Mono and hybrid nanofluid analysis over shrinking surface with thermal radiation: A numerical approach

Case Studies in Thermal Engineering 54 (2024) 104023 Available online 17 January 2024 2214-157X/© 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Contents lists available at ScienceDirect Case Studies in Thermal Engineering journal homepage: www.elsevier.com/locate/csite Mono and hybrid nanofluid analysis over shrinking surface with thermal radiation: A numerical approach S. Saleema, Bilal Ahmadb, Azra Naseemb, Muhammad Bilal Riazc,d, Tasawar Abbasb,* aDepartment of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia bDepartment of Mathematics, University of Wah, Wah Cantt, 47040 Pakistan cIT4Innovations, VSB –Technical University of Ostrava, Ostrava, Czech Republic dDepartment of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon ARTICLE INFO Handling Editor: Huihe Qiu Keywords: Hybrid nanofluid Shrinking surface MHD Thermal radiation Viscous dissipation ABSTRACT The study of magnetohydrodynamics (MHD) incompressible flow of a fluid having hybrid nanoparticles making the colloidal combination with base fluid is presented in this research. Comparative analysis is carried out for the nanofluids Al2O3/Kerosene and ZnO/Kerosene oil with the hybrid nanofluid Al2O3–ZnO/Kerosene oil. The subject flows are influenced with thermal radiation and viscous dissipation. A numerical technique Keller box is employed to examine the envision mathematical model. For computational procedure MATLAB software will be used. Tabulated and graphical outcomes of different effects are presented for the appraisal of velocity and temperature distributions. It is comprehend that the thermal radiation and viscous dissipation parameters possess up surging trends for the hybrid and nanofluid temperature profile but opposite trend has been observed for different volume fractions of nanoparticles. 1. Introduction The boundary layer flow caused by a shrinking or stretching sheet has been investigated and discussed by various researchers because of its widespread uses in industries. The rate at which the surface contracts or stretches and the rate of cooling (exchange of heat) during this process determine the surface's actual behaviour. The flow of fluid brought about by a shrinking sheet was first studied and a numerical solution was given by Miklavcic and Wang [1]. For shrinking sheet they found dual solution. Wang [2] examined the fluid flow caused by shrinking of a sheet near the stagnation point region and as a result obtained dual solutions. Shoukat Ahmed, Maryam Ahmed [3] carried out the study on mixed convective MHD flow and concluded that thermal flow behaviour expands with rising radiation factor, Eckert number, radiation and strength of heat source. The analytical solution for the flow of boundary layer brought on by the stretching of a sheet was provided by M. Hassani [4]. Lok et al. [5] considered the hydromagnetic flow of liquid on a shrinking sheet close to stagnation point region. Dual solution flow cases in area of stagnation point through a shrinking/stretching sheet were studied by Norfifah Bachok, Anaur Ishak [6]. The slip condition's impact on stagnation point flow was studied by Bhattacharyya et al. [7]. By employing shooting method for the