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OPEN Shape efect on MHD fow of time fractional Ferro‑Brinkman type nanofuid with ramped heating Muhammad Saqib1, Ilyas Khan2*, Sharidan Shafe1* & Ahmad Qushairi Mohamad1
The colloidal suspension of nanometer-sized particles of Fe3O4 in traditional base fuids is referred to as Ferro-nanofuids. These fuids have many technological applications such as cell separation, drug delivery, magnetic resonance imaging, heat dissipation, damping, and dynamic sealing. Due to the massive applications of Ferro-nanofuids, the main objective of this study is to consider the MHD fow of water-based Ferro-nanofuid in the presence of thermal radiation, heat generation, and nanoparticle shape efect. The Caputo-Fabrizio time-fractional Brinkman type fuid model is utilized to demonstrate the proposed fow phenomenon with oscillating and ramped heating boundary conditions. The Laplace transform method is used to solve the model for both ramped and isothermal heating for exact solutions. The ramped and isothermal solutions are simultaneously plotted in the various fgures to study the infuence of pertinent fow parameters. The results revealed that the fractional parameter has a great impact on both temperature and velocity felds. In the case of ramped heating, both temperature and velocity felds decreasing with increasing fractional parameter. However, in the isothermal case, this trend reverses near the plate and gradually, ramped, and isothermal heating became alike away from the plate for the fractional parameter. Finally, the solutions for temperature and velocity felds are reduced to classical form and validated with already published results.
Enhanced heat transfer is signifcant due to its industrial and engineering applications. Tere are certain def- ciencies in heat transfer due to the poor thermophysical properties of the working fuid. Recent advancements in nanotechnology result to develop a modern class of heat transfer fuid referred to nanofuids prepared by dis- persing nanometer-sized particles 10–50 nm (nanoparticles) of metals, non-metals, and carbide in the working fuid water, oil, and alcohol, etc.1–6. For the frst time, the term nanofuid was used by Choi and Eastman 7. Te addition of nanoparticles in traditional host fuids has the capacity to signifcantly improve the heat transfer rate which can be utilized in numerous areas such as, transportation industry, nuclear reactor, cooling applications, electronics, cancer therapy, and drug delivery 8. Te innovative nanofuid performs as a next generation of heat transfer fuid for novel applications in engineering and industry including aerospace, transportation, electron- ics, tribology, buildings, medicines. Farshad and Sheikholeslami 9 investigated exergy loss in heat transfer in the 10 turbulence fow of Al2O3-H2O through a solar collector using a fnite volume method. Sadiq et al. relatively analyzed the stagnation point oscillatory fow of Cu-H2O and Al 2O3-H2O micropolar nanofuid using the ffh- order R-K Fehlberg method. Alamri et al.11 studied heat transfer in a channel Poiseuille fow of nanofuid using Buongiorno’s nanofuids model with Stefan blowing, slip, and magnetic feld efects. Ali et al.12 investigated the MHD fow of water-based Brinkman type nanofuid near an infnite rigid plate with variable velocity. Tey determined the exact analytical solutions vie the Laplace transform method. Safarian et al.13 investigated the fow of Al 2O3-H2O and CuO-H2O nanofuids in two U-shaped wavy pipes of the same length over a fat plate solar collector. It was indicated that CuO-H2O and wavy pipe enhance heat transfer rate by 78.25% and change in fow direction taken place with a higher heat transfer coefcient. Tese days, the research community focuses on magnetic nanofuids (MNFs) known as Ferro-nanofuid, mainly because of its exceptional performance in the improvement of heat transfer productivity; these fuids have been utilized in numerous areas of science such as medicine, transformer cooling, nuclear fusion, and chemical engineering. Te MNFs exhibits many characteristics including the controlling of thermal properties and fuid fow by means of the external magnetic feld which leads to a more comprehensive thermo-magnetic convection in contrast to traditional gravitational convection. Furthermore, the MNFs are used in rods separation
1Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia JB, 81310 Johor Bahru, Johor, Malaysia. 2Department of Mathematics, College of Science Al‑Zulf, Majmaah University, Al‑Majmaah 11952, Saudi Arabia. *email: [email protected]; [email protected]
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Figure 1. Physical sketch and coordinate system.
