Rakesh Kumar, Ravinder Kumar, Reena Koundal, Sabir Ali Shehzad,and Mohsen Sheikholeslami

1Department of Mathematics,Central University of Himachal Pradesh,Dharamshala,India

2Department of Mathematics,COMSATS University Islamabad,Sahiwal 57000,Pakistan

3Department of Mechanical Engineering,Babol Noshirvani University of Technology,Babol,Iran

Abstract In this problem,simultaneous effects of Joule and viscous dissipation in three-dimensional flow of nanoliquid have been addressed in slip flow regime under time dependent rotational oscillations.Silver nanoparticles are submerged in the base fluid (water) due to their chemical and biological features.To increment the novelty,effects of cubic autocatalysis chemical reactions and radiative heat transfer have been incorporated in the related boundary layer equations.Dimensionless partial differential system is solved by employing the proposed implicit finite difference approach.Convergence conditions and stability criteria are obtained to ensure the convergence and accuracy of solutions.A comparative analysis is proposed for no-slip nanofluid flow (NSNF) and slip nanofluid flow (SNF).Variations in skin-friction coefficients,Sherwood and Nusselt numbers against physical parameters are tabulated.It is investigated that velocity slip and temperature jump significantly control drag forces and rate of heat transfer.

Key words: cubic autocatalysis reactions,viscous dissipation,Joule heating,rotating-fluctuating environment,slip flow regime

Nomenclature

(Continued)

1 Introduction

In recent years,scientists and engineers who worked in the field of nanoscience and nanotechnology have revived their interest in exploring the new applications of nanomaterials (1 nm–100 nm) and nanostructures.The use of these materials can provide solutions to numerous environmental and technological challenges such as solar energy conversion,medicine,catalysis and water purification as pointed out by Anstas and Warner.[1]In this direction,silver colloids have a point of attraction because of their unique features include good conductivity,chemical stability,antibacterial nature and catalytic activity.[2]There exists number of method for the preparation and synthesis of silver colloids like chemical,photochemical,and physical but the disadvantages like hazardous waste,expensiveness and toxic substances have unmotivated researchers to use these techniques.Therefore,the focus of researchers in present time is to use green synthesis methods through nontoxic chemicals and environment friendly solvents (water).Sharmaet al.[3]in their review report elaborated different green synthesis techniques (Tollens method and polysaccharide method,biological method,irradiation method,and polyoxometalates method) for Ag-NPs.They also incorporated Ag-NPs into other materials (silver-doped hydroxyapatite,polymer-Ag-NPs,poly(vinul alcohol)-Ag-NPs,Ag-NPs on TiO2),and illustrated their applications in antibacterial water filter,antimicrobial air filter and treatment of HIV.

Silver nanoparticles find their common applications in bone cement,wound dressing,house cleaning chemicals,washing machines,bactericidal coatings in water filters,sprays,respirators and many more (Prabhu and Poulose[4]).Tranet al.[5]described the use of silver nanoparticles in controlling antibacterial,antifungal,antiviral activities,and environmental treatments such as air disinfection,water disinfection (drinking water,groundwater,biological waste water),surface disinfection (plastic catheters antimicrobial surface functionalization,antimicrobial paints,food preservation through antimicrobial packing of paper and silver-impregnated fabrics for clinical clothing).He further demonstrated that in future silver nanoparticles can prove powerful in controlling and preventing microbial infections,waterbrone diseases through magnetic disinfectant system,environmental pollution through effective sorbent and catalyst etc.Carboneet al.[6]explained the role of silver nanoparticles for fresh food packaging in the polymeric matrices.Zhanget al.[7]illustrated various properties of Ag-NPs including antiangiogenic and anti-cancer agents.Seyhanet al.[8]presented an experimental analysis to examine the influence of functionalized silver nanopaticles over thermal conductivity of water,hexane and ethylene glycol.Peristaltic movement of silver nanofluid through a symmetric channel was analyzed by Abassiet al.[9]Zinet al.[10]reported silver nanofluid flow induced by the oscillation of vertical plate.