solution of self-similar equations they obtained dual solutions. Begawada and Nandeppanavar [8] examined the impact of thermal radiation on micropolar fluid flow through vertical porous medium. The researcher also considered the slip condition and obtained numerical solution by the use of Runge-Kutta-Fehlberg (RKF) method. On a shrinking/stretching porous surface stagnation point flow was studied by Bachok [9]. For a shrinking sheet dual solutions were ob- * Corresponding author. E-mail address: [email protected] (T. Abbas). https://doi.org/10.1016/j.csite.2024.104023 Received 29 September 2023; Received in revised form 10 January 2024; Accepted 12 January 2024 Case Studies in Thermal Engineering 54 (2024) 104023 2 S. Saleem et al. Table 1 Thermal and physical properties of base fluid and nanomaterials [29]. Materials ρ(kg/m3) 𝜎((Ω.m))−1 KCp Kerosene oil 783 6 × 10−10 0.15 2090 ZnO 5700 10–1 × 10−325 523 Al2O33970 1 × 10−10 40 765 Table 2 Properties of nanofluid and hybrid nanofluid [29]. Properties Nanofluid Hybrid Nanofluid Heat Capacity (ρCp)nf = (ρCp)f(1- ∅1) +(ρCp)s1∅1. (ρCp)hnf = (ρCp)nf (1- ∅2) +(ρCp)s2∅2. Thermal Conductivity knf =kf2kf+ks1−2kf−ks1∅1 2kf+ks1+kf−ks1∅1 khnf =knf 2knf +ks2−2knf −ks2∅2 2knf +ks2+knf −ks2∅2 Electrical Conductivity 𝜎nf =𝜎f(𝜎s1(1+2∅1)+2(1−∅1)𝜎f 𝜎s1(1−∅1)+(2+∅1)𝜎f 𝜎hnf =𝜎nf (2𝜎f(1−∅2)+(2∅2+1)𝜎s2 𝜎f(2+∅2)+(1−∅2)𝜎s2 Density ρnf = (1 − ∅1)ρf+ρs1∅1ρhnf = (1 − ∅2)ρnf +ρs2∅2 Dynamic Viscosity 𝜇nf =𝜇f (1−∅1)2.5 𝜇hnf =𝜇nf (1−∅2)2.5 tained and in case of stretching sheet unique solution was obtained by him. By using different physical conditions some more researchers also obtained the dual solutions. Maxwell [10] examined the impact on the thermal conductivity of fluid using various materials with better conductivity. SP Samrat and MG Reddy [11] investigated the magnetohydrodynamic free convective flow along the upper region of a paraboloid of revolution while keeping in check the effects of Brownian motion and thermophoresis. Nanofluid was introduced by Choi & Eastman [12] to intensify the conductivity of a fluid. They observed that when nanoparticles are mixed in a base fluid then nanofluid is obtained. With the advancement in the field of heat transfer by means of nanotechnology, nanofluid is characterized as a mixture of nanoparticles in a base fluid. The insertion of nanoparticles improves fluids thermal conductivity and ability of cooling becomes limited [13–15]. Some nanoparticles used are carbon metal oxides and metals. Nanofluids are widely used in industries therefore researchers are interested in the study of these fluids. The establishment of hybrid nanofluid which is obtained by blending nanoparticles in a base fluid has improved heat transfer and other features of the fluid [16–20]. The introduction of this fluid attracted many researches towards the extension work. Amalraj and Michael [21] proved the hybrid nanofluid Al2O3/CuO as a better coolant for the solar panel. Thermal conductivity of two nanofluids CuO/Water and Al2O3/Water and hybrid nanofluid Al2O3–CuO/Water was investigated by S. Senthilraja [22]. The increase in thermal conductivity was 8%, 6.1% and 9% respectively. The findings demonstrated that in comparison to the other two nanofluids the conductivity rate of hybrid nanofluid was high. Flow