systems, X-rays tubes, oil lubricant bearing, sealing of hard dick damper processes. Tese fuids are utilized in Wi-Fi speakers, controller in electronic motors. Te MNFs are considerably used in MHD based equipment such as audiometer, sensor systems, densitometer, electromechanical converter, pressure transducer, and silent printers14. In view of this revolutionary importance, MNF was initiated by Guptha and Guptha15. Many studies considered MNFs fow in diferent fow regimes such as Li et al.16 considered the fow of Ferro-nanofuid under the infuence of Lorentz forces to investigate the efect of anisotropic thermal conductivity on the fuid fow and heat transfer. Shah et al.17 investigated micropolar Ferro-nanofuid fow over a dynamic stretching sheet in the presence of a magnetic feld and thermal radiation. Kumar et al.18 studied the fow of hybrid Ferro-nanofuid of Fe 3O4-CoFe2O4 into H2O-C2H6O2 (50–50%). It was detected that the Nusselt number of hybrid Ferro-Nanofuid is higher than Ferro-Nanofuid. Abro et al.19, Khan et al.20, Bezaatpour and Rostamzadeh21, Jamaluddin et al.22, and Aly and Ahmad 23 focused on Ferro-nanofuid in their investigations. Te literature survey indicates that many researchers had concentrated on constant wall temperature. But in various real-world circumstances, follow variable thermal conditions at the boundary. Te convection heat transfer studies are efcient to examine with step-change thermal boundary conditions. Te studies with ramped wall thermal boundary conditions can be employed in thin-flm photovoltaic devices to accomplish a certain fnish of the system 24. Ramped wall temperature is also necessary for heat management in buildings such as air conditioning where the constant wall thermal conditions lead to noticeable error. Motivated from the importance of step-change thermal boundary conditions (Ramped wall thermal boundary conditions), this study examines convection heat transfer with ramped boundary conditions. To the best of the author’s knowledge and from the literature survey, it is noticed that Ferro-nanofuid with a time-fractional Brinkman type fuid model with ramped heating is not reported yet. To fll the research gap, the main objective of the study is to consider the fow of Ferro-nanofuid over a vertical plate. As Ferro-nanofuid is electrically conducting thereby an external magnetic feld is employed normal to the fow direction. Te Caputo- Fabrizio fractional operator 25 is used to fractionalized the Brinkman type fuid model. As Ali et al.26 utilized the Caputo-Fabrizio fractional derivative to investigate the MHD fow simultaneously with heat and mass transfer in Walters’-B fuid in the present magnetic feld and porous medium. Khan et al.27–29 analyzed the fow of Casson and H 2O-CNTs nanofuid in a microchannel using a fractional derivatives approach. Tey determine the exact solutions by using the Laplace transform method. In a similar way, the proposed model is solved for exact analyti- cal solutions vie the Laplace transform method. Tese solutions are presented for temperature and velocity felds for both ramped and isothermal heating. Te ramped and isothermal solutions are simultaneously plotted in the various fgures to study the infuence of embedded fow parameters with the physical explanation. Eventually, by making α 1 , the classical solutions are recovered for temperature and velocity feld for both ramped and → isothermal heating from