On the other hand,nanofluid flows over deformable surfaces with time dependent rotational oscillations have been the concern of several researchers due to the broader spectrum of their applications in food processing,fiber technology,polymer industry,and metallurgical sciences.Matched asymptotic expansion scheme for higher suction was exploited by Wang[11]to understand the flow created by the oscillatory stretching velocities of the sheet.Slip effects in oscillatory environment were investigated by Abbaset al.[12]Impacts of Dufour and Soret on fluctuating flows were analyzed through homotopy analysis method by Zhenget al.[13]Keller box method was employed by Javedet al.[14]to discuss the oblique stagnant flow due to fluctuating plate.Khanet al.[15]accomplished numerical solution for the Bodewadt nanofluid flow caused by a deformable disk.Kumar and Sood[16]explored the unidirectional surface stretching conditions under rotating,oscillatory,and radiating and reacting environment.Impact of rotational oscillations in stretchable nanofluid flow under cubic auto-catalysis reaction environment is illustrated by Kumaret al.[17]

In addition to this,work done by velocity against viscous stresses (transformation of mechanical energy into thermal energy) is termed as viscous heating of energy.Viscous dissipation mechanism is dominating in those regions where large velocity gradients exist,for example,in boundary layers or where large variations in buoyancy force exists or devices which move at high speeds.Also,dissipation term cannot be ignored in the flow of highly viscous fluids (polymers,oils),and further,presence of fluctuating velocities on the surface can significantly alter the dissipative heat.Therefore,it becomes important to examine the influence of rotational oscillations on the dissipative heat through the Eckert number.The pioneering work of Brinkman on viscous dissipation led the researchers to investigate the various aspects of the influence of dissipation.On the other hand,when the energy is transferred from conduction electrons to conductor’s atoms via the collision process,heat is always generated.This type of transformation gives rise to an important term called as Joule heating.It plays a healthy role in the design of numerous devices.Singh and Kumar[18]elaborated the combined importance of viscous heating in hydromagnetic fluid flow over hot vertical surface.Ramet al.[19]discussed the work of Ref.[18]by adding the aspect of solar radiation.Simultaneous impacts of viscous and Joule heating in nanofluid flow near the stretchable stagnation region are discussed by Nandkeolyaret al.[20]Ahmad and Iqbal[21]inspected the reactions of reactions(heterogeneous/homogeneous) in ferrofluid flow with viscous heating.Recently,Khanet al.[22]presented a modified model for heterogeneous and homogeneous reactions in flow of MHD fluid with viscous/Ohmic dissipations.

Though no-slip boundary conditions have been widely used to analyze Navier-Stokes equations yet there are some situations where these conditions fail to predict the exact dynamics of flow field such as thin film flow of light oil relative to the moving plates,thick monolayer of hydrophobic octadecyltrichlorosilane coated on surface,multiple interface problems,non-Newtonian fluid (polymeric materials) flows and flows involving the suspensions of nanoparticles.Rao and Rajagopal[23]represented different models available for the description of slip flows,and they observed that rectilinear and fully developed flow is impossible if velocity has strong dependence on normal stresses.Mehmood and Ali[24]inspected slip condition effects on oscillatory flow in a planer channel.Singh and Kumar[25]investigated fluctuating flow of of reacting and radiating liquid along vertical porous plate with slip-flow.Kumar and Chand[26]revealed the influences of Hall currents and velocity slip interface conditions on viscoelastic fluid flow over flat surface.Darcy and rarefaction impacts on radiative-reactive flows inside channel were examined by Chandet al.[27]For more relevant works,readers can look through (Refs.[28–36]).

The role of Joule and viscous dissipation in the design of various equipments and devices motived us to examine their effects on the nanofluid flow past surfaces for rotational oscillations (time dependent) under slip interface condition environment.To enhance the impact of the analysis,radiation(heat transfer)and cubic autocatalysis chemical reaction effects are also added to the energy and mass diffusion equations respectively.

2 Geometrical Description and Governing Equations

Here we focus ourselves on an unsteady flow of convective nanofluid which is viscous,electrically conducting and incompressible.The nanofluid is formed by the suspension of silver nanoparticles and the flow is subjected to time dependent rotational oscillations of the deforming sheet.In the flow field we consider isothermal cubic homogeneous chemical reaction and on the catalyst surface we consider single isothermal first order heterogeneous chemical reaction (Fig.1).

Fig.1 (Color online)Schematic description of the problem.

The homogeneous chemical reaction is defined as:

and heterogeneous reaction on the catalyst surface as

wherea,bdenote the chemical species.

