𝑥,𝑡)). (2.11) The dynamic boundary conditions are imposed on the fluid by the symmetric motion of the flexible walls, which can be written as [18]𝜕𝐿(𝜂)𝜕𝑥=−𝜌[𝜕𝑢𝜕𝑡+𝑢𝜕𝑢𝜕𝑥+𝜐𝜕𝑢𝜕𝑦]+(2𝜇_𝐵(1+𝛾⁻¹)+𝑘2)∇²𝑢+𝑘𝜕𝑁_𝜃,𝜕𝑦at𝑦=±(𝑑+𝜂(𝑥,𝑡)), (2.12) where 𝜕𝐿(𝜂)𝜕𝜕𝑥=−ℓ³𝜂𝜕𝑥³𝜕+𝑚³𝜂𝜕𝑡²𝜕𝜕𝑥+𝑛³𝜂𝜕𝑡𝜕𝑥. (2.13) Here L(𝜂) is the pressure at the walls, ℓ is the tension in the membrane, m is the mass per unit area, and n is the coefficient of viscous damping force.

The plate temperature starts oscillating about a nonzero mean temperature. Under these physical conditions, the temperature and concentration at the upper wall give𝑇=𝑇_𝑤+𝑎𝑑(𝑇_𝑤−𝑇_𝑚)cos(2𝜋(𝑥−𝑐𝑡)𝜆),𝐶=𝐶_𝑤+𝑎𝑑(𝐶_𝑤−𝐶_𝑚)cos(2𝜋(𝑥−𝑐𝑡)𝜆),at𝑦=𝑑+𝜂(𝑥,𝑡), (2.14) where C_m is the mean fluid concentration, T_w and C_w are the uniform temperature and concentration at the lower wall, that is, 𝑇=𝑇_𝑤,𝐶=𝐶_𝑤,at𝑦=−(𝑑+𝜂(𝑥,𝑡)). (2.15) Equation (2.3) allows the use of the stream function 𝜓(𝑥,𝑦,𝑡) such that 𝑢=𝜕𝜓𝜕𝑦,𝜐=−𝜕𝜓𝜕𝑥. (2.16) By introducing the following nondimensional quantities: 𝑥^′=𝑥𝑑,𝑦^′=𝑦𝑑,𝑡^′=𝑡𝑐𝑑,𝑢^′=𝑢𝑐,𝜐^′=𝜐𝑐,𝑁_𝜃^′=𝑑𝑁_𝜃𝑐,𝑃^′=𝑃𝜌𝑐²,𝜂^′=𝜂𝑑,𝜓^′=𝜓𝑐𝑑,𝐽^′=𝐽𝑑²,𝑇^′=𝑇−𝑇_𝑚𝑇_𝑤−𝑇_𝑚,𝐶^′=𝐶−𝐶_𝑚𝐶_𝑤−𝐶_𝑚𝑎,𝜀=𝑑,𝛼=2𝜋𝑑𝜆,𝜇₁=𝑘𝜇_𝐵,Re=𝜌𝑐𝑑𝜇_𝐵𝜇,𝑁=(¹2+𝜇₁)^1/2,𝜇𝑁=(¹2(1+𝛾⁻¹)+𝜇₁)^1/2,𝜇𝑀=2𝑑(^𝐵𝛾)^1/2,𝑀𝑀=(²(1−𝑁²)(1−𝑁²))^1/2,𝑀Γ=𝑀,𝑅_ℓ=4𝜇𝜌𝑐𝑑𝐽𝛾(2𝜇_𝐵+𝑘),𝐾₂=𝑛𝑑𝜇_𝐵,𝐾₃=ℓ𝜌𝑑𝜇_𝐵²,𝑚₁=𝑚𝜇𝜌𝑑,Pr=^𝐵𝐶_𝜌𝑘_𝑐,𝑐Ec=²𝐶_𝜌(𝑇_𝑤−𝑇_𝑚)𝛼,af=^𝑐𝑐𝜇_𝐵𝐶_𝜌𝑑𝜇,Sc=^𝐵𝑘𝐷𝜌,Sr=^𝑇(𝑇_𝑤−𝑇_𝑚)𝜌𝐷𝑇_𝑚(𝐶_𝑤−𝐶_𝑚)𝜇_𝐵, (2.17) substituting from (2.17) into (2.3)–(2.8), eliminating P between (2.4) and (2.5), and dropping the accent mark for simplicity, the governing equations may be written as𝜕∇𝜕𝑡²𝜓+𝜓_𝑦∇²𝜓_𝑥−𝜓_𝑥∇²𝜓_𝑦=2(1+𝛾⁻¹)+𝜇₁2Re[∇²∇²𝜇𝜓]+¹∇Re²𝑁_𝜃,𝑅_ℓ(𝜕𝑁_𝜃𝜕𝑡+𝜓_𝑦𝜕𝑁_𝜃𝜕𝑥−𝜓_𝑥𝜕𝑁_𝜃𝜕𝑦)=2(1−𝑁²)∇²𝑁_𝜃−𝑁²𝑀²(∇²𝜓+2𝑁_𝜃),Re⋅Pr(𝜕𝑇𝜕𝑡+𝜓_𝑦𝜕𝑇𝜕𝑥−𝜓_𝑥𝜕𝑇𝜕𝑦)=∇²𝑇+2(1+𝛾⁻¹)+𝜇₁2Pr⋅Ec×[4𝜓²_𝑥𝑡+(𝜓_𝑦𝑦−𝜓_𝑥𝑥)²]+2𝜇₁⋅Pr⋅Ec(𝑁_𝜃²+𝑁_𝜃∇²4𝜓)+𝑀²Pr⋅Ec×[(𝜕𝑁_𝜃)𝜕𝑥²+(𝜕𝑁_𝜃)𝜕𝑦²]+af⋅Pr[𝜕𝑇𝜕𝑥𝜕𝑁_𝜃−𝜕𝑦𝜕𝑇𝜕𝑦𝜕𝑁_𝜃𝜕𝑥],Sc⋅Re(𝜕𝐶𝜕𝑡+𝜓_𝑦𝜕𝐶𝜕𝑥−𝜓_𝑥𝜕𝐶𝜕𝑦)=∇²𝐶+Sc⋅Sr⋅∇²𝑇. (2.18)(2.19)(2.20)(2.21) Also, the boundary conditions at 𝑦=±(1+𝜂(𝑥,𝑡)), when 𝜂(𝑥,𝑡)=𝜀cos(𝛼(𝑥−𝑡)), are𝜓_𝑦=𝑁_𝜃=0,𝜕𝐿(𝜂)𝜕𝑥=−[𝜓_𝑦𝑡+𝜓_𝑦𝜓_𝑥𝑦−𝜓_𝑥𝜓_𝑦𝑦]+2(1+𝛾⁻¹)+𝜇₁2Re[∇²𝜓_𝑦𝜇]+¹Re𝜕𝑁_𝜃,𝜕𝑦 (2.22) where𝜕𝐿(𝜂)𝐾𝜕𝑥=−³Re²𝜕³𝜂𝜕𝑥³+𝑚₁𝜕³𝜂𝜕𝑡²+𝐾𝜕𝑥²𝜕Re²𝜂𝜕𝑡𝜕𝑥, (2.23) while𝑇=1,𝐶=1,at𝑦=−(1+𝜂(𝑥,𝑡)),𝑇=1+𝜀cos(𝛼(𝑥−𝑡)),𝐶=1+𝜀cos(𝛼(𝑥−𝑡))at𝑦=1+𝜂(𝑥,𝑡). (2.24) The parameter 𝜀 represents the amplitude ratio, 𝛼 is wave number, Re is the Reynolds number, 𝜇₁ is the ratio between the viscosity coefficient for micropolar fluids and the classical viscosity coefficient, and M is the micropolar fluid parameter characterizing spin-gradient viscosity. The parameter M can be thought of as a fluid property depending upon the size of the microstructure [18]. We note that 𝑅_ℓ is the modified Reynolds number and involves the quantity J. In this paper, we considered that the effect of microinertia is neglected and 𝑅_ℓ is taken to be zero [24]. The nondimensional quantities Pr are the Prandtl number, Ec is Eckert number, af is the dimensionless heat conduction coefficient of the micropolar fluid, Sc is the Schmidt number, Sr is the Soret number and represents the thermal-diffusion effect, 𝐾₂ and 𝐾₃ represent the dissipative and rigiditive feature of walls, and 𝑚₁ indicates the stiffness property of walls.

3. Method of Solution

Assume the amplitude ratio 𝜀 of the wave to be small. As given by Muthu et al. [25], we seek for an approximate solution as a power series in terms of 𝜀 in the form𝜓(𝑥,𝑦,𝑡)=𝜓₀𝜀(𝑦)+2(𝜙₁(𝑦)𝑒^{𝑖𝛼(𝑥−𝑡)}+𝜙^∗₁𝑒^{−𝑖𝛼(𝑥−𝑡)})+𝜀²2(𝜙₂₀(𝑦)+𝜙₂₂(𝑦)𝑒^{2𝑖𝛼(𝑥−𝑡)}+𝜙^∗₂₂(𝑦)𝑒^{−2𝑖𝛼(𝑥−𝑡)})+𝑜(𝜀³𝑁),_𝜃(𝑥,𝑦,𝑡)=𝑁₀𝜀(𝑦)+2(𝜉₁(𝑦)𝑒^{𝑖𝛼(𝑥−𝑡)}+𝜉^∗₁𝑒^{−𝑖𝛼(𝑥−𝑡)})+𝜀²2(𝜉₂₀(𝑦)+𝜉₂₂(𝑦)𝑒^{2𝑖𝛼(𝑥−𝑡)}+𝜉^∗₂₂(𝑦)𝑒^{−2𝑖𝛼(𝑥−𝑡)})+𝑜(𝜀³),𝑇(𝑥,𝑦,𝑡)=𝑇₀𝜀(𝑦)+2(𝑇₁(𝑦)𝑒^{𝑖𝛼(𝑥−𝑡)}+𝑇^∗₁𝑒^{−𝑖𝛼(𝑥−𝑡)})+𝜀²2(𝑇₂₀(𝑦)+𝑇₂₂(𝑦)𝑒^{2𝑖𝛼(𝑥−𝑡)}+𝑇^∗₂₂(𝑦)𝑒^{−2𝑖𝛼(𝑥−𝑡)})+𝑜(𝜀³),𝐶(𝑥,𝑦,𝑡)=𝐶₀𝜀(𝑦)+2(𝐶₁(𝑦)𝑒^{𝑖𝛼(𝑥−𝑡)}+𝐶^∗₁𝑒^{−𝑖𝛼(𝑥−𝑡)})+𝜀²2(𝐶₂₀(𝑦)+𝐶₂₂(𝑦)𝑒^{2𝑖𝛼(𝑥−𝑡)}+𝐶^∗₂₂(𝑦)𝑒^{−2𝑖𝛼(𝑥−𝑡)})+𝑜(𝜀³). (3.1)(3.2)(3.3)(3.4) Here the asterisk denotes complex conjugate. The solutions for 𝜓₀(𝑦) and 𝑁₀(𝑦) are obtained as follows:𝑑𝜓₀𝑑𝑦=𝐾^′(𝑦²−1)(1+𝑁²Γ²(1−𝑁²))+2𝐾^′𝑁²Γ²(1−𝑁²)(cosh(𝑁𝑀)−cosh(𝑁𝑀𝑦)𝑁𝑀sinh(𝑁𝑁𝑀)),₀=𝐾^′Γ²(1−𝑁²)(sinh(𝑁𝑀𝑦)−𝑦sinh(𝑁𝑀)sinh(𝑁𝑀)), (3.5) where 𝐾^′=(Re/(2(1+𝛾⁻¹)+𝜇₁))(𝑑𝑃/𝑑𝑥)₀ is the poiseuille flow parameter for the micropolar fluid. For pure peristalsis, which means that the flow is generated by wall motion only. The pressure gradient (𝑑𝑃/𝑑𝑥)₀=0; this implies that 𝐾^′=0, which gives 𝜓₀(𝑦)=0 and 𝑁₀(𝑦)=0. The expressions of 𝜙₁ and 𝜉₁ for the case of free pumping can be written as 𝜙₁(𝑦)=𝐴₃sinh𝛼𝑦+𝐴₄sinh𝛽𝑦+𝑎₁𝐴₁sinh𝑟₁𝑦+𝑎₂𝐴₂sinh𝑟₂𝜉𝑦,₁(𝑦)=𝐴₁sinh𝑟₁𝑦+𝐴₂sinh𝑟₂𝑦, (3.6)(3.7) where 𝛽²=𝛼²−2𝑖𝛼Re/(2(1+𝛾⁻¹)+𝜇₁), 𝜆₁=𝑁²𝑁²𝑀²/(1−𝑁²), 𝜆₂=𝑁²𝑀²/(1−𝑁²),𝑟₁1=[2{𝜆₂−𝜆₁+(𝛼²+𝛽²)+((𝜆₁−𝜆₂)−(𝛼²+𝛽²))²−4(𝛼²𝛽²−𝜆₁𝛼²+𝜆₂𝛽²)}]^1/2,𝑟₂1=[2{𝜆₂−𝜆₁+(𝛼²+𝛽²)−((𝜆₁−𝜆₂)−(𝛼²+𝛽²))²−4(𝛼²𝛽²−𝜆₁𝛼²+𝜆₂𝛽²)}]^1/2,𝑎₁=−2𝑁²𝑟²₁−𝛽²,𝑎₂=−2𝑁²𝑟²₂−𝛽²,𝐴₁=Re𝑁²𝑀²sinh𝑟₂2𝑑₁,𝐴₂=−Re𝑁²𝑀²sinh𝑟₁2𝑑₁,𝑑₁=(𝑟³₁+(𝜆₁−𝜆₂−𝛼²)𝑟₁)sinh𝑟₂cosh𝑟₁−(𝑟³₂+(𝜆₁−𝜆₂−𝛼²)𝑟₂)sinh𝑟₁cosh𝑟₂,𝐴₃=1𝛼(𝛽²−𝛼²)cosh𝛼(−2Re𝛿₁2(1+𝛾⁻¹)+𝜇₁+2𝑁²(𝑟₁𝐴₁cosh𝑟₁+𝑟₂𝐴₂cosh𝑟₂)+𝑎₁𝐴₁𝑟₁(𝑟²₁−𝛽²)cosh𝑟₁+𝑎₂𝐴₂𝑟₂(𝑟²₂−𝛽²)cosh𝑟₂𝐴),₄=1𝛽(𝛽²−𝛼²()cosh𝛽2Re𝛿₁2(1+𝛾⁻¹)+𝜇₁−2𝑁²(𝑟₁𝐴₁cosh𝑟₁+𝑟₂𝐴₂cosh𝑟₂)−𝑎₁𝐴₁𝑟₁(𝑟²₁−𝛼²)cosh𝑟₁+𝑎₂𝐴₂𝑟₂(𝑟²₂−𝛼²)cosh𝑟₂𝛿),₁𝐾=𝑖(³𝛼³Re²−𝑚₁𝛼³)+𝐾₂𝛼².Re (3.8)

