# Slip Effects on Unsteady Oblique Stagnation Point Flow of Nanofluid in a View of Inclined Magnetic Field

Authors: Rizwana Rizwana; Azad Hussain; S. Nadeem
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This study may be applicable in heavy power engine and cooling of a nuclear reactor, insulation for buildings, petroleum reservoir operations, and magnetic material processing solar energy collectors. In this manuscript, the slip results are evaluated for the non-Newtonian fluid on the oblique stagnation point flow of induced magnetic field over the oscillating surface. The valuation of heat flux is examined through the Fourier law of heat transfer. The metallic nanoparticle CopperCu is within the base fluid, and water is utilized in the analysis. Nanofluids have benefits such as steadiness of the working fluid, decreasing blockage, clogs, decreasing prices, decreasing the friction coefficient, and decreasing the size of the heat transfer system. Similarity variables are utilized to convert the developed flow into higher nonlinear coupled ordinary differential equations (ODE) which are tackled numerically using a mathematical technique such as the bvp4c method in Maple and Matlab software. According to the present geometry, the flow behavior of the operating nanofluid has analyzed by stream lines. Disparities in velocity and temperature profile are demonstrated by graphs to describe the effects of controlling parameters. The Casson fluid parameter enhances the velocity of the fluid. The system heats up by the impact of Joule heating and dissipation.


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## Body

## 1. Introduction

Due to the small size of the nanoparticles, these fluids behave like normal fluids first introduced by Choi and Eastman [1]. Nanofluids are essentially utilized to improve heat conduction and have many applications in several business processes such as heat exchanger equipment and electronic cooling system radiators given by Khan et al. [2]. Due to narrative properties of nanofluids, it is potentially valuable in numerous applications in heat transfer, such as car engines, microelectronics, engine cooling/vehicle thermal management, chiller, nuclear reactor, heat exchanger, domestic refrigerator, fuel cells, pharmaceutical processes, coolant, in grinding, defense, in boiler flue gas temperature decline, in space technology, machining, and ship given by Turkyilmazoglu [3]. Nanofluids have benefits such as steadiness of the working fluid, decreasing blockage, clogs, decreasing prices, decreasing the friction coefficient, decreasing the size of the heat transfer system, improving steadiness related to other colloids, decreasing the power required for fluid pumping, increasing heat transfer capability, and improving the heat transfer. Several researchers have studied the nanofluid flow [4–13].Hiemenz [14] was the first person who studied the stagnation point flow by finding the exact solution to the Navier–Stokes equations. Stagnation point flows can be unsteady or steady, inviscid or viscous, oblique or normal, two or more dimensional, and opposite or onward. Alike flow frequently compacts with the motion of fluid close to the stagnation region of a firm surface preserved in moving fluid given by Khan et al. [15]. Stagnation point flows beside the heat transfer characteristic are fairly manifested in rotating filaments, paper manufacture, melt spinning process, crystal puffing, and continuous molding given by Abbas et al. [16]. A great number of researchers have put their attention on stagnation point flows [17–26].Non-Newtonian fluids have a huge range of industrial applications, such as high molecular weight systems, most multiphase mixtures, foods, and solutions. When air is involved in such procedures as agitation, mixing, dispersion, and kneading, then gas bubbles are isolated and assorted in fluids given by Zhan et al. [27]. Materials displaying non-Newtonian flow characteristics include natural products, food products, magmas and lavas, biological fluids, dairy wastes, and agricultural wastes, building materials, soap solutions, polymer melts and solutions, multiphase mixtures, and personal care products including cosmetics and toiletries, given by Ijaz et al. [28]. Non-Newtonian fluids are of excessive importance in engineering sciences, medical field, and industry. The increasing applications of non-Newtonian fluids are in demonstrating of extrusion of polymers, steel substances, petroleum drilling, and blasting of glasses. Few polymers are utilized in communication appliances, medical, and agriculture given by Hussain et al. [29]. Many researchers have studied the non- Newtonian fluid flows [30–40].In the present study, we describe the non-Newtonian fluid for oblique stagnation point flow with the induced magnetic field of nanofluid over the oscillatory and slip surface. Heat transfer is exposed to Cu nanoparticles. The coupled differential equations are solved by mathematical techniques such as bvp4c in MatLab and Maple. Graphical results are achieved for velocity and temperature profiles. Moreover, the flow behavior is also demonstrated by streamlines.