The stretchable sheet in this problem executes oscillations and preserves velocity jumps in normal direction.This effect is mathematically defined as

The sheet temperature is considered to vary periodically along difference of mean temperature of surface and free-stream temperature.the temperature jump is assumed in normal direction.Mathematical form of such temperature is

Following Tiwari and Das[37]and Merkin,[38]the governing radiating and reacting boundary layer equations subjected to velocity/temperature jumps and viscous/Ohmic dissipations under time dependent rotational oscillations can be written as:

Further,the conditions (initial/boundary) have the following mathematical presentation:

For the complete construction of mathematical model elaborating the flow,and heat and mass transfer mechanisms under rotational oscillatory environment,following models are employed (Tiwari and Das model,[37]Hamilton-Crosser model,[39]Makinde and Animasaun,[40]Makinde and Animasaun[41]):

In the dynamics of fluid flows we are interested in those results which are independent from dimensions and thus we introduce the following non-dimensional quantities which will reduce Eqs.(5)–(10) along with Eq.(11)into dimensionless form:

After utilization of these dimensionless quantities,following set of coupled partial differential equations is achieved:

where

The dimensionless transformations convert the initial and boundary conditions into

The emerging dimensionless parameters are defined as follows:

The coefficients which attract engineers most are skinfrictions,Nusselt number and reactant Sherwood numbers.These have the following respective forms in fluid dynamics:

where

These coefficients in dimensionless pattern can be readily decided as:

3 Numerical Scheme

An explicit finite difference approach has been exploited to compute the solutions of involve mathematical expressions.Using explicit finite difference approximation,Eqs.(14)–(19) are written in the following finite difference form:

The corresponding boundary conditions are

In the above proposed scheme,l,m,n(subscripts)andk(superscripts) represent the grid nodes in the directions ofX,Y,Zandt′respectively.We selected(Xmax,Ymax,Zmax,t′max)=(100,100,10,10) after confirming that the conditions may not change outside this range.Here the step-size is considered to be(∆t′,∆X,∆Y,∆Z)=(0.005,10,10,0.1).The algorithm is presented as follows:

(i) Initially (t′=0),we assume thatare known.

(iii) At first,we findat every node.

(iv)At second,at every node is computed using

(vi)Lastly,are calculated through the equation of continuity (22).

4 Stability and Convergence

To obtain the stability conditions,we take Fourier expansions for all the quantities (U,V,θ,A,andB) at timet′and constant grid sizes as(Von Neumann stability analysis):

After single time step,these quantities have the forms:

Now,we substitute these assumptions into Eqs.(22)–(27),and obtain the following equations after trivial simplifications:

Further simplification gives rise to a following set of equations:

which in matrix form can be written asAX=BBB.The entries of the coefficient matrixAAAare:

HereWis non-positive andUandVare non-negative everywhere.Eigenvalues of this matrix (AAA) areλ1=(A21+A22)/A1,λ2=A1,λ3=A4,λ4=A7,andλ5=A8.For stability of present scheme,we should receive|λi|≤1 fori=1,...,5.Letp1,p2,andp3are odd integers,maximum modulus of above mentioned eigenvalues takes place whenα3△Z=p3π,α2△Y=p2π,andα1△X=p1π.To satisfy|λi| ≤1,the most negative admissible value isλi=−1.Hence stability conditions are

Hereχ=U(△t′/△X) +V(△t′/△Y) +|W|(△t′/△Z).Keeping in mind these stability conditions,computations have been carried out to accomplish numerical quantities (U,V,W,θ,A,B) and (cfX,cfY,NuX,ShAX,ShBX)which will be presented and analyzed in next section.

5 Results and Discussion

To acquaint with the mechanisms of heat and mass transportation in the nanofluid flow subjected to viscous and Joule dissipation past deforming surfaces in slip flow regime,numerical quantities obtained in previous section are analyzed through plots and tables for different values of emerging parameters.Thermophysical qualities of silver (Ag) and water (H2O) nanoparticles are presented in Table 1.Accuracy/validation of present outcomes have been portrayed in Table 2.Results of primary and secondary velocity(U,V),temperature(θ),homogeneous concentration (A) and heterogeneous concentration (B) are presented graphically.The skin-friction(coefficients) inXandYdirections (cfXandcfY),local Nusselt number (NuX) and reactant Sherwood numbers(ShAX,ShBX) evaluated through tables.Figures and tables are prepared to exhibit the relative importance of two cases to name as NSNF (no-slip nanofluid flow) and SNF (slip nanofluid flow).If not mentioned differently,then values assigned to different parameters are:ω=0.1,ϕ=0.1,n=3,Ec=0.01,M=3,K=3,R=1,Ra=0.4,Ks=1,Kc=0.5,Vs=1,Ts=1,Gr=15,Sc=3.5,δ=1.2,andPr=6.07.

