Ratio Mathematica Volume 45, 2023 Analytical Study of Mixed Convective Flow and Heat Transfer in Vertical Channel Filled with Immiscible Viscous Fluids Daimi Syeda Mariya Begum* Sharad Kumar Jagtap† Abstract In this paper investigation of mixed convective flow and heat transfer in vertical channel filled with immiscible viscous fluids has been carried out. The governing differential equations are solved analytically by regular perturbation method. The impact of governing parameters on velocity and temperature fields namely Grash of number, Brinkman number, perturbation parameter, viscosity ratio, width ratio, conductivity ratio, Nusselt number are investigated and represented graphically. Keywords: mixed convective flow, heat transfer, perturbation method. 2010 AMS subject classification: 76D05, 35Q30‡ * Research student at Department of Mathematics, SRTM University, Maharashtra 431606, India; daimimariya@gmail.com. † HOD Mathematics, Shivaji college, Udgir, Maharashtra, India 413517; sharadvjagtap@gmail.com. ‡ Received on July 10th, 2022. Accepted on October 15th, 2022. Published on January 30, 2023.doi: 10.23755/rm. v45i0.1015. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY license agreement. 194 mailto:daimimariya@gmail.com mailto:sharadvjagtap@gmail.com Mariya Daimi1, S. Jagtap2 1. Introduction Mixed convective flow and heat transfer has importance from researchers due to their diverse application in engineering, automobile sector and various technical fields. This includes geothermal mining, nuclear reactors, heat and cold storage, wielding equipment, aircrafts Desai and Vafai [5]. The study of laminar fully developed mixed convection in a vertical channel with uniform wall temperature was one of the first attempt of Tao [10]. Recently Hamadah and Wirtz [11], Bratetta [3], Aung and Worku [1] assumed symmetric and asymmetric heating of walls of the vertical channel. Prathap kumar et al [7] studied the chemical reaction effects on mixed convection flow in vertical channel with immiscible fluids analytical as well as numerically. Umavati and Chamkha [6] analyzed mixed convection in presence of heat source or heat sink in a vertical channel. Jha and Oni [2] assumed temperature dependent viscosity to study the mixed convection flow in a vertical channel where they found that increase in viscosity parameter increases fluid velocity. With hall and ion-slip effects Srinivasacharya and Shafeeurrahaman [4] analyzed mixed convection flow in a vertical channel filled with nanofluid where they found with the increment in magnetic parameter decrease in temperature, velocity, nanoparticle concentration were occurred. Prathap kumar et al [8] analyzed and found impact of different governing parameter on mixed convective flow in a vertical channel using third kind of boundary condition where channel is filled with porous media using differential transfer method as well as perturbation method. In fluid dynamics regular perturbation method is