## 2. Formulation

Assume the two-phase flow model for oscillatory, unsteady, and incompressible two-dimensional state flow. The slip results are evaluated for the non-Newtonian fluid on the oblique stagnation point flow of induced magnetic field over the oscillating surface. The stream of nanofluid occupies obliquely on a plate vacillates in its own plan. The fluid isy≥0, the upper half of the plane. The flow equations are given as(1)∂u∂x+∂v∂y=0,(2)∂u¯∂t+u¯∂u¯∂x¯+v¯∂u¯∂y¯=−1ρdpdx¯+vnf1+1β∂2u¯∂x¯2+∂2v¯∂y¯2−4a2σnfρnfβo24a2+b2u¯,(3)∂v¯∂t+u¯∂v¯∂x¯+v¯∂v¯∂y¯=−1ρdpdy¯+vnf1+1β∂2u¯∂x¯2+∂2v¯∂y¯2−4a2σnfρnfβo24a2+b2v¯,(4)αnf∂2T∂x¯2+∂2T∂y¯2=∂T∂t+u¯∂T∂x¯+v¯∂T∂y¯,with boundary conditions(5)u¯=Ucosωt¯+λ1μnfμf∂u¯∂y¯,v¯=0,T¯=T¯w+εT¯w−T¯∞Reeiωt¯+λ2knfkf∂T¯∂y¯,y¯=0,u¯=aγy¯+x¯,v¯=−ay¯,y¯⟶∞,where ρnf,B0,γnf,αnf,vnf,μnf,λ1,λ2,p¯=p¯x¯,y¯,t¯,andV¯=u¯x¯,y¯,t¯,v¯x¯,y¯,t¯,0 are the density of nanofluid, magnetic field, thermal diffusivity of nanofluid, kinematic viscosity, dynamic viscosity, slip parameters, velocity field, and pressure, respectively.Suppose fluid obliquely affects via the planey¯=A and the velocity components are(6)u¯=ax¯+by¯−B,v¯=−ay¯−A,A,BParameters.Now, the stagnation point developsA,b/aB−A, and the streamlines are hyperbolas whose asymptotes are(7)y¯=−2abx¯−A+2B,y¯=A.Moreover, the magnetic field must be equivalent to distributing streamline due to presence of oblique stagnation flow as(8)H¯0=B04a2+b2−bi^+2aj^,where B0 is the uniform magnetic field and aandb are constants.Consider the velocity component as(9)u¯=axf¯′y¯+bg¯y¯,v¯=−af¯y¯.The streamlines’ flow becomes hyperbolic nature when considering oblique stagnation flow, and their asymptotes may have the equation(10)y¯=0,y¯=−2abx¯.The boundary conditions are given by(11)f¯0=0,f¯′0=λ1μnfμff¯″0,g¯0,t¯=ReUbeiωt¯+λ1μnfμfg¯′0,t¯,(12)f¯′y¯=1,g¯′y¯,t¯=1,y¯⟶∞.From (12), the behavior of f¯ and g¯ asymptotically at infinity is given by(13)f¯∼y¯−A,g¯∼y¯−B,y¯⟶∞.The continuity equation (1) deduces the existence of a stream function ψ such that(14)u=∂ψ∂y¯,v=−∂ψ∂x¯.Applying equation (14) in (2) to (4), eliminating the pressure using this pxy=pyx yields, also making use of equation (10) in system (2) to (4), and using boundary conditions (11) and (12), in equations (14) to (17), we obtain(15)1avnf1+1βd3f¯dy¯3+f¯d2f¯dy¯2−df¯dy¯df¯dy¯−4a2σnfρnfβo24a2+b2=−4a2σnfρnfβo24a2+b2−1,(16)1avnf1+1β∂2g¯∂y¯2+f∂g¯∂y¯−∂g¯∂y¯df¯dy¯−∂g¯∂t¯+4a2σnfρnfβo24a2+b2f¯−g¯=1+4a2σnfρnfβo24a2+b2B−A.We define(17)fy¯=VfkFy,gt¯,y¯=VfkGoy+εG1yeiτ,y=kVfy¯,Ω=wk,τ=wt,ε=UVfk,θoy+εθ1y=T−T∞Tw−T∞.In the light of equation (21), governing equations (4), (15), and (16) become(18)μnfμfρfρnf1+1βF‴−F′2+FF″+1+ρfρnfM21−F′=0,(19)μnfμfρfρnf1+1βGo″+Go′F−GoF′−ρfρnfM2f−Go−1+ρfρnfM2δ−α=0,(20)μnfμfρfρnf1+1βG1″+G1′F−G1F′−iΩG1′−ρfρnfM2G1′=0,(21)knfkfθo″+prρCpSρCpfFθo′=0,(22)knfkfθ1″+prρCpSρCpfFθ1′−iΩprρCpSρCpfθ1=0,(23)f0=0,f′0=N1μnfμff″0,g00=N1μnfμfg0′0,g10=1+N1μnfμfg1′0,(24)θ00=1+N2knfkfθ0′0,θ10=1+N2knfkfθ1′0,(25)θ1y=0,f∼y−α,g∼y−β≤y¯⟶∞.In whichϕ,ρnf,ρf,ρs,αnf,knf,ks,kf,ρnf,ρCpnf,Pr,Ω,M,andϵ symbolized for nanoparticles volume fraction, the heat capacity of nanofluid, the thermal conductivity of solid fractions fluid, thermal conductivity of nanofluid, densities of fluid and solid fractions, Prandtl number, dimensionless frequency, Hartmann number, and amplitude of the wave, respectively, which are defined as(26)μnf=μnf1−φ2.51−φρf+φρs,αnf=knfρCpnf,ρnf=φρs+1−φρf,knfkf=ks+2kf−2φks−kfks+2kf−φks−kf,ρCpnf=φρCps+1−φρCpf,δ=avfB,α=avfA,Ω=ωa,N1=avfλ1,M2=4aσfρfB024a2+b2,Pr=νfkf,N2=avfλ2.The Skin friction number can be extolled as(27)Cf=τw1/2ρfUw2,and the shear stress is declared by(28)τw=μnf∂u¯∂y¯y¯=0,Nondimensional form of equation (39) takes the form(29)12Rex¯Cf=μnfμfρfρnfRef″0+bag0′0−εg1′0eit,where Re=kx2/υf is the local Reynolds number.The nondimensional stream function is(30)ψ∗=ψVf=xFη+Gη.Displayed in Figure1, the fluid flow forms an angle α with the plate, and the slope of the straight line can be found by putting ψ∗=0 as ψ=1/2γy2+xy, where η=−2/γx which gives slope = −2/γ. Hence, the γ (shearing parameter) and α (impinging angle) relationship is(31)α=tan−1−2γ.Figure 1
Geometry of the flow problem.The Nusslet numberNu is presented as(32)Nu=xqwkfTw−T∞,and heat flux is computed by(33)qw=−knf∂T∂yy=0.Dimensionless form of equation (40):(34)Rex−1/2Nu=−knfkfθ0′0+εθ10eiτ.