of hybrid nanofluid Al2O3–CuO/water over a stretching surface in three dimensions under the effect of Lorentz force was studied by Devi and Devi [23] and proved that by using different nanoparticles heat transfer rate can be maximized. Effect of nonlinear radiation on MHD flow of Casson hybrid nanofluid caused by a curved stretching sheet was investigated by N Sandeep et al. [24]. SP Samrat et al. [25] examined the heat transfer and flow characteristics of MHD flow of dusty nano and dusty hybrid nanoliquids caused by a stretching surface. Recently Khan et al. [26] investigated bioconvective catlized Casson hybrid nanofluid over vertical cone. Different studies on hybrid nanofluid's flow over a shrinking or stretching surface by employing different physical conditions are [27,28]. Based on above literature review it is indicated that a study which includes comparative analysis of different types of nanofluids and hybrid nanofluid keeping the same effects is missing. This study indicates the Idiosyncratic behaviour of monoand hybrid nanofluids along with their applications in various thermal systems including solar thermal systems, automotive cooling systems, heat sinks, or thermal energy storage. Here we consider three different types of fluids viz., Al2O3/Kerosene oil and ZnO/Kerosene oil nanofluid and their mixrure Al2O3–ZnO/Kerosene oil named as Hybrid nanofluid over a shrinking sheet. The important objectives of this work are. •To analyze the flow behaviour for mono and hybrid nanofluids against the magnetic field. •To investigate the thermal aspects of fluid in the presence of thermal radiation and viscous dissipation •Justification of the anticipated solution for the heat transfer phenomena by comparing results with previous studies. The model includes the nonlinear Partial differential equations which are transformed into ordinary differential equations by application of suitable similarity transformation. Keller box methodology will be used to adopt numerical solutions. The impact of parameters involved in the modeled equations will be studied on the behaviour of velocity and temperature profiles of the understudy fluids and the findings will be shown graphically. 2. Mathematical model and formulation Here two dimensional boundary layer flow of three different fluids near the region of stagnation point is investigated. The flow is developed by Stretching/  S hrinking surface. Kerosene oil is taken as base fluid while nanoparticles used are aluminum oxide and zinc Case Studies in Thermal Engineering 54 (2024) 104023 3 S. Saleem et al. Fig. 1. Problem's geometry. Fig. 2. Graph of velocity profile for various values of M. oxide. Thermophysical properties of these nanoparticles are given in Tables 1 and 2. The direction of surface is along the horizontal axis ( x ) while vertical axis ( y ) is perpendicular to it. The rate at which surface is stretched or shrunk is given by uw(x)=ax , where a is negative for surfaces that shrink and positive for surfaces that stretch. The velocity for the orthogonal flow of stagnation point is given by ue(x)=bx where b is positive and gives the strength of stagnation flow. A magnetic field of strength B0is also applied normal to the surface as shown in Fig. 1. With the additional effect of viscous dissipation and convective boundary the steady state continuity, momentum and energy equations of the modeled problem are [29]. 