To obtain the mass and heat transfer, we shall use the perturbation scheme (3.3), (3.4) in (2.20) and (2.21), and equating the zero-order terms on both sides, we obtain the following set of equations:𝑑²𝑇₀𝑑𝑦²𝑑=−Ec⋅Pr[²𝜓₀𝑑𝑦²(2𝜇₁𝑁₀+2(1+𝛾⁻¹)+𝜇₁2𝑑²𝜓₀𝑑𝑦²)+2𝜇₁𝑁²₀+4𝑀²(𝑑𝑁₀)𝑑𝑦²𝑑],₂𝐶₀𝑑𝑦²𝑑=−Sc⋅Sr²𝑇₀𝑑𝑦²𝑇₀,(±1)=1,𝐶₀(±1)=1. (3.9)(3.10)(3.11)

In case of plane Poiseuille flow of micropolar fluid (𝜀=0), the governing (3.9) and (3.10) may be solved in the light of the boundary conditions (3.11) with the constant pressure gradient (𝜕𝑃/𝜕𝑥)₀. At this stage, the temperature and concentration distributions 𝑇₀ and 𝐶₀, respectively, are given by𝑇₀(𝑦)=1−2Pr⋅Ec⋅𝐾^′21×[6(𝑦⁴1−1)+𝑀²sinh²𝑁𝑀1×{4(𝑦²−1)(4sinh²𝑁𝑀+𝑀²(𝑁²(2−𝑁²)−𝜇₁(1−𝑁²)²))−4sinh𝑁𝑀(𝑦sinh𝑁𝑀𝑦−sinh𝑁+𝑀)4sinh𝑁𝑀𝑁𝑀(cosh𝑁𝑀𝑦−cosh𝑁+1𝑀)8𝑁²(𝑁²(2+𝑁²)+𝜇₁(1−𝑁²)²)×(cosh2𝑁𝑀𝑦−cosh2𝑁𝐶𝑀)}],₀(𝑦)=1+ScSr(1−𝑇₀(𝑦)). (3.12)

Let us consider the case in which the pressure gradient (𝜕𝑃/𝜕𝑥)₀ vanishes. In this case there will be no flow if the wall motion stops. Hence 𝜓₀=𝑁₀=0 and 𝑇₀=𝐶₀=1, and equating the first-order terms on both sides of (2.20) and (2.21), we get the following set of equations:𝑑²𝑇₁𝑑𝑦²−𝛾²₁𝑇₁𝑑=0,₂𝐶₁𝑑𝑦²−𝛾²₂𝐶₁=𝑖𝛼Sc⋅Sr⋅Pr⋅Re⋅𝑇₁, (3.13) where 𝛾²₁=𝛼²−𝑖𝛼⋅Pr⋅Re,𝛾²₂=𝛼²−𝑖𝛼⋅Sc⋅Re. (3.14) The boundary conditions are𝑇₁(−1)=𝐶₁(−1)=0,𝑇₁(1)=𝐶₁(1)=1. (3.15) The solutions of (3.13) with boundary conditions (3.15) are𝑇₁(𝑦)=sinh(𝛾₁+𝛾₁𝑦)sinh2𝛾₁,𝐶₁(𝑦)=(1−Sc⋅Sr⋅Pr)(Sc−Pr)sinh(𝛾₂+𝛾₂𝑦)sinh2𝛾₂+Sc⋅Sr⋅Pr(Sc⋅Pr)sinh(𝛾₁+𝛾₁𝑦)sinh2𝛾₁. (3.16)(3.17) In this case, the governing equations for 𝑇₂₀ and 𝐶₂₀ reduce to𝑑²𝑇₂₀𝑑𝑦²=𝑖𝛼2𝑑Pr⋅Re𝑑𝑦(𝑇₁𝜙^∗₁−𝜙₁𝑇^∗₁)−Pr⋅Ec[2(1+𝛾⁻¹)+𝜇₁2×(4𝛼²𝑑𝜙₁𝑑𝑦𝑑𝜙^∗₁+𝑑𝑑𝑦²𝜙₁𝑑𝑦²𝑑²𝜙^∗₁𝑑𝑦²+𝛼⁴𝜙₁𝜙^∗₁+𝛼²(𝜙₁𝑑²𝜙^∗₁𝑑𝑦²+𝜙^∗₁𝑑²𝜙₁𝑑𝑦²))+2𝜇₁(𝜉₁𝜉^∗₁+12(𝜉₁𝑑²𝜙^∗₁𝑑𝑦²+𝜉^∗₁𝑑²𝜙₁𝑑𝑦²−𝛼²(𝜉₁𝜙^∗₁+𝜉^∗₁𝜙₁+4)))𝑀²(𝑑𝜉₁𝑑𝑦𝑑𝜉^∗₁𝑑𝑦+𝛼²𝜉₁𝜉^∗₁)]−𝑖𝛼2𝑑af⋅Pr𝑑𝑦(𝑇₁𝜉^∗₁−𝜉₁𝑇^∗₁𝑑),₂𝐶₂₀𝑑𝑦²=𝑖𝛼2𝑑Sc⋅Re𝑑𝑦(𝐶₁𝜙^∗₁−𝜙₁𝐶^∗₁𝑑)−Sc⋅Sr²𝑇₂₀𝑑𝑦², (3.18) with the boundary conditions𝑇₂₀1(±1)=∓2(𝑇_1𝑦(±1)+𝑇^∗_1𝑦(±1)),𝐶₂₀1(±1)=∓2(𝐶_1𝑦(±1)+𝐶^∗_1𝑦(±1)). (3.19) The solutions of (3.18) are𝑇₂₀(𝑦)=𝑖𝛼41Pr⋅Re[sinh2𝛾₁(𝐴^∗₃𝑔₁(𝛾₁,𝛼)+𝐴^∗₄𝑔₁(𝛾₁,𝛽^∗)+𝐴^∗₁𝑎^∗₁𝑔₁(𝛾₁,𝑟^∗₁)+𝐴^∗₂𝑎^∗₂𝑔₁(𝛾₁,𝑟^∗₂−1))sinh2𝛾^∗₁(𝐴₃𝑔₁(𝛾^∗₁,𝛼)+𝐴₄𝑔₁(𝛾^∗₁,𝛽)+𝐴₁𝑎₁𝑔₁(𝛾^∗₁,𝑟₁)+𝐴₂𝑎₂𝑔₁(𝛾^∗₁,𝑟₂))]−Ec⋅Pr[2(1+𝛾⁻¹)+𝜇₁4×(2𝛼²𝐴₃𝐴^∗₃cosh2𝛼𝑦+4𝛼²(𝛼𝐴₃𝑓₁+𝑟₁𝐴₁𝑎₁𝑓₂+𝑟₂𝐴₂𝑓₃+𝛽𝐴₄𝑓₄)+2𝛼²𝐴₃𝑓₅+(𝛼²+𝑟²₁)𝐴₁𝑎₁𝑓₆+(𝛼²+𝑟²₂)𝐴₂𝑎₂𝑓₇+(𝛼²+𝛽²)𝐴₄𝑓₈𝑀)+(²𝜇₁+2𝛼²𝑀²)×(𝐴₁(𝐴^∗₁𝑔₃(𝑟₁,𝑟^∗₁)+𝐴^∗₂𝑔₃(𝑟₁,𝑟^∗₂))+𝐴₂(𝐴^∗₁𝑔₃(𝑟₂,𝑟^∗₁)+𝐴^∗₂𝑔₃(𝑟₂,𝑟^∗₂+2))𝑀²(𝑟₁𝐴₁(𝑟^∗₁𝐴^∗₁𝑔₂(𝑟₁,𝑟^∗₁)+𝑟^∗₂𝐴^∗₂𝑔₂(𝑟₁,𝑟^∗₂))+𝑟₂𝐴₂(𝑟^∗₁𝐴^∗₁𝑔₂(𝑟₂,𝑟^∗₁)+𝑟^∗₂𝐴^∗₂𝑔₂(𝑟₂,𝑟^∗₂+𝜇)))₁2(𝐴₁𝑓₉+𝐴₂𝑓₁₀+𝐴₄(𝛽²−𝛼²)(𝐴^∗₁𝑔₃(𝛽,𝑟^∗₁)+𝐴^∗₂𝑔₃(𝛽,𝑟^∗₂−)))]𝑖𝛼41af⋅Pr[sinh2𝛾₁(𝐴^∗₁𝑔₁(𝛾₁,𝑟^∗₁)+𝐴^∗₂𝑔₁(𝛾₁,𝑟^∗₂−1))sinh2𝛾^∗₁(𝐴₁𝑔₁(𝛾^∗₁,𝑟₁)+𝐴₂𝑔₁(𝛾^∗₁,𝑟₂))]+𝐴₅𝑦+𝐴₆𝐶₂₀,(𝑦)=𝐴₈+𝐴₇𝑦−Sc⋅Sr⋅𝑇₂₀+𝑖𝛼4Sc⋅Re×{(1−Sc⋅Sr⋅Pr)1(Sc−Pr)sinh2𝛾₂×(𝐴^∗₃𝑔₁(𝛾₂,𝛼)+𝐴^∗₄𝑔₁(𝛾₂,𝛽^∗)+𝐴^∗₁𝑎^∗₁𝑔₁(𝛾₂,𝑟^∗₁)+𝐴^∗₂𝑎^∗₂𝑔₁(𝛾₂,𝑟^∗₂+))Sc⋅Sr⋅Pr(Sc−Pr)sinh2𝛾₁×(𝐴^∗₃𝑔₁(𝛾₁,𝛼)+𝐴^∗₄𝑔₁(𝛾₁,𝛽^∗)+𝐴^∗₁𝑎^∗₁𝑔₁(𝛾₁,𝑟^∗₁)+𝐴^∗₂𝑎^∗₂𝑔₁(𝛾₁,𝑟^∗₂))+(1−Sc⋅Sr⋅Pr)1(Sc−Pr)sinh2𝛾^∗₂×(𝐴₃𝑔₁(𝛾^∗₂,𝛼)+𝐴₄𝑔₁(𝛾^∗₂,𝛽)+𝐴₁𝑎₁𝑔₁(𝛾^∗₂,𝑟₁)+𝐴₂𝑎₂𝑔₁(𝛾^∗₂,𝑟₂−))Sc⋅Sr⋅Pr1(Sc−Pr)sinh2𝛾^∗₁(𝐴₃𝑔₁(𝛾^∗₁,𝛼)+𝐴₄𝑔₁(𝛾^∗₁,𝛽)+𝐴₁𝑎₁𝑔₁(𝛾^∗₁,𝑟₁)+𝐴₂𝑎₂𝑔₁(𝛾^∗₁,𝑟₂))}, (3.20)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) where 𝑔₁−𝑔₃, 𝑓₁−𝑓₁₀  𝐾₃ and 𝐴₅−𝐴₈ are given in the appendix.