## 3. Solution Process

Numerical solution of equations (18)–(22) is achieved using BVP solution technique built in Matlab and Maple software. It can be perceived that leading classification (18)–(22) has one-way coupling, i.e., fy effects G0y,G1y,θ0y, and θ1y but not vice versa. Also, for minor values of Ω, a series solution of equations (20) and (22) has been obtained followed by(35)G1y=∑n=0∞iΩnχny,θ1y=∑n=0∞iΩnϑ1ny.We are taking the solution of the real part only, as the form(36)G1y=χ0y−Ω2χ2y+Ω4χ4y.Using equation (35) in equations (20) and (23), we obtain(37)μnfμfρfρnf1+1βχo″+χo′F−χoF′−ρfρnfM2χo=0,(38)μnfμfρfρnf1+1βχn‴′+χn″F−χn′F′−ρfρnfM2χo=χn−1,(39)χo0=0,χo′0=1,χo′∞=1,(40)χn0=0,χn′0=0,χn′∞=0,n=1,2,3,….For equation (22), the series solution can also be explained in a similar way:(41)θ1y=∑n=0∞iΩnϑ1ny.For minor values ofΩ (considering the real part only), we obtain(42)θ1y=ϑ10y−Ω2ϑ12y+Ω4ϑ14y.We have observed thatϑ10y=θ0y, and solving θ0y by direct integration, we obtain(43)θ0y=Jnf∞,Pr−Jnfy,PrJnf∞,Pr,Jnfy,Pr=∫0yExp−PrkfknfρCpnfρCpf∫0sfηdηds.Finally, the series form solution of the temperature equation (22) becomes(44)1prknfkfϑ1n+″ρCpnfρCpffϑ1n′−ϑ1n−1=0,ϑ1n0=1,ϑ1n∞=0,n=1,2,3,….Thermophysical properties of base fluid and nanoparticle are composed by [21].


