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

Case S udies in The mal Enginee ing 54 (2024) 104023 A ailable online 17 Janua y 2024 2214-157X/© 2024 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/). Con en s lis s a ailable a ScienceDi ec Case S udies in The mal Enginee ing jou nal homepage: www.else ie .com/loca e/csi e Mono and hyb id nano luid analysis o e sh inking su ace wi h he mal adia ion: A nume ical app oach S. Saleema, Bilal Ahmadb, Az a Naseemb, Muhammad Bilal Riazc,d, Tasawa Abbasb,* aDepa men o Ma hema ics, College o Science, King Khalid Uni e si y, Abha 61413, Saudi A abia bDepa men o Ma hema ics, Uni e si y o Wah, Wah Can , 47040 Pakis an cIT4Inno a ions, VSB –Technical Uni e si y o Os a a, Os a a, Czech Republic dDepa men o Compu e Science and Ma hema ics, Lebanese Ame ican Uni e si y, Byblos, Lebanon ARTICLE INFO Handling Edi o : Huihe Qiu Keywo ds: Hyb id nano luid Sh inking su ace MHD The mal adia ion Viscous dissipa ion ABSTRACT The s udy o magne ohyd odynamics (MHD) incomp essible low o a luid ha ing hyb id nanopa icles making he colloidal combina ion wi h base luid is p esen ed in his esea ch. Compa a i e analysis is ca ied ou o he nano luids Al2O3/Ke osene and ZnO/Ke osene oil wi h he hyb id nano luid Al2O3–ZnO/Ke osene oil. The subjec lows a e in luenced wi h he - mal adia ion and iscous dissipa ion. A nume ical echnique Kelle box is employed o examine he en ision ma hema ical model. Fo compu a ional p ocedu e MATLAB so wa e will be used. Tabula ed and g aphical ou comes o di e en e ec s a e p esen ed o he app aisal o eloci y and empe a u e dis ibu ions. I is comp ehend ha he he mal adia ion and iscous dissipa- ion pa ame e s possess up su ging ends o he hyb id and nano luid empe a u e p o ile bu opposi e end has been obse ed o di e en olume ac ions o nanopa icles. 1. In oduc ion The bounda y laye low caused by a sh inking o s e ching shee has been in es iga ed and discussed by a ious esea che s be- cause o i s widesp ead uses in indus ies. The a e a which he su ace con ac s o s e ches and he a e o cooling (exchange o hea ) du ing his p ocess de e mine he su ace's ac ual beha iou . The low o luid b ough abou by a sh inking shee was i s s ud- ied and a nume ical solu ion was gi en by Mikla cic and Wang [1]. Fo sh inking shee hey ound dual solu ion. Wang [2] examined he luid low caused by sh inking o a shee nea he s agna ion poin egion and as a esul ob ained dual solu ions. Shouka Ahmed, Ma yam Ahmed [3] ca ied ou he s udy on mixed con ec i e MHD low and concluded ha he mal low beha iou expands wi h ising adia ion ac o , Ecke numbe , adia ion and s eng h o hea sou ce. The analy ical solu ion o he low o bounda y laye b ough on by he s e ching o a shee was p o ided by M. Hassani [4]. Lok e al. [5] conside ed he hyd omagne ic low o liquid on a sh inking shee close o s agna ion poin egion. Dual solu ion low cases in a ea o s agna ion poin h ough a sh inking/s