Also, the equations governing for 𝑇₂₂ and 𝐶₂₂ are𝑑²𝑇₂₂𝑑𝑦²−𝛾²₃𝑇₂₂=𝑖𝛼2Pr⋅Re(𝑇₁𝑑𝜙₁𝑑𝑦−𝜙₁𝑑𝑇₁𝑑𝑦)−Pr⋅Ec[2(1+𝛾⁻¹)+𝜇₁4𝑑((²𝜙₁𝑑𝑦²)²+𝛼⁴𝜙²₁+2𝛼²(𝜙₁𝑑²𝜙₁𝑑𝑦²−2(𝑑𝜙₁)𝑑𝑦²))+𝜇₁(𝜉²₁+𝜉₁𝑑²𝜙₁𝑑𝑦²−𝛼²𝜉₁𝜙₁2)+𝑀²((𝑑𝜉₁)𝑑𝑦²−𝛼²𝜉²₁−)]𝑖𝛼2af⋅Pr(𝑇₁𝑑𝜉₁𝑑𝑦−𝜉₁𝑑𝑇₁𝑑𝑑𝑦),₂𝐶₂₂𝑑𝑦²−𝛾²₄𝐶₂₂=𝑖𝛼2Sc⋅Re(𝐶₁𝑑𝜙₁𝑑𝑦−𝜙₁𝑑𝐶₁𝑑𝑑𝑦)−Sc⋅Sr(²𝑇₂₂𝑑𝑦²−4𝛼²𝑇₂₂), (3.21)(20)(21)(22)(23) where𝛾²₃=4𝛼²−2𝑖𝛼Pr⋅Re,𝛾³₄=4𝛼²−2𝑖𝛼Sc⋅Re, (3.22) with boundary conditions 𝑇₂₂1(±1)=∓2𝑇_1𝑦(±1),𝐶₂₂1(±1)=∓2𝐶_1𝑦(±1). (3.23) Using boundary conditions (3.23), the expressions of 𝑇₂₂ and 𝐶₂₂ are𝑇₂₂(𝑦)=𝐴₉cosh𝛾₃𝑦+𝐴₁₀sinh𝛾₃𝑦+𝑖𝛼41Prsinh2𝛾₁×[Re(𝛼𝐴₃𝑔₄(𝛾₁,𝛼)+𝛽𝐴₄𝑔₄(𝛾₁,𝛽)−𝛾₁𝐴₃𝑔₅(𝛾₁,𝛼)−𝛾₁𝐴₄𝑔₅(𝛾₁,𝛽))+(𝑎₁Re−af)(𝑟₁𝐴₁𝑔₄(𝛾₁,𝑟₁)−𝛾₁𝐴₁𝑔₅(𝛾₁,𝑟₁))+(𝑎₂Re−af)(𝑟₂𝐴₂𝑔₄(𝛾₁,𝑟₂)−𝛾₁𝐴₂𝑔₅(𝛾₁,𝑟₂))]−Pr⋅Ec{2(1+𝛾⁻¹)+𝜇₁8(4𝛼⁴𝐴²₃𝑔₈(𝛼)+(𝑟²₁+𝛼²)²𝐴²₁𝑎²₁𝑔₈(𝑟₁)+(𝑟²₂+𝛼²)²𝐴²₂𝑎²₂𝑔₈(𝑟₂)+(𝛽²+𝛼²)²𝐴²₄𝑔₈(𝛽)+4𝛼²𝐴₃𝑓₁₁+2𝐴₁𝑎₁(𝛼²+𝑟²₁)𝑓₁₂+2𝐴₂𝑎₂(𝛼²+𝑟²₂)𝑓₁₃−4𝛼²(𝛼²𝐴²₃𝑔₉(𝛼)+𝑟²₁𝐴²₁𝛼²₁𝑔₉(𝑟₁)+𝑟²₂𝐴²₂𝑎²₂𝑔₉(𝑟₂)+𝛽²𝐴²₄𝑔₉(𝛽)+2𝛼𝐴₃𝑓₁₄+2𝑟₁𝐴₁𝑎₁𝑓₁₅+2𝑟₂𝐴₂𝑎₂𝑓₁₆+𝜇))₁2((1+(𝑟²₁−𝛼²)𝑎₁)𝐴²₁𝑔₈(𝑟₁)+(1+(𝑟²₂−𝛼²)𝑎₂)𝐴²₂𝑔₈(𝑟₂)+𝐴₁𝑓₁₇+𝐴₂𝐴₄(𝛽²−𝛼²)𝑔₆(𝑟₂+1,𝛽))𝑀²(𝑟²₁𝐴²₁𝑔₉(𝑟₁)+𝑟²₂𝐴²₂𝑔₉(𝑟₂)+2𝑟₁𝑟₂𝐴₁𝐴₂𝑔₇(𝑟₁,𝑟₂)−𝛼²(𝐴²₁𝑔₈(𝑟₁)+𝐴²₂𝑔₈(𝑟₂)+2𝐴₁𝐴₂𝑔₆(𝑟₁,𝑟₂𝐶)))},₂₂(𝑦)=𝐴₁₁cosh𝛾₄𝑦+𝐴₁₂sinh𝛾₄𝑦+𝑖𝛼4Sc⋅Re{(1−Sc⋅Sr⋅Pr)𝑓(Sc−Pr)sinh¹⁸2𝛾₂+Sc⋅Sr⋅Pr𝑓(Sc−Pr)sinh¹⁹2𝛾₁}+Pr⋅Sc⋅Sr(Sc−Pr)(𝐴₉cosh𝛾₃𝑦+𝐴₁₀sinh𝛾₃𝑦)−𝑖𝛼4𝑓Sc⋅Sr⋅Pr⋅Re²⁰sinh2𝛾₁+Pr⋅Ec⋅Sc⋅Sr×[2(1+𝛾⁻¹)+𝜇₁8(32𝛼⁶𝐴²₃(𝛾₃𝛾₄)²+(𝑟²₁+𝛼²)²𝐴²₁𝑎²₁𝑔₁₄(𝑟₁)+(𝑟²₂+𝛼²)²𝐴²₂𝑎²₂𝑔₁₄(𝑟₂)+(𝛽²+𝛼²)²𝐴²₄𝑔₁₄(𝛽)+4𝛼²𝐴₃𝑓₂₁+2(𝛼²+𝑟²₁)𝐴₁𝑎₁𝑓₂₂+2(𝛼²+𝑟²₂)𝐴₂𝑎₂𝑓₂₃−4𝛼²(𝑟²𝐴²₁𝑎²₁𝑔₁₅(𝑟₁)+𝑟²₂𝐴²₂𝑎²₂𝑔₁₅(𝑟₂)+𝛽²𝐴²₄𝑔₁₅(𝛽)+2𝛼𝐴₃𝑓₂₄+2𝑟₁𝐴₁𝑎₁𝑓₂₅+2𝑟₂𝐴₂𝑎₂𝑓₂₆+𝜇))₁2((1+(𝑟²₁−𝛼²)𝑎₁)𝐴²₁𝑔₁₄(𝑟₁)+(1+(𝑟²₂−𝛼²)𝑎₂)𝐴²₂𝑔₁₄(𝑟₂)+𝐴₁𝑓₂₇+𝐴₂𝐴₄(𝛽²−𝛼²)𝑔₁₆(𝑟₂+1,𝛽))𝑀²(𝑟²₁𝐴²₁𝑔₁₅(𝑟₁)+𝑟²₂𝐴²₂𝑔₁₅(𝑟₂)+2𝑟₁𝑟₂𝐴₁𝐴₂𝑔₁₇(𝑟₁,𝑟₂)−𝛼²(𝐴²₁𝑔₁₄(𝑟₁)+𝐴²₂𝑔₁₄(𝑟₂)+2𝐴₁𝐴₂𝑔₁₆(𝑟₁,𝑟₂+)))]𝑖𝛼4𝑓af⋅Pr⋅Sc⋅Sr⋅²⁸sinh2𝛾₁. (3.24)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35)(36)(3.25) Hence 𝑔₄−𝑔₁₇, 𝑓₁₁−𝑓₂₈ and 𝐴₉−𝐴₁₂ are given in the appendix.

Using the expressions for Ψ₁, 𝑇₁, 𝐶₁, 𝑇₂₀, 𝐶₂₀, 𝑇₂₂, and 𝐶₂₂ in (3.1), (3.3), and (3.4), we can determine the stream function Ψ(𝑥,𝑦,𝑡), temperature 𝑇(𝑥,𝑦,𝑡), and concentration 𝐶(𝑥,𝑦,𝑡) of micropolar fluid for the case of free pumping as𝜀𝜓(𝑥,𝑦,𝑡)=2(𝜙₁(𝑦)𝑒^{𝑖𝛼(𝑥−𝑡)}+𝜙^∗₁𝑒^{−𝑖𝛼(𝑥−𝑡)})+𝑜(𝜀²𝜀),𝑇(𝑥,𝑦,𝑡)=1+2(𝑇₁(𝑦)𝑒^{𝑖𝛼(𝑥−𝑡)}+𝑇^∗₁(𝑦)𝑒^{−𝑖𝛼(𝑥−𝑡)})+𝜀²2(𝑇₂₀(𝑦)+𝑇₂₂𝑒^{2𝑖𝛼(𝑥−𝑡)}+𝑇^∗₂₂(𝑦)𝑒^{−2𝑖𝛼(𝑥−𝑡)}𝜀),𝐶(𝑥,𝑦,𝑡)=1+2(𝐶₁(𝑦)𝑒^{𝑖𝛼(𝑥−𝑡)}+𝐶^∗₁(𝑦)𝑒^{−𝑖𝛼(𝑥−𝑡)})+𝜀²2(𝐶₂₀(𝑦)+𝐶₂₂𝑒^{2𝑖𝛼(𝑥−𝑡)}+𝐶^∗₂₂(𝑦)𝑒^{−2𝑖𝛼(𝑥−𝑡)}), (3.26)(3.27)(3.28) where 𝜙^∗₁,𝑇^∗₁(𝑦),𝐶^∗₁(𝑦),𝑇^∗₂₂(𝑦), and 𝐶^∗₂₂(𝑦) are conjugate to (3.6), (3.16), (3.17), (3.24), and (3.25), respectively.

4. Results and Discussion

To discuss the effect of the parameters of micropolar, non-Newtonian fluid, and viscoelastic membrane properties on the solutions of the considered problem. A numerical results are calculated from formula (3.6), (3.27), and (3.28) for the temperature 𝑇(𝑥,𝑦,𝑡), concentration 𝐶(𝑥,𝑦,𝑡) and stream function 𝜓(𝑥,𝑦,𝑡) and shown by Figures 2–24. Take into account the parameters t and x have the numerical value 2.0, while parameter 𝛼 and amplitude ratio 𝜀 take the value 0.5 and 0.01, respectively, but 𝛼 takes the value 2 for the stream function.

Figure 2 

The variation of temperature distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\gamma =0.8\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$$M=10\text{,}$$\text{af}=1\text{,}$$K_2$ = $m_1$ = 0, and $K_3$ = 10 for various values of ${\mu }_1$.

Figure 3 

The variation of temperature distribution versus y, with fixed $\epsilon \mathrm{\hspace{0.17em}\hspace{0.17em}}=0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\gamma =0.8\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$${\mu }_1=10\text{,}$$\text{af}=1\text{,}$$K_2$ = $m_1$ = 0, and $K_3$ = 10 for various values of M.

Figure 4 

The variation of temperature distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\gamma =0.8\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$${\mu }_1=10\text{,}$$M=10\text{,}$$K_2$ = $m_1$ = 0, and $K_3$ = 10 for various values of af.

Figure 5 

The variation of temperature distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\gamma =0.8\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$${\mu }_1=10\text{,}$$M=10\text{,}$$\text{af}=1\text{,}$$m_1$ = 0, and $K_3$ = 10 for various values of $K_2$.

Figure 6 

The variation of temperature distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\gamma =0.8\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$${\mu }_1=10\text{,}$$M=10\text{,}$$\text{af}=1$ and $K_2$ = $m_1$ = 0 for various values of $K_3$.

Figure 7 

The variation of temperature distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\gamma =0.8\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$${\mu }_1=10\text{,}$$M=10\text{,}$$\text{af}=1\text{,}$$K_2$ = 0 and $K_3$ = 10 for various values of $m_1$.

Figure 8 

The variation of temperature distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\gamma =0.8\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$${\mu }_1=10\text{,}$$M=10\text{,}$$\text{af}=1\text{,}$$K_2$ = $m_1$ = 0 and $K_3$ = 10 for various values of $\mathrm{Re}$.

Figure 9 

The variation of temperature distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\gamma =0.8\text{,}$$\text{Ec}=0.5\text{,}$${\mu }_1=10\text{,}$$M=10\text{,}$$\text{af}=1\text{,}$$K_2$ = $m_1$ = 0 and $K_3$ = 10 for various values of $\text{Pr}$.

Figure 10 

The variation of temperature distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\gamma =0.8\text{,}$$\text{Pr}=5\text{,}$${\mu }_1=10\text{,}$$M=10\text{,}$$\text{af}=1\text{,}$$K_2$ = $m_1$ = 0 and $K_3$ = 10 for various values of $\text{Ec}$.

Figure 11 

The variation of temperature distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$${\mu }_1=10\text{,}$$M=10\text{,}$$\text{af}=1\text{,}$$K_2$ = $m_1$ = 0 and $K_3$ = 10 for various values of $\gamma$.