## 4. Discussion and Results

The problem is solved by a mathematical technique such as bvp4c method in Matlab and maple and expresses the results by various parameter such as Hartmann numberM, Casson parameter β, slip parameter N1, and thermal slip parameter N2. We have illuminated graphically the behavior of flow parameters. Figures 2–7 exposed the velocity profiles commonly, and the nanofluid flow on the vacillating plate is operated through the mutual action of free stream velocity and magnetic field. At the plate surface, the nanofluid velocity is zero and rises slowly till it achieves the free stream value, satisfying the given boundary conditions. Figure 2 expresses the action of the velocity gradient Fy,F′y,F″y, which is described for M=10−7andβ=0.5. Figure 3 shows the impact of Hartmann number M on the velocity profile. When M rises, then the velocity field declines and the viscosity of the boundary layer increases. On the fluid in the boundary layer, a resistive nature force produced by a magnetic field drops the motion of the fluid and the existing phenomena arise when it can describe the conductive fluid. Hence, lastly, it is expected that the magnetic field is utilized to curb boundary layer departure. Figure 4 demonstrates the effect of slip parameter N1 on the velocity field. When N1 rises, the velocity profile enhances and the boundary layer declines. Figure 5 explored the effect of Casson parameter β on the velocity profile. When Casson parameter β increases, then the velocity field increases and the thickness of the boundary layer declines. On the other way, for Casson fluid, it is observed that the velocity boundary layer thickness is greater as compared to that of Newtonian fluid. Figures 6 and 7 expose the effects of the velocity profile G′y rises and the boundary layer drops when the values of Casson parameter β and Hartmann number M increases. Figures 8 and 9 show the temperature profiles θy when Pr=6.2; then, choose the values of thermal slip parameter N2 and Hartmann number M. Thermal slip and hartmann number decrease the fluid temperature. The boundary layer drops, and the temperature gradient at the surface also decreases. Physically, the momentum thickness and thermal boundary layer are functioned by Prandtl number. Low Prandtl number fluids have a greater rate of heat conduction; thus, heat can spread faster than for higher Pr fluid. Figures 10–21 demonstrate the effect of M, β, and N1 on the stream line. Table 1 is made to examine the impact of M and γ on Skin friction; when the slip parameter N1 rises, then the Skin friction enhances; when the Hartmann number M increases, then Skin friction declines; and when Casson parameter increases, then the Skin friction also drops. Table 2 is made to explore the effect of Pr and γ on Nusselt number; when thermal slip parameter N1 and volume friction φ increases, then the Nusselt number enhances.Figure 2
ExposingFy,F′y,andF″y for M=1.5,β=0.5,andγ=0.5.Figure 3
Influence ofM on F′.Figure 4
Variation ofN1 on F′.Figure 5
Influence ofβ on velocity component F′.Figure 6
Inspiration ofM on velocity component G′.Figure 7
Result ofN1 on G′y.Figure 8
Influence ofN2 on θ.Figure 9
Influence ofM on θ.Figure 10
Stream lines forM=0.Figure 11
Stream lines forM=0.3.Figure 12
Stream lines forM=0.5.Figure 13
Stream lines forM=0.7.Figure 14
Stream lines forβ=0.Figure 15
Stream lines forβ=1.Figure 16
Stream lines forβ=2.Figure 17
Stream lines forβ=3.Figure 18
Stream lines forN1=0.Figure 19
Stream lines forN1=0.5.Figure 20
Stream lines forN1=1.0.Figure 21
Stream lines forN1=1.5.Table 1
Effects ofN1, M, and β on skin friction.
N1Mβ1/2CfRe=F′0+F″00.10.10.1−0.4227600.2−0.4065190.3−0.3908210.4−0.3757700.1−0.4227600.4−0.4330070.7−0.45464691.0−0.4745690.1−0.4227600.11−0.4401390.12−0.4565590.13−0.472115Table 2
Effects ofγ and Pr on Nusselt number.
N2φ−θη0.10.10.6649140.120.8218940.141.0819420.10.10.6649140.20.8562560.30.887956


## 5. Conclusion

In this section, we demonstrate the MHD oblique stagnation point flow of nanofluid to an oscillatory plate which contains copper (Cu) as nanoparticle within the base fluid water. Also, consider the effect of thermal jump and velocity slip. The resulting opinions are worth citing:Because it is perceived at a variant distance from the plate, and the behavior of angular velocity increases by rising volume friction parameter and slip parameter.The heat transfer rate declines at the surface with rising Hartmann number, whereas it rises with raising values of nanoparticle volume fraction and thermal slip parameter. The stagnation point and zero skin friction depend on.By expanding the Hartmann number and thermal stratification, the thermal boundary layer is dejected.



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