these solutions and validated with previously published work. Description of the problem Assume heat transfer in MHD fow of time-fractional Ferro-Brinkman type nanofuid near an infnite vertical plate along the x-axis and y-axis is selected transverse to it. Magnetic nanoparticles of diferent shapes (blade, brick, spherical, and platelet) are dispersed into the water as base fuid to form Ferro-Brinkman type nanofuid. Termal radiation, heat generation, and ramped wall heating are also considered. At t 0 , the fuid and plate ≤ are set in rest with u y,0 0 and T y,0 T where T is the ambient temperature. Aferward at t 0+ , = = ∞ ∞ = the velocity and temperature felds are switched to u(0, t) U0H(t) cos (ωt) and T(0, t) T0 (TW T )t/t0 = = + − ∞ if 0 < t < t0 or T(0, t) TW if t > t0 respectively. At this phase, the fuid starts fowing in x-direction as pre- = sented in Fig. 1. Te Ferro-Brinkman type nanofuid experience magnetic force because the fuid is electrically
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conducting, thereby, an external magnetic feld is employed normally to the fow direction. Te governing equa- tions of the proposed model are derived in the following section. Mathematese formulation In accordance with Rajagopal30 and Fetecau, Fetecau31, the linear momentum equation for Brinkman type fuid can be written as DV ρ T ρF I0, (1) Dt =∇· + − where D ∂ u ∂ v ∂ w ∂ refers to material time derivatives, V is the velocity vector, T represents the Dt = ∂t + ∂x + ∂y + ∂z Cauchy stress tensor, ρF depicts the body forces, and I0 exhibits the interaction force of porous medium which can be expressed as I α V, 0 = d (2) where αd is a positive coefcient of drag force which yields Eq. (1) to the following form ∂V ρ (V. )V .T ρF αdV. (3) ∂t + ∇ =∇ + − In the case of Brinkman type fuid, the constitutive equation of Cauchy stress tensor is expressed by32 T pI µA , =− + 1 (4) where p is the scalar pressure, I is the identity tensor, µ is the dynamic viscosity, and A1 is the Rivlin–Ericksen tensor determined by
A V ( V)T , (5) 1 =∇ + ∇ where the superscript T refers to the matrix transpose and V represents the gradient of the velocity. In the case ∇ of the proposed problems, the unsteady, incompressible, unidirectional, and one-dimensional fow is considered thereby, the velocity vector is defned as V u y, t ,0,0 u y, t i. = = (6) Bearing in mind, Eqs. (4) and (6), the T is determined as ∇· ∂p ∂2u y, t µ T 2 , (7) ∇· =−∂x + ∂ y whereas, the fuid fow is considered in x-direction, therefore, p p y, z which lead ∂p/∂y 0 ∂p/∂z 0 . = = = Introducing Eq. (7) into Eq. (3) and bearing in mind Eq. (6) which yield to ∂u y, t ∂p ∂2u y, t ρ µ ρ α 2 F du y, t . (8) ∂t =−∂x + ∂ y + − Based on Jaluria33, the body forces ρF for convection fow of electrically conducting Brinkman type fuid is given by ρF J B ρg, = × + (9) where J B is the Lorentz force, J is the current density, B B0 b is the magnetic fux intensity, B0 is applied × = + magnetics fled acting in y-direction, b is the induced magnetic feld and g g,0,0 is the gravitation accel- = − eration. Te Lorentz force can be defned by Maxwell’s set of equations as34 .B 0, ∇ = B µmJ, ∇× = , (10) ∂B E , ∇× =−∂t where µm is the magnetic permeability and E is electric feld intensity. Te current density J is described by the generalized Ohm’s law as 35 J σ(E V B), = + × (11) where σ is the electrical conductivity. Meanwhile, the magnetic Reynolds number is assumed small enough so that the induced feld b is neglected compared to the applied feld B0 . Furthermore, it is assumed that there is no polarization and applied voltages thereby, the electric feld E is ignored. Hence, Eq. (11) takes the following form i jk J σ(V B0) σ u 00 0, 0, σB0u y, t . (12) = × = = 0 B0 0