e ching shee we e s udied by No i ah Bachok, Anau Ishak [6]. The slip condi ion's impac on s agna ion poin low was s udied by Bha - acha yya e al. [7]. By employing shoo ing me hod o he solu ion o sel -simila equa ions hey ob ained dual solu ions. Begawada and Nandeppana a [8] examined he impac o he mal adia ion on mic opola luid low h ough e ical po ous medium. The e- sea che also conside ed he slip condi ion and ob ained nume ical solu ion by he use o Runge-Ku a-Fehlbe g (RKF) me hod. On a sh inking/s e ching po ous su ace s agna ion poin low was s udied by Bachok [9]. Fo a sh inking shee dual solu ions we e ob- * Co esponding au ho . E-mail add ess: asawa [email protected] (T. Abbas). h ps://doi.o g/10.1016/j.csi e.2024.104023 Recei ed 29 Sep embe 2023; Recei ed in e ised o m 10 Janua y 2024; Accep ed 12 Janua y 2024 Case S udies in The mal Enginee ing 54 (2024) 104023 2 S. Saleem e al. Table 1 The mal and physical p ope ies o base luid and nanoma e ials [29]. Ma e ials ρ(kg/m3) 𝜎((Ω.m))−1 KCp Ke osene oil 783 6 × 10−10 0.15 2090 ZnO 5700 10–1 × 10−325 523 Al2O33970 1 × 10−10 40 765 Table 2 P ope ies o nano luid and hyb id nano luid [29]. P ope ies Nano luid Hyb id Nano luid Hea Capaci y (ρCp)n = (ρCp) (1- ∅1) +(ρCp)s1∅1. (ρCp)hn = (ρCp)n (1- ∅2) +(ρCp)s2∅2. The mal Conduc i i y kn =k 2k +ks1−2k −ks1∅1 2k +ks1+k −ks1∅1 khn =kn 2kn +ks2−2kn −ks2∅2 2kn +ks2+kn −ks2∅2 Elec ical Conduc i i y 𝜎n =𝜎 (𝜎s1(1+2∅1)+2(1−∅1)𝜎 𝜎s1(1−∅1)+(2+∅1)𝜎 𝜎hn =𝜎n (2𝜎 (1−∅2)+(2∅2+1)𝜎s2 𝜎 (2+∅2)+(1−∅2)𝜎s2 Densi y ρn = (1 − ∅1)ρ +ρs1∅1ρhn = (1 − ∅2)ρn +ρs2∅2 Dynamic Viscosi y 𝜇n =𝜇 (1−∅1)2.5 𝜇hn =𝜇n (1−∅2)2.5 ained and in case o s e ching shee unique solu ion was ob ained by him. By using di e en physical condi ions some mo e e- sea che s also ob ained he dual solu ions. Maxwell [10] examined he impac on he he mal conduc i i y o luid using a ious ma- e ials wi h be e conduc i i y. SP Sam a and MG Reddy [11] in es iga ed he magne ohyd odynamic ee con ec i e low along he uppe egion o a pa aboloid o e olu ion while keeping in check he e ec s o B ownian mo ion and he mopho esis. Nano luid was in oduced by Choi & Eas man [12] o in ensi y he conduc i i y o a luid. They obse ed ha when nanopa icles a e mixed in a base luid hen nano luid is ob ained. Wi h he ad ancemen in he ield o hea ans e by means o nano echnology, nano luid is cha ac- e ized as a mix u e o nanopa icles in a base luid. The inse ion o nanopa icles imp o es luids he mal conduc i i y and abili y o cooling becomes limi ed [13–15]. Some nanopa icles used a e ca bon me al oxides and me als. Nano luids a e widely used in indus- ies he e o e esea che s a e in e es ed in he s udy o hese luids. The es ablishmen o hyb id nano luid which is ob ained by blending nanopa icles in a base luid has imp o ed hea ans e and o he ea u es o he luid [16–20]. The in oduc ion o his luid a ac ed many esea ches owa ds he ex ension wo k. Amal aj and Michael [21] p o ed he hyb id