Table 1 Properties of H2O and Ag at 25◦C.

Table 2 Comparison of the variability of skin friction coefficients(cfX/cfY)and local Nusselt number NuX for different Ra at ω=0,Vs=1,Ts=0,R=0.5,M=1,K=1.5,Ks=3,Kc=0.5,Ec=0.001,ϕ=0.1,and Gr=10.

5.1 Effect of Oscillation Frequencies (ω)

Fig.2 (Color online) Outcomes of frequency of oscillation (ω).

Figures 2(a)–2(e) illustrate the influence ofωon the profiles of (U,V),θandAandB.Under both the cases,Uandθare reduced withωbutVis enhanced with it.As magnitude of frequency of oscillation is raised,primary velocity starts to behave as secondary velocity and reverse flow patterns start to appear inUcomponent.Thus to curtail reverse flow,optimal value ofωcan be exploited.Further,with the increasingω,negative temperature profiles appear owing to inverted Boltzmann distribution.Same negative profiles have been obtained by Kumar and Sood.[16]In the vicinity of the sheet velocity is higher in SNF situation in comparison to NSNF case.Therefore in SNF case,more heat will be transferred by this augmented velocity component and the result will be reduced temperature.That is why,temperature profiles are lowered in SNF case.Homogeneous concentration (A) is reduced whereas a growth is noticed in heterogeneous concentrationBwith increasing oscillation frequency.

Fig.3 (Color online) Outcomes of frequency of oscillation (ω) against t′.

Figures 3(a)–3(e)portray the control of frequency of oscillation oncfX,cfY,NuX,ShAX,andShBX.As expected,fluctuating nature ofcfX,cfY,andNuXis encountered when time constraint is varied.Whereas prominent oscillations inShAXandShBXare disclosed fort′>2.Interesting part prevails atω=(52/100)πwhere|cfX|is not altered by time,but minima ofcfYis augmented for higher values oft′.Peak points ofNuXexist atω=(53/100)π,which get uplifted witht′by maintaining constant minima.

5.2 Effect of Eckert Number (Ec)

Figures 4(a)–4(c) determine the effect of Eckert number on (U,V) andθin the boundary layer.Magnitudes of primary and secondary velocities as well as temperature are elevated with the heightened values ofEcin both the cases.Since,Eckert number is the ratio of the product of stretching rate and kinematic viscosity to temperature difference,therefore asEcis boosted up,physically fluid velocity should increase.On the other hand,higher values ofEcincrease viscous dissipation impact therefore fluid temperature is augmented.

5.3 Effect of Volume Fraction of Nanoparticles (ϕ)

Figures 5(a)–5(c) elucidate variations in the profiles ofU,V,andθwith respect toϕin SNF and NSNF cases.It is noted that magnitude ofVandθare strengthened withϕeverywhere inside the boundary layer,howeverUis diminished in the vicinity of sheet.The behavior ofUis altered after crossing the point of inflections in the flow field.Physically this is true because increasing values ofϕraise nanofluid viscosity which hampers the resultant velocity,that is why,Uis compressed.

Fig.4 (Color online) Outcomes of Eckert number (Ec).

Fig.5 (Color online) Outcomes of nanoparticles volume fraction (ϕ).

5.4 Effect of Homogeneous/Heterogeneous Reaction Parameters (Kc/Ks)

Figures 6(a)–6(d) highlight the importance ofKcandKson the concentrations profilesAandB.It is examined that homogeneous concentration enhances and heterogeneous concentration retards with the increasing generative homogeneous reactions (Kc<0).However,opposite trends are noticed with respect to destructive homogeneous reactions(Kc>0).It is also deduced from mentioned figures that constructive and destructive heterogeneous reactions(Ks<0,Ks>0) have similar effects onAandBasKc<0 andKc>0 have onAandB.

5.5 Effect of Schmidt Number (Sc)

Figures 7(a)–7(d) accentuate the relevance of Schmidt numberScin the distribution of homogeneous and heterogeneous concentrations profiles.Schmidt number tends to raise the homogeneous concentration profiles (A) in both the cases of generative and destructive homogeneous reaction parameters,on the other hand,it tends to lower the heterogeneous concentration profiles (B).An interesting effect ofSchas been noticed in the case of heterogeneous reactions.Homogeneous concentration profiles (A) are diminished with respect toKs<0 and elevated withKs>0.However,opposite behavior has been detected in the case of heterogeneous concentration profiles (B).