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Keeping in mind Eq. (12), the Lorentz force J B became × ij k 2 J B 00σB0u σB0u y, t ,0,0 . (13) × = = − 0 B0 0 Introducing Eq. (13) into Eq. (9) and then into Eq. ( 8) which gives the following ∂u ∂p ∂2u y, t ρ µ σ 2 α ρ 2 B0u y, t du y, t g. (14) ∂t =−∂x + ∂ y − + − Referred to Jaluria33, the pressure p in Eq. (14) can be written in the following form p p p , = h + d (15) where ph is the hydrostatic pressure pd is the dynamic pressure. Te proposed problem is considered for convec- 36 tion heat transfer therefore, pd.can be neglected. According to White , the hydrostatic pressure ph in the case of convection heat transfer can be written as
∂ph ρ g, (16) ∂x =− ∞ where ρ is the ambient density of the fuid. Introducing Eq. (16) into Eq. (14) yield to ∞ ∂u ∂2u y, t ρ µ σ 2 α (ρ ρ) 2 B0u du g. (17) ∂t = ∂ y − + + ∞ −
Assuming that βT is the volumetric thermal expansion of the fluid then according to Boussinesq’s 33 approximation , βT can be written as 1 ∂ρ 1 �ρ 1 ρ ρ βT ∞ − , (18) =−ρ ∂T p ≈−ρ �T =−ρ T T y, t ∞ − or
ρβT T y, t T ρ ρ. (19) − ∞ = ∞ − Introducing Eq. (19) into Eq. (17) yield to ∂u ∂2u y, t ρ µ σ 2 α ρβ 2 B0u y, t du y, t g T T y, t T , (20) ∂t = ∂ y − + + − ∞ or ∂u y, t ∂2u y, t σB2u y, t υ 0 β β 2 ∗u y, t g T T y, t T , (21) ∂t = ∂ y − ρ − + − ∞ where β α /ρ is the Brinkman type fuid parameter which corresponds to the drag force of highly non- ∗ = d Darcy’s porous medium. Te energy equation together with thermal radiation and heat generation is given by37 ∂T y, t ∂2T y, t ∂q ρ r Cp k 2 Q0 T y, t T . (22) ∂t = ∂ y − ∂y + − ∞ 38 Te radiative heat fux qr in Eq. (22) is formulated by using via the Roseland approximation as 4 4σ1 ∂T qr . (23) =−3k1 ∂y Te T4 is expanded along T by using the Taylor series as ∞ 3 2 4 4 4T 12T 2 24T 3 T T ∞ (T T ) ∞ (T T ) ∞ (T T ) (24) = ∞ + 1 − ∞ + 2 − ∞ + 3 − ∞ ··· ! ! ! Te temperature gradient is assumed to be small enough so, the higher-order terms are neglected which yield to 3 4 4 4T T T ∞ (T T ). (25) ≈ ∞ + 1 − ∞ ! Te further simplifcation of Eq. (25) yield to the following 4 3 4 T 4T T 3T . (26) ≈ ∞ − ∞ Te simplifed form of T4 from Eq. (26), is used in Eq. (23) which yield to
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3 16σ1T ∂T qr ∞ . (27) =− 3k1 ∂y Diferentiating Eq. (27) with respect to “ y ” yield to the following 3 2 ∂qr 16σ1T ∂ T ∞ 2 . (28) ∂y =− 3k1 ∂y Incorporating Eq. (28) into Eq. (22) yield to the following 3 2 ∂T y, t 16σ1T ∂ T y, t ρCp k 1 ∞ Q0 T y, t T . (29) ∂t = + 3k k ∂y2 + − ∞ 1 For enhanced heat transfer, the Fe 3O4 has been dispersed into the water as base fuid to form water-Ferro- nanofuid. As refer to Khanafar et al.39 and Tiwari and Das 40 Eqs. (21 and (29) can be written for Ferro-nanofuid fow as
∂u y, t ∂2u y, t ρ β µ σ 2 (ρβ ) nf ∗u y, t nf 2 nf B0u y, t g T nf T y, t T , (30) ∂t + = ∂ y − + − ∞ 3 2 ∂T y, t 16σ1T ∂ T y, t ρCp knf 1 ∞ Q0 T y, t T , (31) nf ∂t = + 3k k ∂y2 + − ∞ 1 nf where ρnf is the density, u y, t is the velocity, β∗ is the Brinkman type fuid parameter, µnf is the dynamic veloc- ity, σnf is the electrical conductivity, B0 is the uniform magnetic feld, g gravitational acceleration, (βT ) is the nf thermal expansion, T y, t is the temperature, Cp nf is the heat capacitance, knf is the thermal conductivity, qr Q is the radiative heat fux and 0 is the heat generation. Te corresponding initial and boundary conditions are given as