nano luid Al2O3/CuO as a be e coolan o he sola panel. The mal conduc i i y o wo nano luids CuO/Wa e and Al2O3/Wa e and hyb id nano luid Al2O3–CuO/Wa e was in es iga ed by S. Sen hil aja [22]. The inc ease in he - mal conduc i i y was 8%, 6.1% and 9% espec i ely. The indings demons a ed ha in compa ison o he o he wo nano luids he conduc i i y a e o hyb id nano luid was high. Flow o hyb id nano luid Al2O3–CuO/wa e o e a s e ching su ace in h ee dimen- sions unde he e ec o Lo en z o ce was s udied by De i and De i [23] and p o ed ha by using di e en nanopa icles hea ans- e a e can be maximized. E ec o nonlinea adia ion on MHD low o Casson hyb id nano luid caused by a cu ed s e ching shee was in es iga ed by N Sandeep e al. [24]. SP Sam a e al. [25] examined he hea ans e and low cha ac e is ics o MHD low o dus y nano and dus y hyb id nanoliquids caused by a s e ching su ace. Recen ly Khan e al. [26] in es iga ed biocon ec i e ca lized Casson hyb id nano luid o e e ical cone. Di e en s udies on hyb id nano luid's low o e a sh inking o s e ching su ace by em- ploying di e en physical condi ions a e [27,28]. Based on abo e li e a u e e iew i is indica ed ha a s udy which includes compa a i e analysis o di e en ypes o nano luids and hyb id nano luid keeping he same e ec s is missing. This s udy indica es he Idiosync a ic beha iou o mono- and hyb id nano luids along wi h hei applica ions in a ious he mal sys ems including sola he mal sys ems, au omo i e cooling sys ems, hea sinks, o he mal ene gy s o age. He e we conside h ee di e en ypes o luids iz., Al2O3/Ke osene oil and ZnO/Ke osene oil nano luid and hei mix u e Al2O3–ZnO/Ke osene oil named as Hyb id nano luid o e a sh inking shee . The impo an objec i es o his wo k a e. •To analyze he low beha iou o mono and hyb id nano luids agains he magne ic ield. •To in es iga e he he mal aspec s o luid in he p esence o he mal adia ion and iscous dissipa ion •Jus i ica ion o he an icipa ed solu ion o he hea ans e phenomena by compa ing esul s wi h p e ious s udies. The model includes he nonlinea Pa ial di e en ial equa ions which a e ans o med in o o dina y di e en ial equa ions by ap- plica ion o sui able simila i y ans o ma ion. Kelle box me hodology will be used o adop nume ical solu ions. The impac o pa a- me e s in ol ed in he modeled equa ions will be s udied on he beha iou o eloci y and empe a u e p o iles o he unde s udy lu- ids and he indings will be shown g aphically. 2. Ma hema ical model and o mula ion He e wo dimensional bounda y laye low o h ee di e en luids nea he egion o s agna ion poin is in es iga ed. The low is de eloped by S e ching/  S h inking su ace. Ke osene oil is aken as base luid while nanopa icles used a e aluminum oxide and zinc Case S udies in The mal Enginee ing 54 (2024) 104023 3 S. Saleem e al. Fig. 1. P oblem's geome y. Fig. 2. G aph o eloci y p o ile o a ious alues o M. oxide. The mophysical p ope ies o hese nanopa icles a e