Fig.6 (Color online) Outcomes of Kc and Ks on concentrations (A and B).

Fig.7 (Color online) Outcomes of Schmidt number (Sc) on concentrations on A and B for different Ks and Kc.

5.6 Effect on Skin-Friction,Reactant Sherwood Numbers and Nusselt Number

Effects ofM,K,andRon skin friction factors (cfX,cfY),reactant Sherwood numbers (ShAX,ShBX) and Nusselt number(NuX)are presented in Table 3 and Table 4 to emphasize the relative importance of slip nanofluid flow(SNF)over no slip nanofluid flow (NSNF).In SNF case,magnitudes ofcfXandcfYare reduced withMandK,whereas,RreducescfXand enhancescfYin magnitudes.NuXis aggrandized byK,butMandRcurtail it.Magnitudes ofShAXandShBXare lowered withM,K,andR.In general,increasingMandKhave the tendency to raise skin friction coefficients,but here time dependent rotational oscillations have played a predominant role in inverting this physical phenomenon.It is pertinent to mention here that in NSNF case,the quality of trends are same as that of SNF case but the major difference is that the magnitudes of skin-friction coefficients are much smaller and rates of heat transfer are much higher in SNF situation in comparison to NSNF case.

Table 3 Variability of skin friction coefficients (cfX/cfY),local Nusselt number NuX and Sherwood numbers(ShAX/ShBX) for different values of M,K and R in SNF case (Vs=Ts ≠0).

Table 4 Variability of skin friction coefficients (cfX/cfY),local Nusselt number (NuX) and Sherwood numbers(ShAX/ShBX) for different values of M,K and R in NSNF case (Vs=Ts=0).

Table 5 Variability of skin friction coefficients (cfX/cfY),local Nusselt number (NuX) and Sherwood numbers(ShAX/ShBX) for different values of Ra,ω and Ec in SNF case (Vs=Ts ≠0).

Influences ofRa,ω,andEcon (cfX,cfY),(NuX),and (ShAX,ShBX) are highlighted in Table 5 and Table 6.In SNF case,magnitudes of (cfX,cfY) and (ShA,ShB) are hiked up byRaandEc,howeverNuXis depressed withRaandEc.Magnitudes of reactant Sherwood numbers (ShAX,ShBX) are also hampered withω.The most interesting phenomenon is created by the rotational oscillations of the stretching surface.For lower values of frequency of oscillations,skin-friction coefficients are raised but for broader values ofωthese coefficients (cfX,cfY) are reduced.Same trends can be seen in Table 6 for NSNF case.It is worthy to indicate that frequency of oscillation (ω=0.2) can be exploited to turn the behaviour of skin friction coefficients and Nusselt number.Maxima of rate of heat transfer(NuX) exists atω=0.2.Here,the point of interest is that magnitudes ofcfX,cfYandShAX,ShBXare smaller,andNuXare larger in SNF case than in NSNF case.

Table 6 Variability of skin friction coefficients (cfX/cfY),local Nusselt number (NuX) and Sherwood numbers(ShAX/ShBX) for different values of Ra,ω and Ec in NSNF case (Vs=Ts=0).

6 Conclusions

Three-dimensional boundary layer equations for the nanofluid flow under time dependent rotational oscillations are proposed considering viscous/ohmic dissipations,and slip conditions on the interface (wall-fluid).For novelty,cubic auto-catalysis chemical reactions in mass diffusion equations and radiative heat transfer in energy equation are incorporated.The assumptions and algorithm for the solution of governing equations are presented.Based on above results captured from this study,following conclusions are made:

•Increasing values ofωtend to reverse the flow behaviour as well as distribution of temperature inside the boundary layer.

•Eckert numberEcaugments nano-fluid velocity as well as its temperature.

•Nano-fluid temperature is lesser in SNF case in contrast to NSNF case,whereas (U,V) are larger in SNF problem as compared NSNF situation.

•HighercfXandcfYdue toMandKcan be curtailed by introducing time dependent rotational oscillations on the surface.

•Interface slip conditions help in reducing skinfriction (coefficients),and in enhancing multiple transfer rates.

•Critical frequencyω=(52/100)πorω=(53/100)πis useful for enhanced transfer rates (heat/mass).

Thus we can say that interface velocity slip and temperature jump in presence of viscous/Ohmic dissipations play a substantial role in controlling skin-friction (coefficients) and heat/mass transportation rates when treated as a function of time.