Figure 12 

The variation of concentration distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\gamma =0.8\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$$M=10\text{,}$$\text{af}=1\text{,}$$K_2$ = $m_1$ = 0, $K_3$ = 10, $\text{Sc}=1$ and $\text{Sr}=1$ for various values of ${\mu }_1$.

Figure 13 

The variation of concentration distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\gamma =0.8\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$${\mu }_1=10\text{,}$$M=10\text{,}$$K_2$ = $m_1$ = 0, $K_3$ = 10, $\text{Sc}=1$ and $\text{Sr}=1$ for various values of af.

Figure 14 

The variation of concentration distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\gamma =0.8\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$${\mu }_1=10\text{,}$$M=10\text{,}$$\text{af}=1\text{,}$$m_1$ = 0, $K_3$ = 10, $\text{Sc}=1$ and $\text{Sr}=1$ for various values of $K_2$.

Figure 15 

The variation of concentration distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\gamma =0.8\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$${\mu }_1=10\text{,}$$M=10\text{,}$$\text{af}=1\text{,}$$K_2$ = $m_1$ = 0, $\text{Sc}=1$ and $\text{Sr}=1$ for various values of $K_3$.

Figure 16 

The variation of concentration distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\gamma =0.8\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$${\mu }_1=10\text{,}$$M=10\text{,}$$\text{af}=1\text{,}$$K_2$ = 0, $K_3$ = 10, $\text{Sc}=1$ and $\text{Sr}=1$ for various values of $m_1$.

Figure 17 

The variation of concentration distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\gamma =0.8\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$${\mu }_1=10\text{,}$$M=10\text{,}$$\text{af}=1\text{,}$$K_2$ = $m_1$ = 0, $K_3$ = 10 and $\text{Sc}=1$ for various values of $\text{Sr}$.

Figure 18 

The variation of concentration distribution versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =2\text{,}$$\mathrm{Re}=10\text{,}$$\text{Pr}=5\text{,}$$\text{Ec}=0.5\text{,}$${\mu }_1=10\text{,}$$M=10\text{,}$$\text{af}=1\text{,}$$K_2$ = $m_1$ = 0, $K_3$ = 10, $\text{Sc}=1$ and $\text{Sr}=1$ for various values of $\gamma$.

Figure 19 

The variation of streamlines versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =0.5\text{,}$$\mathrm{Re}$ = 10, $\gamma =0.8\text{,}$ M = 10, $K_2$ = $m_1$ = 0 and $K_3$ = 10 for various values of ${\mu }_1$.

Figure 20 

The variation of streamlines versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =0.5\text{,}$$\mathrm{Re}$ = 10, $\gamma =0.8\text{,}$${\mu }_1=1\text{,}$$K_2$ = $m_1$ = 0 and $K_3$ = 10 for various values of M.

Figure 21 

The variation of streamlines versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =0.5\text{,}$$\gamma =0.8\text{,}$${\mu }_1=1\text{,}$ M = 10, $K_2$ = $m_1$ = 0 and $K_3$ = 10 for various values of $\mathrm{Re}$.

Figure 22 

The variation of streamlines versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =0.5\text{,}$$\mathrm{Re}$ = 10, $\gamma =0.8\text{,}$${\mu }_1=1\text{,}$ M = 10, $m_1$ = 0 and $K_3$ = 10 for various values of $K_2$.

Figure 23 

The variation of streamlines versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =0.5\text{,}$$\mathrm{Re}$ = 10, $\gamma =0.8\text{,}$${\mu }_1=1\text{,}$ M = 10 and $K_2$ = $m_1$ = 0 for various values of $K_3$.

Figure 24 

The variation of streamlines versus y, with fixed $\epsilon =0.01\text{,}$$\alpha =0.5\text{,}$$\mathrm{Re}$ = 10, ${\mu }_1$ = 1, M = 10, $K_2$ = $m_1$ = 0 and $K_3$ = 10 for various values of $\gamma$.

Figures 2–4 illustrate the effects of 𝜇₁=(𝑘/𝜇_𝐵), 𝑀=(2𝑑(𝜇_𝐵/𝛾)^0.5), and af, in order, on the temperature distribution with taking into account that the elastic wall (𝐾₂ = 0). The ratios 𝑘/𝜇_𝐵 and 𝛾/𝜇_𝐵 measure the relative strengths of the vortex viscosity coefficient to the viscosity coefficient and the spin gradient viscosity coefficient to the viscosity coefficient, respectively. These coefficients 𝜇_𝐵, 𝑘, and 𝛾 may be greater or equal to zero. For example, the blood 50% hematocrit has the values 𝜇_𝐵 = 0.0029, 𝑘 = 0.000232, and 𝛾 = 0.000001 [27]. When the viscous effects are much larger than the spin gradient viscosity effects, (𝛾/𝜇_𝐵) may tend to be zero, and the parameter 𝑀 will tend to infinity. Ahmadi [28] has stated that the parameter 𝜇₁ depends on the shape and concentration of the microelements while the parameter M signifies the size of microstructure. In this context, small (large) value of 𝜇₁ means low (high) concentration of the microelements. Similarly, a given small (large) value of M is related to small (large) size of the particles. It is noted from Figure 2 that the temperature distribution decreases with the increasing of 𝜇₁ near the lower wall, while it increases as 𝜇₁ increases when 𝑦≥−0.7. In Figure 3, it can be seen that the temperature distribution increases as M increases. Also, Figure 4 shows that the temperature increases as af increases but at 𝑦>0.8, an opposite effect is noticed near the upper wall. The results in our study are consistent with those which are obtained by Agarwal et al. [24] in case of isothermal wall.

The choice 𝐾₂ = 0 implies that the walls move up and down with no damping force on them and hence indicates that the membrane is treated as an elastic wall. The parameter 𝐾₂ depends upon the wall tension and represents the rigid nature of the walls. Figures 5, 6, and 7 illustrate the effect of the parameters of the viscoelastic wall properties 𝐾₂, 𝐾₃, and 𝑚₁ on the temperature distribution, respectively. It is found that the temperature distribution increases as 𝐾₂ increases as shown in Figure 5, but near the lower wall, it decreases with the increase of 𝐾₂. The temperature distribution decreases with the increase of both 𝐾₃ and 𝑚₁, but when 𝑦≥−0.7, it starts increasing as both 𝐾₃ and 𝑚₁ increase for the case of elastic walls (𝐾₂ = 0), as shown in Figures 6 and 7. This phenomenon reflects the fact that the increasing of the values of tension or dissipative for the membrane leads to loss of heat from the walls and gives it to the fluid. Therefore, the temperature distribution of fluid will increase.

In Figure 8, we see the effect of Reynolds number Re on the temperature distribution. It is observed that the temperature distribution behaves a dual role with the variation of Re. The variations of temperature distribution for various values of Prandtl number Pr and Eckert number Ec are displayed in Figures 9 and 10, respectively. The graphical results of Figures 9 and 10 indicate that the temperature of fluid increases as Pr and Ec increase. While for |𝑦|>0.7, near the lower and upper wall, the temperature decreases with the increase of Pr and Ec. In the case when the temperature of the plate is periodic, Das et al. [29] showed that the temperature decreases with the increase of the Prandtle number of Newtonian fluid. In Figure 11, the temperature is plotted against y for various values of the upper limit apparent viscosity coefficient 𝛾; it is observed that an increase of 𝛾 increases the temperature.

Figures 12–18 reveal the influence of physical parameters entering in the problem on the concentration distribution of fluid. The effect of the coupling parameter 𝜇₁ on concentration is indicated in Figure 12; we find that the concentration distribution of fluid decreases with the increase of 𝜇₁. It has been observed that the concentration distribution decreases with the increase of af, but an opposite effect is seen at 𝑦>0.8. This is noticed in Figure 13. Also, in Figures 12 and 13, for small values of 𝜇₁ and af, its effects are rather insignificant. In Figures 14, 15, and 16, it is found that the increase of the dissipative, rigid nature of the walls and stiffness in the walls causes a decrease of the concentration distribution. The behavior of the concentration distribution with the nondimensional distance y for various values of Soret number Sr is depicted in Figure 17. It is clear, from Figure 17, that the effect of Sr on the concentration can be neglected near the lower wall. When −0.8<𝑦<0.5, the concentration decreases with the increase of Sr, but at 𝑦>0.5, an increase in Sr leads to increase the concentration distribution. The effect of the thermal-diffusion parameter Sr on the concentration distribution, in case of non-Newtonian fluid, has been studied by El Dabe et al. [23]. The result in Figure 17, at the upper wall, is in agreement with those obtained by El Dabe et al. [23]. Figure 9 illustrates the effects of the upper limit apparent viscosity coefficient 𝛾 on the concentration distribution. It is found that the concentration decreases near lower and upper walls, but it increases with the increase of 𝛾 at the middle of the channel.

The effects of various physical parameters on the stream function 𝜓 are indicated in Figures 19–24. In these figures the stream function is plotted versus the dimensionless distance 𝑦, and it is clear that there is an inversely non-linear relation between y and 𝜓 and that the obtained curves of all figures will intersect at the origin where 𝑦=0, 𝜓=0. Figure 19 shows the variation of 𝜓 with y for different values of the coupling parameter 𝜇₁; it is observed that, in the region −1≤𝑦<0, the stream function decreases by increasing 𝜇₁, while in the region 0<𝑦≤1, 𝜓 increases by increasing 𝜇₁. Note that, when 𝜇₁≥1, the peristaltic pumping region becomes slightly wider as 𝜇₁ increases, and if 𝜇₁<1, it is slightly narrow as 𝜇₁ decreases. Figure 20 illustrates the stream function 𝜓 for different values of 𝑀. From this figure, it is observed that the effect of M on 𝜓 is similar to the effect of 𝜇₁ on 𝜓 illustrated in Figure 19. In Figures 21 and 22, the effects of Reynolds number Re and the parameter of dissipates of walls 𝐾₂ on the stream function 𝜓, respectively, are reverse to the effect of 𝜇₁ and M on 𝜓, that is, in the region −1≤𝑦<0, 𝜓 increases by increasing both of Re and 𝐾₂, while in the region 0<𝑦≤1, 𝜓 decreases by increasing both of Re and 𝐾₂. Note that the peristaltic pumping region becomes slightly wider as Re and 𝐾₂ increase. Figures 23 and 24 show that the behaviors of the stream function 𝜓 with the dimensionless distance y for different values of the parameter of rigidities feature of walls 𝐾₃ and upper limit apparent viscosity coefficient 𝛾, respectively, are similar to the behaviors of 𝜓 for different values of 𝜇₁ and M illustrated in Figures 19 and 20, with the only difference that the peristaltic pumping will disappear in these cases.

5. Conclusion

In this paper, we study the peristaltic flow of a micropolar biviscosity fluid in channel with dynamic boundary condition. Also, we take into considerations that the temperature and concentration of lower wall are constant while they are periodic for the upper wall. This study is an extension of the work of Muthu et al. [25]. The analytical expressions are constructed for the stream function, microrotation velocity, temperature, and concentration distributions as a power series in terms of small amplitude ratio in case of free pumping. The effects of various physical parameters acting on the problem are discussed by a set of graphs. The micropolar parameters 𝜇₁, M, and af and wall parameters 𝐾₂, 𝐾₃, and 𝑚₁ which describe the viscoelastic behaviors of the flexible wall are discussed through numerical computations. The obtained results can be outlined and summarized as follows.(1)The temperature increases with the increase each of 𝐾₂, 𝐾₃, and 𝑚₁, but near the lower wall it decreases.(2)The temperature for different values of 𝐾₂, 𝐾₃, and 𝑚₁ becomes greater with increasing the dimensionless distance 𝑦 and reaches maximum at 𝑦=1.(3)The concentration decreases with the increase each of 𝐾₂, 𝐾₃, and 𝑚₁.(4) The temperature and concentration increase as 𝛾 increases, but the concentration decreases near the lower and upper walls.(5)The streamlines intersect at a critical value of 𝑦 (=0) after which all the obtained curves will have an opposite behavior.(6)For any value 𝑦<0.8, the temperature increases with the increase of af, while for any value 𝑦≥0.8, all the obtained lines will behave in an opposite manner to these behavior when 𝑦<0.8.(7)The concentration has an opposite behavior compared to the temperature with af (i.e., for any value 𝑦<0.8, the concentration decreases with the increase of af, while for any value 𝑦≥0.8, it increases).