u y,0 0, T y,0 T , y 0, (32) = = ∞ ∀ ≤ u(0, t) U H(t) cos (ωt) t 0+, = 0 ;∀≥ t T0 (TW T ) if 0 < t < t0, T(0, t) + − ∞ t0 ; . = (33) T if t > t W ; 0 u y, t 0 and T y, t T if y , → → ∞; →∞ � � � � Te terms ρnf , µnf , σnf , (βT )nf , Cp nf and knf appeared in Eqs. (30) and (31) for the enhanced thermophysical 41 properties nanofuid with diferent shapes (blade, brick, spherical, and platelet) nanoparticles defned as . ρ (1 φ)ρ φρ , nf = − f + s (34)
2 µnf µf 1 aφ bφ , (35) = + + σ σ 3 s 1 φ nf σf − 1 , (36) σf = + σs 2 σ s 1 φ σf + − σf − (ρβ ) (1 φ)(ρβ ) φ(ρβ ) , T nf = − T f + T s (37)
ρC (1 φ) ρC φ ρC , p nf = − p f + p s (38) and
Ks (n 1)Kf (n 1) Ks Kf φ Knf Kf + − + − − , (39) = K (n 1)K K K φ s + − f − s − f where the subscript nf is used for nanofuid, f for base fuid water, and s for solid nanoparticles Fe3O4. Further- more, in Eq. (35) a and b correspond to shape constant which afects the density factor of nanofuid and in Eq. (39) n is the experimental shape constituent. n can be evaluated as 3 n , = ψ (40)
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Nanoparticle type Sketch a b ψ
Blade 14.6 123.3 0.36
Brick 1.9 471.4 0.81
Cylindrical 13.5 909.4 0.62
Platelet 37.1 612.6 0.52
Table 1. Diferent shapes of nanoparticles with corresponding values of a , b and ψ.
where ψ is the sphericity of nanoparticles which infuences the thermal conductivity. Te diferent shape nano- particles and the corresponding values of a , b and ψ are presented in Table 142,43 In this study magnetic nanoparticle (Fe3O4) of diferent shapes is dissolved in water (H2O) as base fuid to form magnetic nanofuid (MNF). Te physical values of the thermal properties of nanoparticles and base are given in Table 144,45 Solutions of the problem Tis section presents the exact solutions for the magnetic fow of time-fractional Ferro-Brinkman type nanofuid under the efect of a normal magnetic feld. In this section, the problem modeled in Sect. 3 is frst transformed to dimensionless form to diminish the units for simplifcation and reduction of number variables. For this purpose, the following dimensionless variables
u U0 t υf T T ∞ u∗ , y∗ y, t∗ , t0 2 , θ − = U0 = υf = t0 = U = TW T 0 − ∞ are implemented into Eqs. (30)–(33) afer dropping the * sign for simplicity yield to the following form
∂u y, t ∂2u y, t φ β φ φ φ θ 0 u y, t 1 2 2Mu y, t 3Gr y, t , (41) ∂t + = ∂ y − + ∂θ y, t φ ∂2θ y, t φ 5 θ 4 2 Q y, t , (42) ∂t = Preff ∂ y + together with the following dimensionless conditions u y,0 0, θ y,0 0, y 0, = = ∀ ≤ (43) u(0, t) H(t) cos (ωt) or sin (ωt) t 0+ = ;∀≥ t if 0 < t < 1 T(0, t) ; , (44) = 1 if t > 1 � ; u y, t 0 and T y, t 0 if y → → ; →∞ and � � � �
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2 2 (β υ) ( ) 3 U0 β∗ υf σf B0 g T f TW T µCp 16σ1T φ5 Pr β , M , Gr − ∞ , Pr , Nr ∞ , Pr , = υ = U2ρ = U3 = k = 3k k eff = φ Nr f 0 f 0 f 1 f 5 + υf Q0 ρ σnf (ρβ ) Q , φ (1 φ) φ s , φ 1 aφ bφ2, φ , φ (1 φ) φ T s , = 2 0 = − + ρ 1 = + + 2 = σ 3 = − + (ρβ ) U0 ρCp f f f T f