𝜕u 𝜕x +𝜕v 𝜕y =0, (1) u𝜕u 𝜕x+v𝜕u 𝜕y= − 1 𝜌hnf ue due dx+ 𝜇hnf 𝜌hnf 𝜕2u 𝜕y2+ 𝜎hnf 𝜌hnf B0 2(ue−u), (2) u𝜕T 𝜕x+v𝜕T 𝜕y=k 𝜌Cphnf 𝜕2T 𝜕y2−1 𝜌Cphnf 𝜕qr 𝜕y+ 𝜇hnf 𝜌Cphnf 𝜕u 𝜕y2 (3) With boundary conditions For y=0∶u=uw,v=0,−khnf 𝜕T 𝜕y=h1Tf−T, (4) For y→∞ ∶ u→ue,v→0,T→T∞, Case Studies in Thermal Engineering 54 (2024) 104023 4 S. Saleem et al. Fig. 3. Graph of temperature distribution for varying values of Prandtl number. Fig. 4. Impact of varying values of R on temperature distribution of three fluids. Here v is component of velocity perpendicular to surface whereas u is velocity component along the surface, ue is free stream velocity and uw is velocity at wall. T, Tw ,T ∞ represents fluid temperature, temperature at wall and free stream temperature respectively. Also μ,σ,k,ρ,qr,Cp,represents the dynamic viscosity, electrical conductivity, thermal conductivity, density of fluid, radiative heat flux and specific heat at constant pressure respectively. Subscripts s1 and s2 represents the solid particles of aluminum oxide and zinc oxide. Subscripts hnf, f and nf stands for hybrid nanofluid, fluid and nanofluid respectively. Using Rosseland approximation [30]qrtakes the form qr= − 4𝜎1 3k1 𝜕T4 𝜕y, (5) here k1represents the absorption coefficient and σ1stands for Stefan-Boltzmann constant. It is supposed that variation of temperature in fluid is such that T4can be written as a function (linear) of T. By applying Taylor series expansion for T4about the point T ∞ we get T4∼ =4TT3 ∞−3T4 ∞, (6) the terms involving higher powers of T ∞ are ignored. By virtue of (5) and (6) equation (3) becomes u𝜕T 𝜕x+v𝜕T 𝜕y=k 𝜌Cphnf 𝜕2T 𝜕y2−1 𝜌Cphnf −16𝜎1T∞ 3 3k1 𝜕2T 𝜕y2+ 𝜇hnf 𝜌Cphnf 𝜕u 𝜕y2 , (7) Case Studies in Thermal Engineering 54 (2024) 104023 5 S. Saleem et al. Fig. 5. Impact of rising values of Econ temperature distribution of three fluids. Fig. 6. Impact of varying values of Bion temperature distribution of three fluids. Using the similarity transformations u=f′(6)b,v= −f(6)√b𝜇f 𝜌f , 𝜃 (6)=T−T∞ Tw−T∞ ,η = √b𝜌f 𝜇f y, (8) Where prime is used to show differentiation with respect to ƞ. By virtue of equation (8) the dimensionless form of equations (2) and (7) is A1 A2 f′′ +1−f′2+MA3 A2 (1−f′)=0, (9) (A4+R)𝜃′′ = −A5Prf𝜃′−EcA1Prf′′2, (10) Together with boundary conditions At η = 0 = 0 : f(η) = 0, (η) = ʎ, (η) = − (1 − 𝜃(η))𝑓′ 𝜃 ′ 𝐵 𝑖 For 6→∞f′→1, 𝜃 →0, (11) where Pr, R, M, Ec,Bi and ʎ stands for Prandtl number, radiation parameter, magnetic parameter, Eckert number, Biott number and velocity ratio parameter respectively and are given by Case Studies in Thermal Engineering 54 (2024) 104023 6 S. Saleem et al. Fig. 7. Impact of enhancing values of ∅1,∅2on velocity of fluids. Fig. 8. Impact of enhancing values of ∅1,∅2on temperature of fluids. M= 𝜎fB0 2 b𝜌f ,Pr= 𝜇f(𝜌Cp)f 𝜌fkf ,R= 16𝜎1T∞ 3 3k1kf , (12) In addition the ratios A1,A2, A3 ,A4and A5 are given as A1= 𝜇hnf 𝜇f ,A2= 𝜌hnf 𝜌f ,A3= 𝜎hnf 𝜎f ,A4= khnf kf ,A5= (𝜌Cp)hnf (𝜌Cp)f , (13) The two quantities the skin friction coefficient (Cf) and local Nusselt number (Nux)are of prime importance from engineering point of view, which are given by [31,32] Cf=𝜏w 𝜌fue 2,Nux= xqw kf(Tw−T∞), (14) Where qwis heat flux