most preferably used, Rashidi and Ganji [9]. Keeping in view of several applications of mixed convective flow in vertical channel, the aim of this paper is to investigate laminar fully developed mixed convection flow and heat transfer in vertical channel with immiscible viscous fluid analytically to extend the studies available in the literature. The governing differential equations are solved using regular perturbation method valid for small values of perturbation parameter. The thermal buoyancy force, viscous dissipation, viscosity ratio, width ratio, conductivity ratio, Nusselt number are considered to investigate their impact on flow field. Notation: b- ratio of thermal expansion coefficients (β2/β1) Br - Brinkman number (μ1 U0 (1)2/k1 ΔT) g- acceleration due to gravity Gr - Grashof number (gβ1D13ΔT/𝜈12) GR - mixed convective parameter (Gr/Re) D - width ratio (D2/D1) D1, D2 - width of regions k - thermal conductivities (k1/k2) m - ratio of viscosities (μ1/μ2) n - ratio of densities (ρ2/ρ1) p - pressure (assuming p1=p2=p) P- =p+ρ0gX, difference between the pressure and hydrostatic pressure 195 Analytical Study of Mixed Convective Flow and Heat Transfer… Re - Reynolds number (D1U0(1)/𝜈1) U0(i) - reference velocity Greek Symbols α1, α2- Thermal diffusivities β1, β2- Coefficients of thermal expansion ΔT- Difference in temperature (T2 ̶ T1) Ɛ- Dimensionless parameter/perturbation parameter (GR Br) Θ- Dimensionless temperature θ1, θ2- Temperatures μ1, μ2- Viscosities 𝜈1, 𝜈2- Kinematics viscosities ρ1, ρ2 - Densities Subscripts i= 1,2 corresponding to region-I and region-II respectively. 2. Preliminaries Consider a steady two dimensional laminar fully developed mixed convection flow in open ended vertical channel filled with immiscible viscous fluids The X axis is taken upward and parallel to the walls and Y axis is normal on it, shown in Fig 1. We consider fluid to be incompressible, and temperature between the plate and fluid is small, so that the fluid properties taken as constant except the density in the buoyancy term of equation of motion. Fig.1. Physical configuration Region −I gβ1(T1 − T0) − 1 ρ1 ⅆP ⅆX + μ1 ρ1 ⅆ2U1 ⅆY2 = 0 (2.1) α1 ⅆ2T1 ⅆY2 + 𝜈1 Cp ( ⅆU1 ⅆY ) 2 = 0 (2.2) Y = D2 2 Y = − D1 2 𝑔 Y Region-I Region-II 196 Mariya Daimi1, S. Jagtap2 Region −II gβ2(T2 − T0) − 1 ρ2 ⅆP ⅆX + μ2 ρ2 ⅆ2U2 ⅆY2 = 0 (2.3) α2 ⅆ2T2 ⅆY2 + 𝜈2 Cp ( ⅆU2 ⅆY ) 2 = 0 (2.4) P depends only on X ⅆP ⅆY = 0 (2.5) In the presence of viscous dissipation, the energy balance equation can be written as Region −I ⅆ2T1 ⅆY2 = −𝜈1 β1g ⅆ4U1 ⅆY4 (2.6) From Eq. (2.2) and Eq. (2.6) ⅆ4U1 ⅆY4 = β1gρ1 k1 ( ⅆU1 ⅆY ) 2 (2.7) Region −II ⅆ2T2 ⅆY2 = −𝜈2 β2g ⅆ4U2 ⅆY4 (2.8) From Eq. (2.4) and Eq. (2.8) ⅆ4U2 ⅆY4 = β2gρ2 k2 ( ⅆU2 ⅆY ) 2 (2.9) On account of Eq. (2.1) and Eq. (2.3) there exists a constant A such that ⅆP ⅆX = A (2.10) The boundary and interface conditions are UI ( −D1 2 ) = 0 = U2 ( −D2 2 ) , U1(0) = U2(0), T0 = T1 + T2 2 , ⅆ2U1 ⅆY2 | Y=− D1 2 = A μ1 + β1g[T2−T1] 2𝜈1 , ⅆ2U2 