ρCp s knf φ4 (1 φ) φ , φ5 = − + ρ = k Cp f f where β is the Brinkman parameter, M is the magnetic number, Gr is thermal Grashof number, Pr is the Prandtl number, Nr is the radiation parameter, Preff is the efective Prandtl number, Q is the heat generation parameter, and φ0 , φ1 , φ2 , φ3 , φ4 , φ5 are constant terms. Te Caputo–Fabrizio time-fractional derivative is used to transform Eqs. (41) and (42) to time-fractional for as ∂2u y, t φ Dα β φ φ φ θ 0 t u y, t u y, t 1 2 2Mu y, t 3Gr y, t , (45) + = ∂ y − + φ ∂2θ y, t φ Dαθ 5 θ 4 t y, t 2 Q y, t . (46) = Preff ∂ y + Dα 25 Te Caputo–Fabrizio time-fractional operator t (, .,) appeared in Eqs. (45) and (46) is defned by t N(α) α(t τ) ∂f y, τ Dαf y, t exp − dτ 0 <α<1, t = 1 α − 1 α ∂τ ; (47) − 0 − where N(α) is the normalization function with the following property N(1) N(0) 1. = = (48) Using Eq. (48) the Laplace transform of Eq. (47) is given by qf y, q f y,0 L Dα t f y, t q − ,0<α<1, (49) = ( 1 α)q α − + which be reduced for integer-order time derivative as
qf y, q f y,0 L Dα lim t f y, t q lim − α 1 = α 1 (1 α)q α → → − + (50) ∂f y, t qf y, q f y,0 L . = − = ∂t
Solution for temperature fled. Tis presents the solutions for temperature fled in both ramped and isothermal heating case.
Solutions for temperature fled with ramped heating. In order to solve the energy equation for the temperature feld, the Laplace transform is employed to Eq. (46), keeping in view Eq. (49) and using initial condition from Eq. (43) which yield to qθ y, q θ y,0 φ d2θ y, q − 5 θ 2 Q y, q , (51) ( 1 α)q α = φ4 Preff dy + − + which gives the following on further simplifcation dθ y, q a q a 1 2 θ 2 − y, q 0, (52) dy − q b1 = + along with the transformed corresponding boundary conditions 1 ∞ q qt qt 1 e− θ 0, q t e− 1 e− − and θ , q 0 , = · + · = q2 ∞ = (53) � � � � 0 1 � � where
φ4 Preff Qφ4 Preff 1 a1 a0b0 Q1, a2 Q1b1, a0 , Q1 , b0 , b1 b0α. = − = = φ = φ = 1 α = 5 5 −
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Te exact analytical solutions of Eq. (52) can be determined by using the transform boundary conditions for Eq. (53) as
a1q a2 a1q a2 1 y q −b q 1 y q −b θ y, q e− + 1 e− e− + 1 , (54) = q2 − q2 which correspond to the solutions of temperature feld for ramped heating in the Laplace transform domain. Equation (52) can be further simplifed as q θ y, q θ y, q e− θ y, q , = Ramp − Ramp (55) where
a1q a2 1 y q −b θ Ramp y, q e− + 1 . (56) = q2 Te inverse Laplace transform is used to invert back Eq. (55) to the time domain as θ y, t θ y, t θ y, t 1 H(t 1), = Ramp − Ramp − − (57) where H(t 1) is the Heaviside unit step function and the term θRamp y, t is defned by − t ∞ y p y2 y√a1 √ 1 b1τ a1s θRamp y, t te− (t τ) e− − 4s − I1 2√p1sτ dsdτ, = − − 2s√πτ (58) 0 0 where p a a b . 1 =− 2 − 1 1
Solutions for temperature fled with isothermal heating. In order to fnd exact solutions for isothermal heating, the boundary condition in the Laplace transform domain is given by 1 θ 0, q , = q (59) Te exact analytical solutions of Eq. (52) is obtained by using Eq. (59) as
a1q a2 1 y q −b θ y, q e− + 1 , (60) = q Te fnal exact solution for isothermal heating is obtained afer applying the inverse Laplace transform to Eq. (60) which yield to t ∞ y p y2 y√a1 √ 1 b1τ a1s θ y, t e− e− − 4s − I1 2√p1sτ dsdτ. = − 2s√πτ (61) 0 0 Te exact solutions corresponding ramped and isothermal heating are respectively depicted in Eqs. (57) and (61) which satisfy the impose conditions in both cases. Te exact solutions for the velocity feld corresponding to ramped and the isothermal heating is presented in the following section.