Appendix

The functions 𝑔₁−𝑔₁₇, 𝑓₁−𝑓₂₈ and the constants 𝐴₅−𝐴₁₂ are given by 𝑔₁(𝑎,𝑏)=sinh(𝑎+(𝑎+𝑏)𝑦)−𝑎+𝑏sinh(𝑎+(𝑎−𝑏)𝑦),𝑔𝑎−𝑏₂(𝑎,𝑏)=cosh((𝑎+𝑏)𝑦)(𝑎+𝑏)²+cosh((𝑎−𝑏)𝑦)(𝑎−𝑏)²,𝑔₃(𝑎,𝑏)=cosh((𝑎+𝑏)𝑦)(𝑎+𝑏)²−cosh((𝑎−𝑏)𝑦)(𝑎−𝑏)²,𝑓₁=𝑟^∗₁𝐴^∗₁𝑎^∗₁𝑔₂(𝛼,𝑟^∗₁)+𝑟^∗₂𝐴^∗₂𝑎^∗₂𝑔₂(𝛼,𝑟^∗₂)+𝛽^∗𝐴^∗₄𝑔₂(𝛼,𝛽^∗𝑓),₂=𝛼𝐴^∗₃𝑔₂(𝑟₁,𝛼)+𝑟^∗₁𝐴^∗₁𝑎^∗₁𝑔₂(𝑟₁,𝑟^∗₁)+𝑟^∗₂𝐴^∗₂𝑎^∗₂𝑔₂(𝑟₁,𝑟^∗₂)+𝛽^∗𝐴^∗₄𝑔₂(𝑟₁,𝛽^∗𝑓),₃=𝛼𝐴^∗₃𝑔₂(𝑟₂,𝛼)+𝑟^∗₁𝐴^∗₁𝑎^∗₁𝑔₂(𝑟₂,𝑟^∗₁)+𝑟^∗₂𝐴^∗₂𝑎^∗₂𝑔₂(𝑟₂,𝑟^∗₂)+𝛽^∗𝐴^∗₄𝑔₂(𝑟₂,𝛽^∗𝑓),₄=𝛼𝐴^∗₃𝑔₂(𝛽,𝛼)+𝑟^∗₁𝐴^∗₁𝑎^∗₁𝑔₂(𝛽,𝑟^∗₁)+𝑟^∗₂𝐴^∗₂𝑎^∗₂𝑔₂(𝛽,𝑟^∗₂)+𝛽^∗𝐴^∗₄𝑔₂(𝛽,𝛽^∗𝑓),₅=(𝛼²+𝑟^∗₁²)𝐴^∗₁𝑎^∗₁𝑔₃(𝛼,𝑟^∗₁)+(𝛼²+𝑟^∗2₁)𝐴^∗₁𝑎^∗₂𝑔₃(𝛼,𝑟^∗₂)+(𝛼²,𝛽^∗2)+𝐴^∗₄𝑔₂(𝛼,𝛽^∗𝑓),₆=2𝛼²𝐴^∗₃𝑔₃(𝑟₁,𝛼)+(𝛼²+𝑟^∗2₁)𝐴^∗₁𝑎^∗₁𝑔₃(𝑟₁,𝑟^∗₁)+(𝛼²,𝑟^∗22₂)𝐴^∗₂𝑎^∗₂𝑔₃(𝑟₁,𝑟^∗₂)+(𝛼²+𝛽^∗2)𝐴^∗₄𝑔₃(𝑟₁,𝛽^∗𝑓),₇=2𝛼²𝐴^∗₃𝑔₃(𝑟₂,𝛼)+(𝛼²+𝑟^∗2₁)𝐴^∗₁𝑎^∗₁𝑔₃(𝑟₁,𝑟^∗₁)+(𝛼²+𝑟^∗2₂)𝐴^∗₂𝑎^∗₂𝑔₃(𝑟₁,𝑟^∗₂)+(𝛼²+𝛽^∗2)𝐴^∗₄𝑔₃(𝑟₂,𝛽^∗𝑓),₈=2𝛼²𝐴^∗₃𝑔₃(𝛽,𝛼)+(𝛼²+𝑟^∗2₁)𝐴^∗₁𝑎^∗₁𝑔₃(𝛽,𝑟^∗₁)+(𝛼²+𝑟^∗2₂)𝐴^∗₂𝑎^∗₂𝑔₃(𝛽,𝑟^∗₂)+(𝛼²+𝛽^∗2)𝐴^∗₄𝑔₃(𝛽,𝛽^∗𝑓),₉=𝐴^∗₁(𝑎^∗₁(𝑟^∗2₁−𝛼²)+𝑎₁(𝑟²₁−𝛼²))𝑔₃(𝑟₁,𝑟^∗₁)+𝐴^∗₂(𝑎^∗₂(𝑟^∗2₂−𝛼²)+𝑎₁(𝑟²₁−𝛼²))𝑔₃(𝑟₁,𝑟^∗₂)+𝐴^∗₄(𝛽^∗2−𝛼²)𝑔₃(𝑟₁,𝛽^∗𝑓),₁₀=𝐴^∗₁(𝑎^∗₁(𝑟^∗2₁−𝛼²)+𝑎₂(𝑟²₂−𝛼²))𝑔₃(𝑟₂,𝑟^∗₁)+𝐴^∗₂(𝑎^∗₂(𝑟^∗2₂−𝛼²)+𝑎₂(𝑟²₂−𝛼²))𝑔₃(𝑟₂,𝑟^∗₂)+𝐴^∗₄(𝛽^∗2−𝛼²)𝑔₃(𝑟₂,𝛽^∗𝛿),₂(𝑎,𝑏)=sinh(2𝑎+𝑏)−𝑎+𝑏sinh(2𝑎−𝑏)−𝑎−𝑏2𝑎sinh𝑏𝑎²−𝑏²,𝛿₃(𝑎,𝑏)=cosh(𝑎+𝑏)(𝑎+𝑏)²+cosh(𝑎−𝑏)(𝑎−𝑏)²,𝛿₄(𝑎,𝑏)=cosh(𝑎+𝑏)(𝑎+𝑏)²−cosh(𝑎−𝑏)(𝑎−𝑏)²,𝛿₅(𝑎,𝑏)=sinh(2𝑎+𝑏)−𝑎+𝑏sinh(2𝑎−𝑏)+𝑎−𝑏2𝑎sinh𝑏𝑎²−𝑏²,𝜆₃=𝑟^∗₁𝐴^∗₁𝛿₃(𝛼,𝑟^∗₁)+𝑟^∗₂𝐴^∗₂𝑎^∗₂𝛿₃(𝛼,𝑟^∗₂)+𝛽^∗𝐴^∗₄𝛿₃(𝛼,𝛽^∗𝜆),₄=𝛼𝐴^∗₃𝜆₃(𝑟₁,𝛼)+𝑟^∗₁𝐴^∗₁𝑎^∗₁𝛿₃(𝑟₁,𝑟^∗₁)+𝑟^∗₂𝐴^∗₂𝑎^∗₂𝛿₃(𝑟₁,𝑟^∗₂)+𝛽^∗𝐴^∗₄𝛿₃(𝑟₁,𝛽^∗𝜆),₅=𝛼𝐴^∗₃𝛿₃(𝑟₂,𝛼)+𝑟^∗₁𝐴^∗₁𝑎^∗₁𝛿₃(𝑟₂,𝑟^∗₁)+𝑟^∗₂𝐴^∗₂𝑎^∗₂𝛿₃(𝑟₂,𝑟^∗₂)+𝛽^∗𝐴^∗₄𝛿₃(𝑟₂,𝛽^∗𝜆),₆=𝛼𝐴^∗₃𝛿₃(𝛽,𝛼)+𝑟^∗₁𝐴^∗₁𝑎^∗₁𝛿₃(𝛽,𝑟^∗₁)+𝑟^∗₂𝐴^∗₂𝑎^∗₂𝛿₃(𝛽,𝑟^∗₂)+𝛽^∗𝐴^∗₄𝛿₃(𝛽,𝛽^∗𝜆),₇=(𝛼²+𝑟^∗2₁)𝐴^∗₁𝑎^∗₁𝛿₄(𝛼,𝑟^∗₁)+(𝛼²+𝑟^∗2₂)𝐴^∗₂𝑎^∗₂𝛿₄(𝛼,𝑟^∗₂)+(𝛼²+𝛽^∗2)+𝐴^∗₄𝛿₄(𝛼,𝛽^∗𝐴),₅1=−4(𝛾₁(1+cosh2𝛾₁)sinh2𝛾₁𝛾1+^∗1(1+cosh2𝛾^∗)sinh2𝛾^∗₁)−𝑖𝛼81Pr⋅𝑅𝑒(sinh2𝛾₁({sinh(2𝛾₁+𝛼)𝛾₁−+𝛼sinh(2𝛾₁−𝛼)𝛾₁+−𝛼2𝛾₁sinh𝛼𝛾²₁−𝛼²}𝐴^∗₃+{sinh(2𝛾₁+𝛽^∗)𝛾₁+𝛽^∗−sinh(2𝛾₁−𝛽^∗)𝛾₁−𝛽^∗+2𝛾₁sinh𝛽^∗𝛾²₁−𝛼²}𝐴^∗₄)1−{sinh2𝛾^∗₁({sinh(2𝛾^∗₁+𝛼)𝛾^∗₁−+𝛼sinh(2𝛾^∗₁−𝛼)𝛾^∗₁+−𝛼2𝛾^∗₁sinh𝛼𝛾^∗2₁−𝛼²}𝐴₃+{sinh(2𝛾^∗₁+𝛽)𝛾^∗₁−+𝛽sinh(2𝛾^∗₁−𝛽)𝛾^∗₁+−𝛽2𝛾^∗₁sinh𝛽𝛾^∗2₁−𝛽²}𝐴₄+))𝑖𝛼81⋅Pr[sinh2𝛾₁×((af−𝑅𝑒𝑎^∗₁)𝐴^∗₁{sinh(2𝛾₁+𝑟^∗₁)𝛾₁+𝑟^∗₁−sinh(2𝛾₁−𝑟^∗₁)𝛾₁−𝑟^∗₁+2𝛾₁sinh𝑟^∗₁𝛾²₁−𝑟^∗2₁}+(af−𝑅𝑒𝑎^∗₂)𝐴^∗₂{sinh(2𝛾₁+𝑟^∗₂)𝛾₁+𝑟^∗₂−sinh(2𝛾₁−𝑟^∗₂)𝛾₁−𝑟^∗₂+2𝛾₁sinh𝑟^∗₂𝛾²₁−𝑟^∗2₂+1})sinh2𝛾^∗₁((𝑎₁𝑅𝑒−af)𝐴₁×{sinh(2𝛾^∗₁+𝑟₁)𝛾^∗₁+𝑟₁−sinh(2𝛾^∗₁−𝑟₁)𝛾^∗₁−𝑟₁+2𝛾^∗₁sinh𝑟₁𝛾^∗2₁−𝑟²₁}+(𝑎₂𝑅𝑒−af)𝐴₂×{sinh(2𝛾^∗₁+𝑟₂)𝛾^∗₁+𝑟₂−sinh(2𝛾^∗₁−𝑟₂)𝛾^∗₁−𝑟₂+2𝛾^∗₁sinh𝑟₂𝛾^∗2₁−𝑟²₂𝜆})],₈=2𝛼²𝐴^∗₃𝛿₄(𝑟₁,𝛼)+(𝛼²+𝑟^∗2₁)𝐴^∗₁𝑎^∗₁𝛿₄(𝑟₁,𝑟^∗₁)+(𝛼²+𝑟^∗2₂)𝐴^∗₂𝑎^∗₂𝛿₄(𝑟₁,𝑟^∗₂)+(𝛼²+𝛽^∗2)𝐴^∗₄𝛿₄(𝑟₁,𝛽^∗𝜆),₉=2𝛼²𝐴^∗₃𝛿₄(𝑟₂,𝛼)+(𝛼²+𝑟^∗2₁)𝐴^∗₁𝑎^∗₁𝛿₄(𝑟₂,𝑟^∗₁)+(𝛼²+𝑟^∗2₂)𝐴^∗₂𝑎^∗₂𝛿₄(𝑟₂,𝑟^∗₂)+(𝛼²+𝛽^∗2)𝐴^∗₄𝛿₄(𝑟₂,𝛽^∗𝜆),₁₀=2𝛼²𝐴^∗₃𝛿₄(𝛽,𝛼)+(𝛼²+𝑟^∗2₁)𝐴^∗₁𝑎^∗₁𝛿₄(𝛽,𝑟^∗₁)+(𝛼²+𝑟^∗2₂)𝐴^∗₂𝑎^∗₂𝛿₄(𝛽,𝑟^∗₂)+(𝛼²+𝛽^∗2)𝐴^∗₄𝛿₄(𝛽,𝛽^∗𝜆),₁₁=𝐴^∗₁(𝑎^∗₁(𝑟^∗2