gi en in Tables 1 and 2. The di ec ion o su ace is along he ho izon al axis ( x ) while e ical axis ( y ) is pe pendicula o i . The a e a which su ace is s e ched o sh unk is gi en by uw(x)=ax , whe e a is nega i e o su aces ha sh ink and posi i e o su aces ha s e ch. The eloci y o he o hogonal low o s agna ion poin is gi en by ue(x)=bx whe e b is posi i e and gi es he s eng h o s agna ion low. A magne ic ield o s eng h B0is also applied no - mal o he su ace as shown in Fig. 1. Wi h he addi ional e ec o iscous dissipa ion and con ec i e bounda y he s eady s a e con inui y, momen um and ene gy equa ions o he modeled p oblem a e [29]. 𝜕u 𝜕x +𝜕 𝜕y =0, (1) u𝜕u 𝜕x+ 𝜕u 𝜕y= − 1 𝜌hn ue due dx+ 𝜇hn 𝜌hn 𝜕2u 𝜕y2+ 𝜎hn 𝜌hn B0 2(ue−u), (2) u𝜕T 𝜕x+ 𝜕T 𝜕y=k 𝜌Cphn 𝜕2T 𝜕y2−1 𝜌Cphn 𝜕q 𝜕y+ 𝜇hn 𝜌Cphn 𝜕u 𝜕y2 (3) Wi h bounda y condi ions Fo y=0∶u=uw, =0,−khn 𝜕T 𝜕y=h1T −T, (4) Fo y→∞ ∶ u→ue, →0,T→T∞, Case S udies in The mal Enginee ing 54 (2024) 104023 4 S. Saleem e al. Fig. 3. G aph o empe a u e dis ibu ion o a ying alues o P and l numbe . Fig. 4. Impac o a ying alues o R on empe a u e dis ibu ion o h ee luids. He e  is componen o eloci y pe pendicula o su ace whe eas u is eloci y componen along he su ace, ue is ee s eam eloci y and uw is eloci y a wall. T, Tw ,T ∞ ep esen s luid empe a u e, empe a u e a wall and ee s eam empe a u e espec i ely. Also μ,σ,k,ρ,q ,Cp, ep esen s he dynamic iscosi y, elec ical conduc i i y, he mal conduc i i y, densi y o luid, adia i e hea lux and speci ic hea a cons an p essu e espec i ely. Subsc ip s s1 and s2 ep esen s he solid pa icles o aluminum oxide and zinc oxide. Subsc ip s hn , and n s ands o hyb id nano luid, luid and nano luid espec i ely. Using Rosseland app oxima ion [30]q akes he o m q = − 4𝜎1 3k1 𝜕T4 𝜕y, (5) he e k1 ep esen s he abso p ion coe icien and σ1s ands o S e an-Bol zmann cons an . I is supposed ha a ia ion o empe a u e in luid is such ha T4can be w i en as a unc ion (linea ) o T. By applying Taylo se ies expansion o T4abou he poin T ∞ we ge T4∼ =4TT3 ∞−3T4 ∞, (6) he e ms in ol ing highe powe s o T ∞ a e igno ed. By i ue o (5) and (6) equa ion (3) becomes u𝜕T 𝜕x+ 𝜕T 𝜕y=k 𝜌Cphn 𝜕2T 𝜕y2−1 𝜌Cphn −16𝜎1T∞ 3 3k1 𝜕2T 𝜕y2+ 𝜇hn 𝜌Cphn 𝜕u 𝜕y2 , (7) Case S udies in The mal Enginee ing 54 (2024) 104023 5 S. Saleem e al. Fig. 5. Impac o ising alues o Econ empe a u e dis ibu ion o h ee luids. Fig. 6. Impac o a ying alues o Bion empe a u e dis ibu ion o h ee luids. Using he simila i y ans o ma ions u= ′(6)b, = − (6)√b𝜇 𝜌 , 𝜃 (6)=T−T∞ Tw−T∞ ,η = √b𝜌 𝜇 y, (8) Whe e p ime is used o show di e en ia ion wi h espec o ƞ. By i ue o equa ion (8) he dimensionless o m o equa ions (2) and (7) is A1 A2 ′′ +1− ′2+MA3 A2 (1− ′)=0, (9) (A4+R)𝜃′′ = −A5P 𝜃′−EcA1P ′′2, (10) Toge he wi h bounda y condi ions A η = 0 = 0 : (η) = 0, (η) = ʎ, (η) = − (1 − 𝜃(η))𝑓′ 𝜃 ′ 𝐵 𝑖 Fo 6→∞ ′→1, 𝜃 →0, (11) whe e P , R, M, Ec,Bi and ʎ s ands o P and l numbe , adia ion pa ame e , magne ic pa ame e , Ecke numbe , Bio