Solution for velocity feld. Tis section presents exact analytical solutions for the velocity feld for both ramped and isothermal heating.
Solutions for velocity feld with ramped heating. Te Laplace transform is applied to Eq. (45) using initial condi- tion from Eq. (43) yield to
qu y, q u y,0 d2u y, q φ − β φ φ φ θ 0 u q, t 1 2 2Mu q, t 3Gr q, t , (62) ( 1 α)q α + = dy − + − + which takes the following form afer simplifcation
d2u y, q q a1q a2 a3q a4 1 e− y q −b + u y, q Gr0 − e− + 1 , (63) dy2 − q b =− q2 1 + along with the transformed velocity boundary conditions q u 0, q and u , q 0, = q2 ω2 ∞ = (64) +
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where
1 1 φ3 a3 (φ0b0 φ0β φ2M), a4 (φ0βb1 φ2Mb1), Gr0 Gr. = φ1 + + = φ1 + = φ1 Te analytical solutions of Eq. (63) can be obtained by using the boundary conditions from Eq. (64) as
a3q a4 Gr q b q a3q a4 a1q a2 q y q +b 0 1 1 e− y q +b y q −b u y, q e− + 1 + − e− + 1 e− + 1 , (65) = q2 ω2 + a q a q2 − 5 6 + − where a a a , a a a . 5 = 1 − 3 6 = 2 + 4 In order to fnd the inverse Laplace transform, Eq. (65) can be written in a more suitable form as
q u y, q uc y, q u1 q u2(Ramp) y, q e− u2(Ramp) y, q = + − (66) u y, q θ y, q , − 1 where
a3q a4 q y q +b uc y, q e− + 1 , (67) = q2 ω2 +
Gr0 q b1 u1 q + , (68) = a5 q a6 −
a3q a4 1 y q +b u2 Ramp y, q e− + 1 , (69) ( ) = q2 and θ y, q is previously defned by Eq. (56). Now, the inverse Laplace transform is applied to Eq. (66) which gives u y, t uc y, t u1(t) u2(Ramp) y, t H(t 1)u2(Ramp) y, t 1 = + ∗ − − − (70) u y , t θ y, t , − 1 ∗ where t ∞ y p y2 y√a1 √ 2 b1τ a1s uc y, q te− cos (t τ) e− − 4s − I1 2√p2sτ dsdτ, = − − 2s√πτ (71) 0 0
a6 1 a t u1(t) a7e 5 δ(t) , (72) = + a 5
t ∞ y p y2 y√a1 √ 2 b1τ a1s u ( ) y, t te− (t τ) e− − 4s − I1 2√p2sτ dsdτ, 2 Ramp = − − 2s√πτ (73) 0 0 and
a6 a5b1 p2 a4 a3b1, a7 Gr0 + . = − = a2 5 Te symbol * presents the convolutions product and θ y, t is depicted in Eq. (57). It is worth highlighting here that Eq. (70) characterize the exact solutions for the velocity feld with ramped heating. Solutions for velocity feld with isothermal heating. Next, Eq. (45) is solved again for isothermal heating as
a3q a4 Gr q b a3q a4 a1q a2 q y q +b 0 1 1 y q +b y q −b uIso y, q e− + 1 + e− + 1 e− + 1 , (74) = q2 ω2 + a q a q − 3 2 + − For convenience in inverse Laplace transform, Eq. (74) can be written in more in suitable form as q u y, q uc y, q u1 q u3(Iso) y, q e− u3(Iso) y, q = + − (75) u y, q θ y, q , − 1
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where in this case, terms θ y, q ,uc y, q , u1 q , and are already defned in Eqs. (60), (67), and (68) respectively. Te term u3(Iso) y, q newly appeared is presented by a1q a2 1 y q +b u3(Iso) y, q e− + 1 . (76) = q Te solution for isothermal heating is obtained by taking the inverse Laplace transform of Eq. (75) which yield to
u y, t uC y, t u1(t) u3(Iso) y, t H(t 1)u3(Iso) y, t 1 = + ∗ − − − (77) u y, t θ y, t , − 1 ∗ where t ∞ y p y2 y√a1 √ 2 b1τ a1s u3(Iso) y, t e− e− − 4s − I1 2√p2sτ dsdτ, = − 2s√πτ (78) 0 0 and in this case, the terms θ y, t uc y, t , u1(t) are previously defned in Eqs. (61), (71) and (72) respectively. Tis completes the solutions for the proposed problem. Limiting cases Tis section presents the limiting solutions by making α 1 in Eqs. (57), (61), (70) and (77), for both velocity → and temperature felds.