₁−𝛼²)+𝑎₁(𝑟²₁−𝛼²))𝛿₄(𝑟₁,𝑟^∗₁)+𝐴^∗₂(𝑎^∗₂(𝑟^∗2₂−𝛼²)+𝑎₁(𝑟²₁−𝛼²))𝛿₄(𝑟₁,𝑟^∗₂)+𝐴^∗₄(𝛽^∗2−𝛼²)𝛿₄(𝑟₁,𝛽^∗𝜆),₁₂=𝐴^∗₁(𝑎^∗₁(𝑟^∗2₁−𝛼²)+𝑎₂(𝑟²₂−𝛼²))𝛿₄(𝑟₂,𝑟^∗₁)+𝐴^∗₂(𝑎^∗₂(𝑟^∗2₂−𝛼²)+𝑎₂(𝑟²₂−𝛼²))𝛿₄(𝑟₂,𝑟^∗₂)+𝐴^∗₄(𝛽^∗2−𝛼²)𝛿₄(𝑟₂,𝛽^∗𝑓),₁₁=𝐴₁𝑎₁(𝑟²₁+𝛼²)𝑔₆(𝛼,𝑟₁)+𝐴₂𝑎₂(𝑟²₂+𝛼²)𝑔₆(𝛼,𝑟₂)+𝐴₄(𝛽²+𝛼²)𝑔₆𝑓(𝛼,𝛽),₁₂=𝐴₂𝑎₂(𝑟²₂+𝛼²)𝑔₆(𝑟₁,𝑟₂)+𝐴₄(𝛽²+𝛼²)𝑔₆(𝑟₁𝑓,𝛽),₁₃=𝐴₄(𝛽²+𝛼²)𝑔₆(𝑟₂𝑓,𝛽),₁₄=𝑟₁𝐴₁𝑎₁𝑔₇(𝛼,𝑟₁)+𝑟₂𝐴₂𝑎₂𝑔₇(𝛼,𝑟₂)+𝛽𝐴₄𝑔₇𝑓(𝛼,𝛽),₁₅=𝑟₂𝐴₂𝑎₂𝑔₇(𝑟₁,𝑟₂)+𝛽𝐴₄𝑔₇(𝑟₁𝑓,𝛽),₁₆=𝛽𝐴₄𝑔₇(𝑟₂𝑓,𝛽),₁₇=(2+(𝑟²₁−𝛼²)𝑎₁+(𝑟²₂−𝛼²)𝑎₂)𝐴₂𝑔₆(𝑟₁,𝑟₂)+(𝛽²−𝛼²)𝐴₄𝑔₆(𝑟₁𝑓,𝛽),₁₈=𝛼𝐴₃𝑔₁₀(𝛾₂,𝛼)+𝛽𝐴₄𝑔₁₀(𝛾₂,𝛽)+𝑟₁𝐴₁𝑎₁𝑔₁₀(𝛾₂,𝑟₁)+𝑟₂𝐴₂𝑎₂𝑔₁₀(𝛾₂,𝑟₂)−𝛾₂(𝐴₃𝑔₁₁(𝛾₂,𝛼)+𝐴₄𝑔₁₁(𝛾₂,𝛽)+𝐴₁𝑎₁𝑔₁₁(𝛾₂,𝑟₁)+𝐴₂𝑎₂𝑔₁₁(𝛾₂,𝑟₂𝑓)),₁₉=𝛼𝐴₃𝑔₁₀(𝛾₁,𝛼)+𝛽𝐴₄𝑔₁₀(𝛾₁,𝛽)+𝑟₁𝐴₁𝑎₁𝑔₁₀(𝛾₁,𝑟₁)+𝑟₂𝐴₂𝑎₂𝑔₁₀(𝛾₁,𝑟₂)−𝛾₁(𝐴₃𝑔₁₁(𝛾₁,𝛼)+𝐴₄𝑔₁₁(𝛾₁,𝛽)+𝐴₁𝑎₁𝑔₁₁(𝛾₁,𝑟₁)+𝐴₂𝑎₂𝑔₁₁(𝛾₁,𝑟₂𝑓)),₂₀=𝛼𝐴₃𝑔₁₂(𝛾₁,𝛼)+𝛽𝐴₄𝑔₁₂(𝛾₁,𝛽)+𝑟₁𝐴₁𝑎₁𝑔₁₂(𝛾₁,𝑟₁)+𝑟₂𝐴₂𝑎₂𝑔₁₂(𝛾₁,𝑟₂)−𝛾₁(𝐴₃𝑔₁₂(𝛾₁,𝛼)+𝐴₄𝑔₁₂(𝛾₁,𝛽)+𝐴₁𝑎₁𝑔₁₂(𝛾₁,𝑟₁)+𝐴₂𝑎₂𝑔₁₂(𝛾₁,𝑟₂𝑓)),₂₁=(𝑟²₁+𝛼²)𝐴₁𝑎₁𝑔₁₆(𝛼,𝑟₁)+(𝑟²₂+𝛼²)𝐴₂𝑎₂𝑔₁₆(𝛼,𝑟₂)+(𝛽²+𝛼²)𝐴₄𝑔₁₆𝑓(𝛼,𝛽),₂₂=(𝑟²₂+𝛼²)𝐴₂𝑎₂𝑔₁₆(𝑟₁,𝑟₂)+(𝛽²+𝛼²)𝐴₄𝑔₁₆(𝑟₁𝑓,𝛽),₂₃=(𝛽²+𝛼²)𝐴₄𝑔₁₆(𝑟₂𝑓,𝛽),₂₄=𝑟₁𝐴₁𝑎₁𝑔₁₇(𝛼,𝑟₁)+𝑟₂𝐴₂𝑎₂𝑔₁₇(𝛼,𝑟₂)+𝛽𝐴₄𝑔₁₇𝑓(𝛼,𝛽),₂₅=𝑟₂𝐴₂𝑎₂𝑔₁₇(𝑟₁,𝑟₂)+𝛽𝐴₄𝑔₁₇(𝑟₁𝐴,𝛽),₆=14(𝛾₁(1−cosh2𝛾₁)sinh2𝛾₁+𝛾^∗₁(1−cosh2𝛾^∗₁)sinh2𝛾^∗₁)+𝑖𝛼8Prsinh2𝛾₁[−𝑅𝑒(𝐴^∗₃𝛿₂(𝛾₁,𝛼)+𝐴^∗₄𝛿₂(𝛾₁,𝛽^∗))+𝐴^∗₁(af−𝑅𝑒𝑎^∗₁)𝛿₂(𝛾₁,𝑟^∗₁)+𝐴^∗₂(af−𝑅𝑒𝑎^∗₂)𝛿₂(𝛾₁,𝑟^∗₂)]+𝑖𝛼8Prsinh2𝛾^∗₁×[𝑅𝑒(𝐴₃𝛿₂(𝛾^∗₁,𝛼)+𝐴₄𝛿₂(𝛾^∗₁,𝛽))+𝐴₁(𝑎₁𝑅𝑒−af)𝛿₂(𝛾^∗₁,𝑟₁)𝐴^∗₂(𝑎₂𝑅𝑒−af)𝛿₂(𝛾^∗₁,𝑟₂)]+Ec⋅Pr[2(1+𝛾⁻¹)+𝜇₁4×(2𝛼²𝐴₃𝐴^∗₃cosh2𝛼+4𝛼²(𝛼𝐴₃𝜆₃+𝑟₁𝐴₁𝑎₁𝜆₄+𝑟₂𝐴₂𝜆₅+𝛽𝐴₄𝜆₆)+2𝛼²𝐴₃𝜆₇+(𝛼²+𝑟²₁)𝐴₁𝑎₁𝜆₈+(𝛼²+𝑟²₂)𝐴₂𝑎₂𝜆₉+(𝛼²+𝛽²)𝐴₄𝜆₁₀2)+𝑀²×(𝑟₁𝐴₁(𝑟^∗₁𝐴^∗₁𝛿₃(𝑟₁,𝑟^∗₁)+𝑟^∗₂𝐴^∗₂𝛿₃(𝑟₁,𝑟^∗₂))+𝑟₂𝐴₂(𝑟^∗₁𝐴^∗₁𝛿₃(𝑟₂,𝑟^∗₁)+𝑟^∗₂𝐴^∗₂𝛿₃(𝑟₂,𝑟^∗₂)))+(𝜇₁+2𝛼²𝑀²)×(𝐴₁(𝐴^∗₁𝛿₄(𝑟₁,𝑟^∗₁)+𝐴^∗₂𝛿₄(𝑟₁,𝑟^∗₂))+𝐴₂(𝐴^∗₁𝛿₄(𝑟₂,𝑟^∗₁)+𝐴^∗₂𝛿₄(𝑟₂,𝑟^∗₂+𝜇)))₁2(𝐴₁𝜆₁₁+𝐴₂𝜆₁₂+𝐴₄(𝛽²−𝛼²)(𝐴^∗₁𝛿₄(𝛽,𝑟^∗₁)+𝐴^∗₂𝛿₄(𝛽,𝑟^∗₂𝐴)))],₇1=−4{(1−Sc⋅Sr⋅Pr(Sc−Pr))[𝛾₂(1+cosh2𝛾₂sinh2𝛾₂)+𝛾^∗₂(1+cosh2𝛾^∗₂sinh2𝛾^∗₂)]+(Pr(Sc−Pr)+1)Sc⋅Sr[𝛾₁(1+cosh2𝛾₁sinh2𝛾₁)+𝛾^∗₁(1+cosh2𝛾^∗₁sinh2𝛾^∗₁−)]}𝑖𝛼8Sc⋅𝑅𝑒{(1−Sc⋅Sr⋅Pr)1(Sc−Pr)×[sinh2𝛾₂(𝐴^∗₃𝛿₅(𝛾₂,𝛼)+𝐴^∗₄𝛿₅(𝛾₂,𝛽^∗)+𝐴^∗₁𝑎^∗₁𝛿₅(𝛾₂,𝑟^∗₁)+𝐴^∗₂𝑎^∗₂𝛿₅(𝛾₂,𝑟^∗₂−1))sinh2𝛾^∗₂(𝐴₃𝛿₅(𝛾^∗₂,𝛼)+𝐴₄𝛿₅(𝛾^∗₂,𝛽)+𝐴₁𝑎₁𝛿₅(𝛾^∗₂,𝑟₁)+𝐴₂𝑎₂𝛿₅(𝛾^∗₂,𝑟₂+)]Sc⋅Sr⋅Pr[1(Sc−Pr)sinh2𝛾₁(𝐴^∗₃𝛿₅(𝛾₁,𝛼)+𝐴^∗₄𝛿₅(𝛾₁,𝛽^∗)+𝐴^∗₁𝑎^∗₁𝛿₅(𝛾₁,𝑟^∗₁)+𝐴^∗₂𝑎^∗₂𝛿₅(𝛾₁,𝑟^∗₂1))−sinh2𝛾^∗₁×(𝐴₃𝛿₅(𝛾^∗₁,𝛼)+𝐴₅𝛿₅(𝛾^∗₁,𝛽)+𝐴₁𝑎₁𝛿₅(𝛾^∗₁,𝑟₁)+𝐴₂𝑎₂𝛿₅(𝛾^∗₁,𝑟₂𝐴))]},₈=14{(1−Sc⋅Sr⋅Pr(Sc−Pr))[𝛾₂(1−cosh2𝛾₂sinh2𝛾₂)+𝛾^∗₂(1−cosh2𝛾^∗₂sinh2𝛾^∗₂)]+(Pr(Sc−Pr)+1)Sc⋅Sr[𝛾₁(1−cosh2𝛾₁sinh2𝛾₁)+𝛾^∗₁(1−cosh2𝛾^∗₁sinh2𝛾^∗₁−)]}𝑖𝛼8Sc⋅𝑅𝑒{(1−Sc⋅Sr⋅Pr)1(Sc−Pr)×[sinh2𝛾₂(𝐴^∗₃𝛿₂(𝛾₂,𝛼)+𝐴^∗₄𝛿₂(𝛾₂,𝛽^∗)+𝐴^∗₁𝑎^∗₁𝛿₂(𝛾₂,𝑟^∗₁)+𝐴^∗₂𝑎^∗₂𝛿₂(𝛾₂,𝑟^∗₂1))−sinh2𝛾^∗₂×(𝐴₃𝛿₂(𝛾^∗₂,𝛼)+𝐴₄𝛿₂(𝛾^∗₂,𝛽)+𝐴₁𝑎₁𝛿₂(𝛾^∗₂,𝑟₁)+𝐴₂𝑎₂𝛿₂(𝛾^∗₂,𝑟₂+))]Sc⋅Sr⋅Pr[1(Sc−Pr)sinh2𝛾₁(𝐴^∗₃𝛿₂(𝛾₁,𝛼)+𝐴^∗₄𝛿₂(𝛾₁,𝛽^∗)+𝐴^∗₁𝑎^∗₁𝛿₂(𝛾₁,𝑟^∗₁)+𝐴^∗₂𝑎^∗₂𝛿₂(𝛾₁,𝑟^∗₂−1))sinh2𝛾^∗₁(𝐴₃𝛿₂(𝛾^∗₁,𝛼)+𝐴₄𝛿₂(𝛾^∗₁,𝛽)+𝐴₁𝑎₁𝛿₂(𝛾^∗₁,𝑟₁)+𝐴₂𝑎₂𝛿₂(𝛾^∗₁,𝑟₂𝑔))]},₄(𝑎,𝑏)=sinh(𝑎+(𝑎+𝑏)𝑦)(𝑎+𝑏)²−𝛾²₃+sinh(𝑎+(𝑎−𝑏)𝑦)(𝑎−𝑏)²