numbe and eloci y a io pa ame e espec i ely and a e gi en by Case S udies in The mal Enginee ing 54 (2024) 104023 6 S. Saleem e al. Fig. 7. Impac o enhancing alues o ∅1,∅2on eloci y o luids. Fig. 8. Impac o enhancing alues o ∅1,∅2on empe a u e o luids. M= 𝜎 B0 2 b𝜌 ,P = 𝜇 (𝜌Cp) 𝜌 k ,R= 16𝜎1T∞ 3 3k1k , (12) In addi ion he a ios A1,A2, A3 ,A4and A5 a e gi en as A1= 𝜇hn 𝜇 ,A2= 𝜌hn 𝜌 ,A3= 𝜎hn 𝜎 ,A4= khn k ,A5= (𝜌Cp)hn (𝜌Cp) , (13) The wo quan i ies he skin ic ion coe icien (C ) and local Nussel numbe (Nux)a e o p ime impo ance om enginee ing poin o iew, which a e gi en by [31,32] C =𝜏w 𝜌 ue 2,Nux= xqw k (Tw−T∞), (14) Whe e qwis hea lux om he pla e and τwis su ace shea s ess along he pla e. They a e compu ed by 𝜏w=𝜇hn (𝜕u 𝜕y)y=0 ,qw= −khn (𝜕T 𝜕y)y=0 (15) By using equa ion (8) we ob ain Case S udies in The mal Enginee ing 54 (2024) 104023 7 S. Saleem e al. Table 3 Hyb id nano luid Al2O3–ZnO/Ke osene oil: Following ables depic he in luence o di e en pa ame e s on − ″(0), −θ′(0) o ∅1, = 0.1, 𝑀=𝑅= 0.5, = 1, ʎ= −1.2,∅2𝑃𝑟𝐸𝑐 = 0.7. ∅1∅2M R P ʎEc−θ′(0) - ″(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 Nano luid Al2O3/Ke osene oil. <! − − Col Coun ∶9− − >∅1 ∅2M R P ʎEc−θ′(0) − ″(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∕2C =A1 ′′ (0),Rex−1∕2Nux= −A4𝜃′(0). (16) He e Rex ep esen s he local Reynolds numbe . 3. Compu a ional me hod The sys em o equa ions (9) and (10) which is a se o nonlinea pa ial di e en ial equa ions wi h i s bounda y condi ions (11) a e compu ed nume ically by employing Kelle Box [33–35] echnique. Le ′=a, (17) a′=b, (18) 𝜃′= , (20) Equa ions (9) and (10) becomes A1 A2 b′+1−a2+ b +MA3 A2 (1−a)=0, (21) Case S udies in The mal Enginee ing 54 (2024) 104023 8 S. Saleem e al. Table 5 Nano luid ZnO/Ke osene oil. <! − − Col Coun ∶9− − >∅1 ∅2M R P ʎEc−θ′(0) − ″(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) ′+A5P +EcA1P b2=0, (22) And bounda y condi ions ake he o m ƞ= 0 𝑎=ʎ,𝑓= 0, 𝑡= − (1 − 𝜃) 𝐵 𝑖 6→∞a→1, 𝜃 →0, By applying ini e di e ence me hod j− j−1− hj 2(aj−aj−1)=0, (23) aj−aj−1− hj 2(bj−bj−1)=0, (24) 𝜃j−𝜃j−1− hj 2( j− j−1)=0, (25) Using (23) o (25) in (21) and (22), we ha e A1 A2bj−bj−1+1−hjaj+aj−1 22 + j+ j−1 2bj+bj−1 2+MA3 A21− aj+aj−1 2=0, (26) A4+R j− j−1+hjA5P  j+ j−1 2 j+ j−1 2+EcA1P bj+bj−1 22=0, (27) New on's me hod o linea iza ion jk+1= jk+𝛿 jk, (28) ajk+1=ajk+𝛿ajk, (29) bjk+1=bjk+𝛿bjk, (30) 𝜃jk+1=𝜃jk+𝛿𝜃jk, (31) jk+1= jk+𝛿 jk, (32) So equa ions (23)–(25) becomes 𝛿 j−𝛿 j−1− hj 2{𝛿aj−𝛿aj−1}=(Q1)j, (33) Case S udies in The mal Enginee ing 54 (2024) 104023 9 S. Saleem e al. Table: 6 Compa ison o he Skin ic ion coe icien wi h R=M=EC =∅1=∅2= 0. ʎ[29] P esen s udy 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{𝛿 j−𝛿 j−1}=(Q3)j, (35) Whe e (Q1)j= j−1− j+hjaj−1 2 ,(Q2)j=aj−1−aj+hjbj−1 2 and (Q3)j=𝜃j−1−𝜃j+hj j−1 2 , So equa ions (26) and (27) becomes C1𝛿aj+C2𝛿aj−1+C3𝛿 j+C4𝛿 j−1+C5𝛿bj+C6𝛿bj−1=(R1)j, (36) D1𝛿 j+D2𝛿 j−1+D3𝛿 j+D4𝛿 j−1+D5𝛿bj+D6𝛿bj−1=(R2)j, (37) Whe e