ⅆY2 | Y= D2 2 = A μ2 − β2g[T2−T1] 2𝜈2 , μ1 ⅆU1 ⅆY = μ2 ⅆU2 ⅆY , at Y = 0 , ⅆ3U1 ⅆY3 = 1 mnbk ⅆ3U2 ⅆY3 , at Y = 0 ⅆ2U1 ⅆY2 = 1 mnb ⅆ2U2 ⅆy2 + A μ1 [1 − 1 nb ] at Y = 0 𝑇1(0) = 𝑇2(0), k1 ⅆT1 ⅆY = k2 ⅆT2 ⅆY , at Y = 0 (2.11) Equations (2.7),(2.9) and (2.11) determine the velocity distribution. They can be written in a non-dimensional form by means of following dimensionless variables u1 = U1 U0 (1) ; u2 = U2 U0 (2) ; y1 = Y1 D1 ; y2 = Y2 D2 ; Gr = gβIΔTD1 3 ν1 2 ; Re = U0 (1) D1 ν1 Br = μ1U0 (1)2 k1ΔT ; U0 (1) = −AD1 2 48𝜇1 ; U0 (2) = − AD2 2 48𝜇2 ; 197 Analytical Study of Mixed Convective Flow and Heat Transfer… GR = Gr Re ; θ1 = T1 − T0 ΔT ; θ2 = T2 − T0 ΔT ; RT = T2 − T1 ΔT (2.12) Eqs. (2.7), (2.9) becomes Region −I ⅆ4u1 ⅆy4 = GRBr ( ⅆu1 ⅆy ) 2 (2.13) Region −II ⅆ4u2 ⅆy4 = GRBr mnbkD 4 ( ⅆu2 ⅆy ) 2 (2.14) The boundary and interface conditions are u1U0 (1) = 0 , at y = − 1 4 ; u1 (− 1 4 ) = 0 = u2 ( 1 4 ) ; u1(0) = mD 2u2(0) ; ⅆ2u1 ⅆy2 = −48 + GRRT 2 , at y = − 1 4 ; ⅆ2u2 ⅆy2 = −48 − GR 𝑛𝑏 RT 2 , at y = 1 4 ; ⅆu1 ⅆy = D ⅆu2 ⅆy , at y = 0 ; ⅆ2u1 ⅆy2 = 1 nb [ ⅆ2u2 ⅆy2 + 48(1 − nb)] , at y = 0 ; ⅆ3u1 ⅆy3 = 1 nbkD ⅆ3u2 ⅆy3 , at y = 0 Where D = D2 D1 , m = μ1 μ2 , n = ρ2 ρ1 , b = β2 β1 , k = k1 k2 (2.15) Solutions Case of Negligible of Viscous Dissipation (𝐁𝐫 = 𝟎) The solution of Eqs. (2.13) and (2.14) can be obtained using Eq. (2.15) in the absence of viscous dissipation, that is, when the parameter, (𝐁𝐫 = 𝟎) is given by Region −I u1 = E1 + E2y + E3y 2 + E4y 3 (3.1) Region −II u2 = E5 + E6y + E7y 2 + E8y 3 (3.2) Using Eq. (2.12) in Eqs. (2.1) and (2.3) the energy balance equations are Region −I θ1 = − 1 GR [48 + ⅆ2u1 ⅆy2 ] (3.3) Region −II θ2 = − 1 nbGR [48 + ⅆ2u2 ⅆy2 ] (3.4) Using the expressions obtained in Eqs. (3.1) and (3.2) the energy balance Eqs. (3.3) and (3.4) becomes Region −I 198 Mariya Daimi1, S. Jagtap2 θ1 = − 1 GR [48 + 2E3 + 6E4y] (3.5) Region −II θ2 = − 1 nbGR [48 + 2E7 + 6E8y] (3.6) Case of Negligible buoyancy Force (GR = 0) When the buoyancy forces are negligible (GR = 0)and viscous dissipation is dominating (Br ≠ 0),so that purely forced convection occurs. For this case, the solutions of Eqs. (2.13) and (2.14) can be obtained using the Eq. (2.15), the velocities are given by Region −I u1 = F1 + F2y + F3y 2 + F4y 3 (3.7) Region −II u2 = F5 + F6y + F7y 2 + F8y 3 (3.8) The energy balance Eqs. (2.6) and (2.8) in non- dimensional form can also be written as Region −I ⅆ2θ1 ⅆy2 = −Br ( ⅆu1 ⅆy ) 2 (3.9) Region −II ⅆ2θ2 ⅆy2 = −Br m k D 4 ( ⅆu2 ⅆy ) 2 (3.10) The boundary and interface conditions are θ1 (− 1 4 ) = − RT 2 ; θ2 ( 1 4 ) = RT 2 ; θ1(0) = θ2(0) ; ⅆθ1 ⅆy = 1 kD ⅆθ2 ⅆy , at y = 0 ; (3.11) Solving Eqs. (3.9) and (3.10) ,using Eqs (3.7) and (3.8) we obtain Region −I θ1 = −Br ( G3y 2 + G4y 3 + G5y 4 G6y 5 + G7y 6 ) + G2y + G1 (3.12) Region −II θ2 = −m k 𝐷 4 Br ( G10y 2+G11y 3+G12y 4 + G13y 5+G14y 6 ) +G9y + G8 (3.13) Combine Effects of Buoyancy Force and Viscous Dissipation By using the perturbation method, we solve Eqs. (2.13) and (2.14) with a dimensionless parameter |𝜀| (< 1) defined as ε = GRBr (3.14) Which is independent of the reference temperature difference ΔT. The solutions are assumed in the form u(y) = u0(y) + εu1(y) + ε 2u2(y) + ⋯ = ∑ ε nun(y) ∞ n=0 (3.15) 199 Analytical Study of Mixed Convective Flow and Heat Transfer… Substituting Eq. (3.15) in Eqs. (2.13) and (2.14) and the coefficients of like powers of ɛ to obtain the zeroth and first order equations as follows Region −I (Zeroth -order equation) ⅆ4u10 ⅆy4 = 0 (3.16) First-order equation ⅆ4u11 ⅆy4 = ( ⅆu10 ⅆy ) 2 (3.17) Region −II (Zeroth-order equation) ⅆ4u20 ⅆy4 = 0 (3.18) First-order equation ⅆ4u21 ⅆy4 = mnbk𝐷4 ( ⅆu20 ⅆy ) 2 (3.19) The corresponding boundary and interface conditions for the zeroth and first order by Eq. (2.15) reduces to u10 (− 1 4 ) = 0 = u20 ( 1 4 ) ; u11 (− 1 4 ) = 0 = u21 ( 1 4 ) ; u10(0) = mD 2u20(0) ; u11(0) = mD 2u21(0) ; ⅆ2u10 ⅆy2 = −48 + GRRT 2 , at y = − 1 4 ; ⅆ2u11 ⅆy2 = 0 , at y = − 1 4 ; ⅆ2u20 ⅆy2 = −48 − nbGRRT 2 , at y = 1 4 ; ⅆ2u21 ⅆy2 = 0, at y = 1 4 ; ⅆu10 ⅆy = D ⅆu20 ⅆy , at y = 0 ; ⅆu11 ⅆy = D ⅆu21 ⅆy , at y = 0 ; ⅆ2u10 ⅆy2 = 1 nb [ ⅆ2u20 ⅆy2 + 48(1 − nb)] , at y = 0 ; ⅆ2u11 ⅆy2 = 1 nb [ ⅆ2u21 ⅆy2 ] , at y = 0 ; ⅆ3u10 ⅆy3 = 1 nbkD ⅆ3u20 ⅆy3 , at y = 0 ; ⅆ3u11 ⅆy3 = 1 nbkD ⅆ3u21 ⅆy3 , at y = 0 ; (3.20) Solutions of zeroth-order Eqs. (3.16) and (3.18) using Eq. (3.20) are u10 = C1 + C2y + C3y 2 + C4y 3 (3.21) u20 = B1 + B2y + B3y 2 + B4y 3 (3.22) Solutions of first-order Eqs. (3.17) and (3.19) using Eq. (3.20) are u11 = P5y 8 + P6y 7 + P7y 6 + P8y 5 +P9y 4 + P1 6 y3 + P2 2 y2 + P3y + P4 (3.23) u21 = Q5y 8 + Q6y 7 + Q7y 6 + Q8y 5 +Q9y 4 + Q1 6 y3 + Q2 2 y2 + Q3y + Q4 (3.24) Using the velocities given by Eqs. (3.21) - (3.24) the energy balance Eqs. (3.3) and (3.4) becomes 200 Mariya Daimi1, S. Jagtap2 Region −I θ1 = − 1 GR [ 48 + 2C3 + 6C4y +ε ( 56P5y 6 + 42P6y 5 + 30P7y 4 +20P8y 3 + 12P9y 2 + P1 y + P2 ) ] (3.25) Region −II θ2 = − 1 nbGR [ 48 + 2B3 + 6B4y +ε ( 56Q5y 6 + 42Q6y 5 + 30Q7y 4 +20Q8y 3 + 12Q9y 2 + Q1 y + Q2 ) ] (3.26) Heat Transfer The wall heat transfer expression in terms of the Nusselt number is Nu− = (1 + D) ⅆθ1 ⅆy , at y = − 1 4 Nu+ = (1 + 1 D ) ⅆθ1 ⅆy , at y = 1 4 Nu− = − (1+D) GR [6C4 − ε( 21 64 P5 − 105 128 P6 + 15 8 P7 − 15 4 P8 + 6P9 − P1)] (3.27) Nu+ = − (1+ 1 D ) nbGR [6B4 + ε( 21 64 Q5 + 105 128 Q6 + 15 8 Q7 + 15 4 Q8 + 6Q9 − Q1)] (3.28) 3. Results and discussions Investigation of laminar mixed convection flow in vertical channel filled with immiscible viscous fluid has been done analytically by using a regular perturbation method taking the product of the thermal Grashof number (GR=Gr/Re) and Brinkman number Br as perturbation parameter. And solution are valid only for small values of perturbation parameter ɛ(<1). Viscous dissipation term is also included in the energy equations. The flow fields are evaluated in case of asymmetric heating (RT=1) and are represented graphically in Fig 2-8. The velocity and temperature fields in the