−𝛾²₃,𝑔₅(𝑎,𝑏)=sinh(𝑎+(𝑎+𝑏)𝑦)(𝑎+𝑏)²−𝛾²₃−sinh(𝑎+(𝑎−𝑏)𝑦)(𝑎−𝑏)²−𝛾²₃,𝑔₆(𝑎,𝑏)=cosh((𝑎+𝑏)𝑦)(𝑎+𝑏)²−𝛾²₃−cosh((𝑎−𝑏)𝑦)(𝑎−𝑏)²−𝛾²₃,𝑔₇(𝑎,𝑏)=cosh((𝑎+𝑏)𝑦)(𝑎+𝑏)²−𝛾²₃+cosh((𝑎−𝑏)𝑦)(𝑎−𝑏)²−𝛾²₃,𝑔₈(𝑎)=cosh(2𝑎𝑦)4𝑎²−𝛾²₃+1𝛾²₃,𝑔₉(𝑎)=cosh(2𝑎𝑦)4𝑎²−𝛾²₃−1𝛾²₃,𝑔₁₀(𝑎,𝑏)=sinh(𝑎+(𝑎+𝑏)𝑦)(𝑎+𝑏)²−𝛾²₄+sinh(𝑎+(𝑎−𝑏)𝑦)(𝑎+𝑏)²−𝛾²₄,𝑔₁₁(𝑎,𝑏)=sinh(𝑎+(𝑎+𝑏)𝑦)(𝑎+𝑏)²−𝛾²₄−sinh(𝑎+(𝑎−𝑏)𝑦)(𝑎+𝑏)²−𝛾²₄,𝑔₁₂(𝑎,𝑏)=(𝑎+𝑏)²−4𝛼²(𝑎+𝑏)²−𝛾²₄sinh(𝑎+(𝑎+𝑏)𝑦)(𝑎+𝑏)²−𝛾²₃+(𝑎−𝑏)²−4𝛼²(𝑎−𝑏)²−𝛾²₄sinh(𝑎+(𝑎−𝑏)𝑦)(𝑎−𝑏)²−𝛾²₃,𝑔₁₃(𝑎,𝑏)=(𝑎+𝑏)²−4𝛼²(𝑎+𝑏)²−𝛾²₄sinh(𝑎+(𝑎+𝑏)𝑦)(𝑎+𝑏)²−𝛾²₃−(𝑎−𝑏)²−4𝛼²(𝑎−𝑏)²−𝛾²₄sinh(𝑎+(𝑎−𝑏)𝑦)(𝑎−𝑏)²−𝛾²₃,𝑔₁₄(𝑎)=4(𝑎²−𝛼²)(4𝑎²−𝛾²₄)cosh(2𝑎𝑦)(4𝑎²−𝛾²₃)+4𝛼²(𝛾₃𝛾₄)²,𝑔₁₅(𝑎)=4(𝑎²−𝛼²)(4𝑎²−𝛾²₄)cosh(2𝑎𝑦)(4𝑎²−𝛾²₃)−4𝛼²(𝛾₃𝛾₄)²,𝑔₁₆(𝑎,𝑏)=(𝑎+𝑏)²−4𝛼²(𝑎+𝑏)²−𝛾²₄cosh(𝑎+(𝑎+𝑏)𝑦)(𝑎+𝑏)²−𝛾²₃−(𝑎−𝑏)²−4𝛼²(𝑎−𝑏)²−𝛾²₄cosh((𝑎−𝑏)𝑦)(𝑎−𝑏)²−𝛾²₃,𝑔₁₇(𝑎,𝑏)=(𝑎+𝑏)²−4𝛼²(𝑎+𝑏)²−𝛾²₄cosh(𝑎+(𝑎+𝑏)𝑦)(𝑎+𝑏)²−𝛾²₃+(𝑎−𝑏)²−4𝛼²(𝑎−𝑏)²−𝛾²₄cosh((𝑎−𝑏)𝑦)(𝑎−𝑏)²−𝛾²₃,𝑓₂₆=𝛽𝐴₄𝑔₁₇(𝑟₂𝑓,𝛽),₂₇=(2+(𝑟²₁−𝛼²)𝑎₁+(𝑟²₂−𝛼²)𝑎₂)𝐴₂𝑔₁₆(𝑟₁,𝑟₂)+𝐴₄(𝛽²−𝛼²)𝑔₁₆(𝑟₁𝑓,𝛽),₂₈=𝑟₁𝐴₁𝑔₁₂(𝛾₁,𝑟₁)+𝑟₂𝐴₂𝑔₁₂(𝛾₁,𝑟₂)−𝛾₁𝐴₁𝑔₁₃(𝛾₁,𝑟₁)−𝛾₁𝐴₂𝑔₁₃(𝛾₁,𝑟₂𝛿),₆(𝑎,𝑏)=sinh(2𝑎+𝑏)−sinh𝑏(𝑎+𝑏)²−𝛾²₃+sinh(2𝑎−𝑏)−sinh𝑏(𝑎−𝑏)²−𝛾²₃,𝛿₇(𝑎,𝑏)=sinh(2𝑎+𝑏)−sinh𝑏(𝑎+𝑏)²−𝛾²₃−sinh(2𝑎−𝑏)+sinh𝑏(𝑎−𝑏)²−𝛾²₃,𝛿₈(𝑎,𝑏)=sinh(2𝑎+𝑏)+sinh𝑏(𝑎+𝑏)²−𝛾²₃+sinh(2𝑎−𝑏)−sinh𝑏(𝑎−𝑏)²−𝛾²₃,𝛿₉(𝑎,𝑏)=sinh(2𝑎+𝑏)+sinh𝑏(𝑎+𝑏)²−𝛾²₃−sinh(2𝑎−𝑏)−sinh𝑏(𝑎−𝑏)²−𝛾²₃,𝛿₁₀(𝑎)=cosh(2𝑎)4𝑎²−𝛾²₃+1𝛾²₃,𝛿₁₁(𝑎)=cosh(2𝑎)4𝑎²−𝛾²₃−1𝛾²₃,𝛿₁₂(𝑎,𝑏)=cosh(𝑎+𝑏)(𝑎+𝑏)²−𝛾²₃+cosh(𝑎−𝑏)(𝑎−𝑏)²−𝛾²₃,𝛿₁₃(𝑎,𝑏)=cosh(𝑎+𝑏)(𝑎+𝑏)²−𝛾²₃−cosh(𝑎−𝑏)(𝑎−𝑏)²−𝛾²₃,𝛿₁₄(𝑎,𝑏)=sinh(2𝑎+𝑏)−sinh𝑏(𝑎+𝑏)²−𝛾²₄+sinh(2𝑎−𝑏)+sinh𝑏(𝑎−𝑏)²−𝛾²₄,𝛿₁₅(𝑎,𝑏)=sinh(2𝑎+𝑏)−sinh𝑏(𝑎+𝑏)²−𝛾²₄−sinh(2𝑎−𝑏)+sinh𝑏(𝑎−𝑏)²−𝛾²₄,𝛿₁₆(𝑎,𝑏)=sinh(2𝑎+𝑏)+sinh𝑏(𝑎+𝑏)²−𝛾²₄+sinh(2𝑎−𝑏)−sinh𝑏(𝑎−𝑏)²−𝛾²₄,𝛿₁₇(𝑎,𝑏)=sinh(2𝑎+𝑏)+sinh𝑏(𝑎+𝑏)²−𝛾²₄−sinh(2𝑎−𝑏)−sinh𝑏(𝑎−𝑏)²−𝛾²₄,𝛿₁₈(𝑎,𝑏)=(𝑎+𝑏)²−4𝛼²(𝑎+𝑏)²−𝛾²₄sinh(2𝑎+𝑏)−sinh𝑏(𝑎+𝑏)²−𝛾²₃+(𝑎−𝑏)²−4𝛼²(𝑎−𝑏)²−𝛾²₄sinh(2𝑎−𝑏)+sinh𝑏(𝑎−𝑏)²−𝛾²₃,𝛿₁₉(𝑎,𝑏)=(𝑎+𝑏)²−4𝛼²(𝑎+𝑏)²−𝛾²₄sinh(2𝑎+𝑏)−sinh𝑏(𝑎+𝑏)²−𝛾²₃−(𝑎−𝑏)²−4𝛼²(𝑎−𝑏)²−𝛾²₄sinh(2𝑎−𝑏)+sinh𝑏(𝑎−𝑏)²−𝛾²₃,𝛿₂₀(𝑎,𝑏)=(𝑎+𝑏)²−4𝛼²(𝑎+𝑏)²−𝛾²₄sinh(2𝑎+𝑏)+sinh𝑏(𝑎+𝑏)²−𝛾²₃+(𝑎−𝑏)²−4𝛼²(𝑎−𝑏)²−𝛾²₄sinh(2𝑎−𝑏)−sinh𝑏(𝑎−𝑏)²−𝛾²₃,𝛿₂₁(𝑎,𝑏)=(𝑎+𝑏)²−4𝛼²(𝑎+𝑏)²−𝛾²₄sinh(2𝑎+𝑏)−sinh𝑏(𝑎+𝑏)²−𝛾²₃−(𝑎−𝑏)²−4𝛼²(𝑎−𝑏)²−𝛾²₄sinh(2𝑎−𝑏)+sinh𝑏(𝑎−𝑏)²−𝛾²₃,𝛿₂₂(𝑎)=4(𝑎²−𝛼²)4𝑎²−𝛾²₄cosh(2𝑎)4𝑎²−𝛾²₃+4𝛼²(𝛾₃𝛾₄)²,𝛿₂₃(𝑎)=4(𝑎²−𝛼²)4𝑎²−𝛾²₄cosh(2𝑎)4𝑎²−𝛾²₃−4𝛼²(𝛾₃𝛾₄)²,𝛿₂₄(𝑎,𝑏)=(𝑎+𝑏)²−4𝛼²(𝑎+𝑏)²−𝛾²₄cosh(𝑎+𝑏)(𝑎+𝑏)²−𝛾²₃+(𝑎+𝑏)²−4𝛼²(𝑎−𝑏)²−𝛾²₄cosh(𝑎−𝑏)(𝑎−𝑏)²−𝛾²₃,𝛿₂₅(𝑎,𝑏)=(𝑎+𝑏)²−4𝛼²(𝑎+𝑏)²−𝛾²₄cosh(𝑎+𝑏)(𝑎+𝑏)²−𝛾²₃−(𝑎+𝑏)²−4𝛼²(𝑎−𝑏)²−𝛾²₄cosh(𝑎−𝑏)(𝑎−𝑏)²−𝛾²₃,𝜆₁₃=𝐴₁𝑎₁(𝑟²₁+𝛼²)𝛿₁₃(𝛼,𝑟₁)+𝐴₂𝑎₂(𝑟²₂+𝛼²)𝛿₁₃(𝛼,𝑟₂)+𝐴₄(𝛽²+𝛼²)𝛿₁₃𝜆(𝛼,𝛽),₁₄=𝐴₂𝑎₂(𝑟²₂+𝛼²)𝛿₁₃(𝑟₁,𝑟₂)+𝐴₄(𝛽²+𝛼²)𝛿₁₃(𝑟₁𝜆,𝛽),₁₅=𝐴₄(𝛽²+𝛼²)𝛿₁₃(𝑟₂𝜆,𝛽),₁₆=𝑟₁𝐴₁𝑎₁𝛿₁₂(𝛼,𝑟₁)+𝑟₂𝐴₂𝑎₂𝛿₁₂(𝛼,𝑟₂)+𝛽𝐴₄𝛿₁₂𝜆(𝛼,𝛽),₁₇=𝑟₂𝐴₂𝑎₂𝛿₁₂(𝑟₁,𝑟₂)+𝛽𝐴₄𝛿₁₂(𝑟₁𝜆,𝛽),₁₈=𝛽𝐴₄𝛿₁₂(𝑟₂𝜆,𝛽),₁₉=(2+(𝑟²₁−𝛼²)𝑎₁+(𝑟²₂−𝛼²)𝑎₂)𝐴₂𝛿₁₃(𝑟₁,𝑟₂)+(𝛽²−𝛼²)𝐴₄𝛿₁₃(𝑟₁𝜆,𝛽),₂₀=𝛼𝐴₃𝛿₁₄(𝛾₂,𝛼)+𝛽𝐴₄𝛿₁₄(𝛾₂,𝛽)+𝑟₁𝐴₁𝑎₁𝛿₁₄(𝛾₂,𝑟₁)+𝑟₂𝐴₂𝑎₂𝛿₁₄(𝛾₂,𝑟₂)−𝛾₂(𝐴₃𝛿₁₅(𝛾₂,𝛼)+𝐴₄𝛿₁₄(𝛾₂,𝛽)+𝐴₁𝑎₁