from the plate and τwis surface shear stress along the plate. They are computed by 𝜏w=𝜇hnf (𝜕u 𝜕y)y=0 ,qw= −khnf (𝜕T 𝜕y)y=0 (15) By using equation (8) we obtain Case Studies in Thermal Engineering 54 (2024) 104023 7 S. Saleem et al. Table 3 Hybrid nanofluid Al2O3–ZnO/Kerosene oil: Following tables depict the influence of different parameters on −f″(0), −θ′(0) for ∅1, = 0.1, 𝑀=𝑅= 0.5, = 1, ʎ= −1.2,∅2𝑃𝑟𝐸𝑐 = 0.7. ∅1∅2M R PrʎEc−θ′(0) -f″(0) 0.1 0.1 0.5 0.5 1 −1.2 0.7 0.99424 2.96009 1 1.09229 3.37927 1.5 1.17353 4.36705 0.5 0.99424 1.5 1.0406 2.5 1.08421 0.8 0.88424 1.7 1.57612 2.5 2.3008 −1.2 0.99424 2.96009 −1.3 1.0758 2.73136 −1.4 1.1574 2.37629 0.1 0.1 0.99424 2.96009 0.4 0.4 1.1796 4.53701 0.7 0.7 2.5452 7.0732 0.7 0.99424 0.9 1.25503 1.2 1.64621 Table 4 Nanofluid Al2O3/Kerosene oil. <! − − Col Count ∶9− − >∅1 ∅2M R PrʎEc−θ′(0) −f″(0) 0.1 0 0.5 0.5 1 −1.2 0.7 0.71743 2.21103 1 0.78355 2.80108 1.5 0.83863 3.27592 0.5 0.7174 1.5 0.7503 2.5 0.7831 0.8 0.66743 1.7 1.1586 2.5 1.7258 −1.2 0.78498 2.21103 −1.3 0.85704 2.04995 −1.4 0.85704 1.08262 0.1 0.71743 2.21103 0.4 0.9903 2.60597 0.7 1.2846 3.12959 0.7 0.71743 0.9 0.90499 1.2 1.18634 Rex 1∕2Cf=A1f′′ (0),Rex−1∕2Nux= −A4𝜃′(0). (16) Here Rex represents the local Reynolds number. 3. Computational method The system of equations (9) and (10) which is a set of nonlinear partial differential equations with its boundary conditions (11) are computed numerically by employing Keller Box [33–35] technique. Let f′=a, (17) a′=b, (18) 𝜃′=t, (20) Equations (9) and (10) becomes A1 A2 b′+1−a2+fb +MA3 A2 (1−a)=0, (21) Case Studies in Thermal Engineering 54 (2024) 104023 8 S. Saleem et al. Table 5 Nanofluid ZnO/Kerosene oil. <! − − Col Count ∶9− − >∅1 ∅2M R PrʎEc−θ′(0) −f″(0) 0 0.1 0.5 0.5 1 −1.2 0.7 0.71150 2.58499 1 0.78768 3.3252 1.5 0.85012 3.91741 0.5 0.71150 1.5 0.75155 2.5 0.78019 0.8 0.67150 1.7 1.10111 2.5 1.5842 −1.2 0.71150 2.58499 −1.3 0.77148 2.45382 −1.4 0.83528 2.25918 0.1 0.71150 2.58499 0.4 0.9219 3.4617 0.7 1.1220 4.6560 0.7 0.71150 0.9 0.88785 1.2 1.15236 (A4+R)t′+A5Prft +EcA1Prb2=0, (22) And boundary conditions take the form ƞ= 0 𝑎=ʎ,𝑓= 0, 𝑡= − (1 − 𝜃) 𝐵 𝑖 6→∞a→1, 𝜃 →0, By applying finite difference method fj−fj−1− hj 2(aj−aj−1)=0, (23) aj−aj−1− hj 2(bj−bj−1)=0, (24) 𝜃j−𝜃j−1− hj 2(tj−tj−1)=0, (25) Using (23) to (25) in (21) and (22), we have A1 A2bj−bj−1+1−hjaj+aj−1 22 +fj+fj−1 2bj+bj−1 2+MA3 A21− aj+aj−1 2=0, (26) A4+Rtj−tj−1+hjA5Prfj+fj−1 2tj+tj−1 2+EcA1Prbj+bj−1 22=0, (27) Newton's method for linearization fjk+1=fjk+𝛿fjk, (28) ajk+1=ajk+𝛿ajk, (29) bjk+1=bjk+𝛿bjk, (30) 𝜃jk+1=𝜃jk+𝛿𝜃jk, (31) tjk+1=tjk+𝛿tjk, (32) So equations (23)–(25) becomes 𝛿fj−𝛿fj−1− hj 2{𝛿aj−𝛿aj−1}=(Q1)j, (33) Case Studies in Thermal Engineering 54 (2024) 104023 9 S. Saleem et al. Table: 6 Comparison of the Skin friction coefficient with R=M=EC =∅1=∅2= 0. ʎ[29] Present study 1 0 0 0.5 0.71330 0.71330 0 1.23258 1.23257 −0.25 1.40224 1.40220 −0.5 1.49567 1.49569 −0.75 1.48930 1.48925 −1 1.32880 1.32881 −1.15 1.08220 1.08220 𝛿aj−𝛿aj−1− hj 2{𝛿bj−𝛿bj−1}=(Q2)j, (34) 𝛿𝜃j−𝛿𝜃j−1− hj 2{𝛿tj−𝛿tj−1}=(Q3)j, (35) Where (Q1)j=fj−1−fj+hjaj−1 2 ,(Q2)j=aj−1−aj+hjbj−1 2 and (Q3)j=𝜃j−1−𝜃j+hjtj−1 2 , So equations (26) and (27) becomes C1𝛿aj+C2𝛿aj−1+C3𝛿fj+C4𝛿fj−1+C5𝛿bj+C6𝛿bj−1=(R1)j, (36) D1𝛿tj+D2𝛿tj−1+D3𝛿fj+D4𝛿fj−1+D5𝛿bj+D6𝛿bj−1=(R2)j, (37) Where