absence of Brinkman number (Br=0) are obtained for different values of thermal Grashof number (GR) and are depicted in Fig.2a, Fig.2b respectively. For negative value of GR velocity field increases in region-I and decreases in region II whereas for positive values of GR velocity decreases in region I and increases in region II. One can also observe that flow was an increasing function for value GR (>0) and decreasing function for GR (<0). But temperature field decreases in both the regions for all different values of GR. 201 Analytical Study of Mixed Convective Flow and Heat Transfer… The dimensionless temperature field θ is obtained and is shown in Fig.3 for different values of Brinkman number Br in case of negligible Buoyancy force (GR=0). As Brinkman number increases temperature field is also increases in both regions. The velocity and temperature fields are obtained at GR=±500 for different values of ɛ and are shown in Fig.4a, Fig,4b respectively. The velocity field is an increasing function of ɛ for upward flow ɛ (>0) and decreasing function of ɛ for downward flow ɛ(<0). Whereas temperature field increases for both ɛ (>0) and ɛ (<0). The perturbation parameter ɛ is more effective on velocity field than temperature field. From Fig.4a it can also pointed out that at cold (left) and hot (right) walls reversal flow occurs for upward and downward flow respectively. The effect of viscosity ratio m, width ratio D and conductivity ratio k on flow field evaluated for the values of GR =100, and ɛ=0.01 in case of asymmetric heating (RT=1). The effect of viscosity ratio m on the velocity and temperature fields are shown in Fig.5a, Fig.5b respectively. As the viscosity ratio m increases the flow field increases in both regions. Temperature increases from cold to hot walls for all values of m. The effect of width ratio D on the velocity and temperature fields are shown in Fig.6a, Fig.6b respectively. Effect of the width ratio is similar to effect of viscosity ratio on flow field as D increases both velocity and temperature fields increases. The effect of conductivity ratio k on the fields of velocity and temperature are depicted in Fig.7a, Fig.7b respectively. As k increases, the velocity and temperature fields reduce in both the regions, this means larger the conductivity ratio smaller the flow rate is. The Nusselt number at the cold wall (Nu-) and hot wall (Nu+) for |ɛ| is shown in Fig.8.The Nu- is an increasing function of |ɛ| and Nu+ is a decreasing function of |ɛ| Fig. 2a. Velocity profiles for different values of GR 202 Mariya Daimi1, S. Jagtap2 Fig. 2b. Temperature profiles for different values of GR Fig. 3. Temperature profile for different values of Br 203 Analytical Study of Mixed Convective Flow and Heat Transfer… Fig. 4a. Velocity profile for different values of ɛ. Fig. 4b. Temperature profile for different values of Ɛ. 204 Mariya Daimi1, S. Jagtap2 Fig. 5a. Velocity profile for different values of m. Fig. 5b. Temperature profile for different values of m. 205 Analytical Study of Mixed Convective Flow and Heat Transfer… Fig. 6a. Velocity profile for different values of D. Fig. 6b. Temperature profile for different values of D. 206 Mariya Daimi1, S. Jagtap2 Fig. 7a. Velocity profile for different values of k Fig. 7b. Temperature profile for different values of k 207 Analytical Study of Mixed Convective Flow and Heat Transfer… Fig. 8. Nusselt number Vs ∣ɛ∣. 