𝛿₁₅(𝛾₂,𝑟₁)+𝐴₂𝑎₂𝛿₁₅(𝛾₂,𝑟₂𝜆)),₂₁=𝛼𝐴₃𝛿₁₄(𝛾₁,𝛼)+𝛽𝐴₄𝛿₁₄(𝛾₁,𝛽)+𝑟₁𝐴₁𝑎₁𝛿₁₄(𝛾₁,𝑟₁)+𝑟₂𝐴₂𝑎₂𝛿₁₄(𝛾₁,𝑟₂)−𝛾₁(𝐴₃𝛿₁₅(𝛾₁,𝛼)+𝐴₄𝛿₁₄(𝛾₁,𝛽)+𝐴₁𝑎₁𝛿₁₅(𝛾₁,𝑟₁)+𝐴₂𝑎₂𝛿₁₅(𝛾₁,𝑟₂𝜆)),₂₂=𝛼𝐴₃𝛿₁₈(𝛾₁,𝛼)+𝛽𝐴₄𝛿₁₈(𝛾₁,𝛽)+𝑟₁𝐴₁𝑎₁𝛿₁₈(𝛾₁,𝑟₁)+𝑟₂𝐴₂𝑎₂𝛿₁₈(𝛾₁,𝑟₂)−𝛾₁(𝐴₃𝛿₁₉(𝛾₁,𝛼)+𝐴₄𝛿₁₉(𝛾₁,𝛽)+𝐴₁𝑎₁𝛿₁₉(𝛾₁,𝑟₁)+𝐴₂𝑎₂𝛿₁₉(𝛾₁,𝑟₂𝜆)),₂₃=(𝑟²₁+𝛼²)𝐴₁𝑎₁𝛿₂₅(𝛼,𝑟₁)+(𝑟²₂+𝛼²)𝐴₂𝑎₂𝛿₂₅(𝛼,𝑟₂)+(𝛽²+𝛼²)𝐴₄𝛿₂₅𝜆(𝛼,𝛽),₂₄=(𝑟²₂+𝛼²)𝐴₂𝑎₂𝛿₂₅(𝑟₁,𝑟₂)+(𝛽²+𝛼²)𝐴₄𝛿₂₅(𝑟₁𝜆,𝛽),₂₅=(𝛽²+𝛼²)𝐴₄𝛿₂₅(𝑟₂𝜆,𝛽),₂₆=𝑟₁𝐴₁𝑎₁𝛿₂₄(𝛼,𝑟₁)+𝑟₂𝐴₂𝑎₂𝛿₂₄(𝛼,𝑟₂)+𝛽𝐴₄𝛿₂₄𝜆(𝛼,𝛽),₂₇=𝑟₂𝐴₂𝑎₂𝛿₂₄(𝑟₁,𝑟₂)+𝛽𝐴₄𝛿₂₄(𝑟₁𝜆,𝛽),₂₈=𝛽𝐴₄𝛿₂₄(𝑟₂𝜆,𝛽),₂₉=(2+(𝑟²₁−𝛼²)𝑎₁+(𝑟²₂−𝛼²)𝑎₂)𝐴₂𝛿₂₅(𝑟₁,𝑟₂)+𝐴₄(𝛽²−𝛼²)𝛿₂₅(𝑟₁𝜆,𝛽),₃₀=𝑟₁𝐴₁𝛿₁₈(𝛾₁,𝑟₁)+𝑟₂𝐴₂𝛿₁₈(𝛾₁,𝑟₂)−𝛾₁𝐴₁𝛿₁₉(𝛾₁,𝑟₁)−𝛾₁𝐴₂𝛿₁₉(𝛾₁,𝑟₂𝜆),₃₁=𝛼𝐴₃𝛿₁₆(𝛾₂,𝛼)+𝛽𝐴₄𝛿₁₆(𝛾₂,𝛽)+𝑟₁𝐴₁𝑎₁𝛿₁₆(𝛾₂,𝑟₁)+𝑟₂𝐴₂𝑎₂𝛿₁₆(𝛾₂,𝑟₂)−𝛾₂(𝐴₃𝛿₁₇(𝛾₂,𝛼)+𝐴₄𝛿₁₇(𝛾₂,𝛽)+𝐴₁𝑎₁𝛿₁₇(𝛾₂,𝑟₁)+𝐴₂𝑎₂𝛿₁₇(𝛾₂,𝑟₂𝜆)),₃₂=𝛼𝐴₃𝛿₁₆(𝛾₁,𝛼)+𝛽𝐴₄𝛿₁₆(𝛾₁,𝛽)+𝑟₁𝐴₁𝑎₁𝛿₁₆(𝛾₁,𝑟₁)+𝑟₂𝐴₂𝑎₂𝛿₁₆(𝛾₁,𝑟₂)−𝛾₁(𝐴₃𝛿₁₇(𝛾₁,𝛼)+𝐴₄𝛿₁₇(𝛾₁,𝛽)+𝐴₁𝑎₁𝛿₁₇(𝛾₁,𝑟₁)+𝐴₂𝑎₂𝛿₁₇(𝛾₁,𝑟₂𝜆)),₃₃=𝛼𝐴₃𝛿₂₀(𝛾₁,𝛼)+𝛽𝐴₄𝛿₂₀(𝛾₁,𝛽)+𝑟₁𝐴₁𝑎₁𝛿₂₀(𝛾₁,𝑟₁)+𝑟₂𝐴₂𝑎₂𝛿₂₀(𝛾₁,𝑟₂)−𝛾₁(𝐴₃𝛿₂₁(𝛾₁,𝛼)+𝐴₄𝛿₂₁(𝛾₁,𝛽)+𝐴₁𝑎₁𝛿₂₁(𝛾₁,𝑟₁)+𝐴₂𝑎₂𝛿₂₁(𝛾₁,𝑟₂𝜆)),₃₄=𝑟₁𝐴₁𝛿₂₀(𝛾₁,𝑟₁)+𝑟₂𝐴₂𝛿₂₁(𝛾₁,𝑟₂)−𝛾₁𝐴₁𝛿₂₁(𝛾₁,𝑟₁)−𝛾₁𝐴₂𝛿₂₁(𝛾₁,𝑟₂𝐴),₉=12cosh𝛾₃1×{2𝛾₁(1−cosh2𝛾₁sinh2𝛾₁)−𝑖𝛼4Prsinh2𝛾₁×[𝑅𝑒(𝛼𝐴₃𝛿₆(𝛾₁,𝛼)+𝛽𝐴₄𝛿₆(𝛾₁,𝛽)−𝛾₁𝐴₃𝛿₇(𝛾₁,𝛼)−𝛾₁𝐴₄𝛿₇(𝛾₁,𝛽))+(𝑎₁𝑅𝑒−af)×(𝑟₁𝐴₁𝛿₆(𝛾₁,𝑟₁)−𝛾₁𝐴₁𝛿₇(𝛾₁,𝑟₁))+(𝑎₂𝑅𝑒−af)(𝑟₂𝐴₂𝛿₆(𝛾₁,𝑟₂)−𝛾₁𝐴₂𝛿₇(𝛾₁,𝑟₂))]+Pr⋅Ec[2(1+𝛾⁻¹)+𝜇₁4×(4𝛼⁴𝐴²₃𝛿₁₀(𝛼)+(𝑟²₁+𝛼²)²𝐴²₁𝑎²₁𝛿₁₀(𝑟₁)+(𝑟²₂+𝛼²)²𝐴²₂𝑎²₂𝛿₁₀(𝑟₂)+(𝛽²+𝛼²)²𝐴²₄𝛿₁₀(𝛽)+4𝛼²𝐴₃𝜆₁₃+2𝐴₁𝑎₁(𝛼²+𝑟²₁)𝜆₁₄+2𝐴₂𝑎₂(𝛼²+𝑟²₂)𝜆₁₅−4𝛼²(𝛼²𝐴²₃𝛿₁₁(𝛼)+𝑟²₁𝐴²₁𝑎²₁𝛿₁₁(𝑟₁)+𝑟²₂𝐴²₂𝑎²₂𝛿₁₁(𝑟₂)+𝛽²𝐴²₄𝛿₁₁(𝛽)+2𝛼𝐴₃𝜆₁₆+2𝑟₁𝐴₁𝑎₁𝜆₁₇+2𝑟₂𝐴₂𝑎₂𝜆₁₈))+𝜇₁((1+(𝑟²₁−𝛼²)𝑎₁)𝐴²₁𝛿₁₀(𝑟₁)+(1+(𝑟²₂−𝛼²)𝑎₂)𝐴²₂𝛿₁₀(𝑟₂)+𝐴₁𝜆₁₉+𝐴₂𝐴₄(𝛽²−𝛼²)𝛿₁₃(𝑟₂+2,𝛽))𝑀²(𝑟²₁𝐴²₁𝛿₁₁(𝑟₁)+𝑟²₂𝐴²₂𝛿₁₁(𝑟₂)+2𝑟₁𝑟₂𝐴₁𝐴₂𝛿₁₂(𝑟₁,𝑟₂)−𝛼²(𝐴²₁𝛿₁₀(𝑟₁)+𝐴²₂𝛿₁₀(𝑟₂)+2𝐴₁𝐴₂𝛿₁₃(𝑟₁,𝑟₂𝐴))]},₁₀1=−2cosh𝛾₃{12𝛾₁(1+cosh2𝛾₁sinh2𝛾₁)+𝑖𝛼4Prsinh2𝛾₁×[𝑅𝑒(𝛼𝐴₃𝛿₈(𝛾₁,𝛼)+𝛽𝐴₄𝛿₈(𝛾₁,𝛽)−𝛾₁𝐴₃𝛿₉(𝛾₁,𝛼)−𝛾₁𝐴₄𝛿₉(𝛾₁,𝛽))+(𝑎₁𝑅𝑒−af)(𝑟₁𝐴₁𝛿₈(𝛾₁,𝑟₁)−𝛾₁𝐴₁𝛿₉(𝛾₁,𝑟₁))+(𝑎₂𝑅𝑒−𝑎𝑓)(𝑟₂𝐴₂𝛿₈(𝛾₁,𝑟₂)−𝛾₁𝐴₂𝛿₉(𝛾₁,𝑟₂𝐴))]},₁₁=12cosh𝛾₄{12[(1−Sc⋅Sr⋅Pr(Sc−Pr))𝛾₂(1−cosh2𝛾₂sinh2𝛾₂𝛾)+¹⋅Sc⋅Sr⋅Pr(Sc−Pr1−cosh2𝛾₁sinh2𝛾₁−)]𝑖𝛼4Sc⋅𝑅𝑒{(1−Sc⋅Sr⋅Pr)𝜆(Sc−Pr)sinh²⁰2𝛾₂+Sc⋅Sr⋅Pr𝜆(Sc−Pr)sinh²¹2𝛾₁}−Pr⋅Sc⋅SrSc−Pr2𝐴₉cosh𝛾₃+𝑖𝛼4𝜆Sc⋅Sr⋅Pr⋅𝑅𝑒²²sinh2𝛾₁−Pr⋅Ec⋅Sc⋅Sr[(1+𝛾⁻¹)+𝜇₁2×(32𝛼⁶𝐴²₃(𝛾₃𝛾₄)²+(𝑟²₁+𝛼²)²𝐴²₁𝑎²₁𝛿₂₂(𝑟₁)+(𝑟²₂+𝛼²)²𝐴²₂𝑎²₂𝛿₂₂(𝑟₂)+(𝛽²+𝛼²)²𝐴²₄𝛿₂₂(𝛽)+4𝛼²𝐴₃𝜆₂₃+2(𝛼²+𝑟²₁)𝐴₁𝑎₁𝜆₂₄+2(𝛼²+𝑟²₂)𝐴₂𝑎₂𝜆₂₅−4𝛼²(𝑟²₁𝐴²₁𝑎²₁𝛿₂₃(𝑟₁)+𝑟²₂𝐴²₂𝑎²₂𝛿₂₃(𝑟₂)+𝛽²𝐴²₄𝛿₂₃(𝛽)+2𝛼𝐴₃𝜆₂₆+2𝑟₁𝐴₁𝑎₁𝜆₂₇+2𝑟₂𝐴₂𝑎₂𝜆₂₈))+𝜇₁((1+(𝑟²₁−𝛼²)𝑎₁)𝐴²₁𝛿₂₂(𝑟₁)+(1+(𝑟²₂−𝛼²)𝑎₂)𝐴²₂𝛿₂₂(𝑟₂)+𝐴₁𝜆₂₉+𝐴₂𝐴₄(𝛽²−𝛼²)𝛿₂₅(𝑟₂+2,𝛽))𝑀²(𝑟²₁𝐴²₁𝛿₂₃(𝑟₁)+𝑟²₂𝐴²₂𝛿₂₃(𝑟₂)+2𝑟₁𝑟₂𝐴₁𝐴₂𝛿₂₄(𝑟₁,𝑟₂)−𝛼²(𝐴²₁𝛿₂₂(𝑟₁)+𝐴²₂𝛿₂₂(𝑟₂)+2𝐴₁𝐴₂𝛿₂₅(𝑟₁,𝑟₂−)))𝑖𝛼4𝜆af⋅Pr⋅Sc⋅Sr⋅³⁰sinh2𝛾₁𝐴]},₁₂=−12cosh𝛾₄{12[(1−Sc⋅Sr⋅Pr(Sc−Pr))𝛾₂(1+cosh2𝛾₂sinh2𝛾₂𝛾)+¹⋅Sc⋅Sr⋅Pr(Sc−Pr1+cosh2𝛾₁sinh2𝛾₁+)]𝑖𝛼4Sc⋅𝑅𝑒[(1−Sc⋅Sr⋅Pr)𝜆(Sc−Pr)sinh³¹2𝛾₂+Sc⋅Sr⋅Pr𝜆(Sc−Pr)sinh³²2𝛾₁]+Pr⋅Sc⋅SrSc−Pr2𝐴₁₀sinh𝛾₃−𝑖𝛼4𝜆Sc⋅Sr⋅Pr⋅𝑅𝑒³³sinh2𝛾₁+𝑖𝛼4𝜆af⋅Pr⋅Sc⋅Sr⋅³⁴sinh2𝛾₁}. (A.1)