4. Conclusions The problem of mixed convective flow and heat transfer in vertical channel filled with immiscible viscous fluids was analyzed analytically by regular perturbation method and represented graphically. The conclusions made are, the flow was an increasing function of perturbation parameter ɛ for upward flow and decreasing function of ɛ for downward flow, viscosity ration m and width ratio D enhance the velocity and temperature fields where as the larger the value of conductivity ratio k, smaller the fields of velocity and temperature. References [1] Aung W., Worku G., Developing flow and flow reversal in mixed convection in vertical channel with asymmetric wall temperatures, J. Heat transfer, 108, 485-488, 1986. [2] Basant K. Jha, Michael O. Oni., Mixed convection flow in vertical channel with temperature dependent viscosity and flow reversal: An exact solution, International Journal of Heat and Technology, 36,607-613,2018. [3] Barletta A., Laminar mixed convection with viscous dissipation in a vertical channel, International Journal of Heat mass transfer, 41,3501-3513, 1998. 208 Mariya Daimi1, S. Jagtap2 [4] D. Srinivasacharya, Md. Shafeeurrahaman, Mixed convention flow of nanofluid in a vertical channel with Hall and Ion-Slip effects, FHMT, 8-11,2017. [5] Desai C., Vafai K., Three-dimensional buoyancy induced flow and heat transfer around the wheel outboard of an aircraft, International Journal of Heat Fluid Flow,13,50-64,1992. [6] J.C. Umavati, Ali j. Chamkha, Fully developed mixed convection in a vertical channel in the presence of heat source or heat sink, international Journal of Energy and technology,3(24),1-9,2011. [7] J. Prathap Kumar, J.U. Umavati and Shreedevi Kalyan, Chemical reaction effects on mixed convection flow of two immiscible viscous fluid in a vertical channel, HMMT, 2(2), 28-46.2014. [8] J. Prathap Kumar, J.U. Umavati and Y. Ramarao, Mixed convective heat transfer of immiscible fluids in a vertical channel with boundary conditions of the third kind, computational thermal Science, 9(5),447-465,2017. [9] M. Rashidi, D. Ganji, Homotopy perturbation combined with Padeꞌ approximation for solving two-dimensional viscous flow in the extrusion process, International Journal of Nonlinear Science,7,387-394,2009. [10] Tao, Combined Free and force convection in channels, ASME Journal of Heat Transfer,82,233-238,1980. [11] T.T Hamadah, R. A. Wirtz, analysis of laminar fully developed mixed convection in a vertical channel with opposing buoyancy, ASME Journal of Heat Transfer,113,507- 510,1991. 209