Limiting solutions for temperature. Tis subsection highlights limiting solutions for ramped and iso- thermal heating for the temperature feld.
Limiting solution for temperature fled with ramped heating. Keeping in mind Eq. (50), employing limα 1 to Eq. (51) which yield to → φ d2θ y, q θ θ 5 θ q y, q y,0 2 Q y, q , (79) − = φ4 Preff dy + where θ y, q is the classical temperature in the Laplace transform domain and θ y,0 is the initial condition. Afer using the boundary conditions from Eq. (53), the analytical solution of Eq. (79) is given by 1 y√a q Q q 1 y√a q Q θ y, q e− 0 − 1 e− e− 0 − 1 , = q2 − q2 (80) which can be written as
Q1 q Q1 θ y, q � y√a0, , q e− � y√a0, , q , (81) = a − a 0 0 where
Q1 Q1 1 y√a0 q a y√a0, , q e− − 0 , (82) a = q2 0 Upon inverting the Laplace transform, Eq. (81) yield to
Q1 Q1 θ y, t � y√a0, , t � y√a0, , t 1 H(t 1), (83) = a − a − − 0 0 where y√a Q ya √a ey√Q1 erfc 0 1 √t t 0 0 Q1 1 2√t + a0 + 2Q1 y√a0, , t � �� � . (84) a0 = 2 y√Q y√a0 Q1 ya0√a0 � � e− 1 erfc √t t + 2√t − a − 2Q � 0 �� 1 � Limiting solution for temperature feld with isothermal heating. To fnd the classical solution for temperature feld in case of isothermal heating, Eq. (79) is analytically solved by using the boundary condition Eq. (59) which gives
Q1 Q1 1 y√a0 q a θ y, q � y√a0, , q e− − 0 , (85) = a = q 0
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Base fuid Nanoparticles
Material H2O F3O4 TiO ρ kg m3 997.1 5200 425
Cp J kg K 4179 670 6862 k W mK 0.613 6 8.9538 5 1 βT 10 (K ) 21 1.3 0.9 × − − σ 0.05 25,000 1 10 12 × − Pr 6.2 – –
Table 2. Physical values of thermal properties of nanoparticles and base fuid.
Figure 2. Comparison of present solutions presented in Eqs. (83), (86) and Nandkeolyar et al. .45, Eq. (19) for t = 0.5, 1.5.
Te inverse Laplace transform is employed to Eq. (85) which yield to
Q 1 1 y√Q1 y√a0 Q1 y√Q1 y√a0 Q1 θ y, t � y√a0, , t e− erfc t e erfc t . = a = 2 2√t − a + 2√t + a 0 0 0 (86) It is worth mentioning here that by assuming the thermal conductivity of Maxwell from41 and using numeri- cal values of thermophysical properties of TiO 2 nanoparticles From Table 2, the solutions obtained in Eq. (83), can be reduced by setting Q 0 , to that of Nandkeolyar et al.45 (Eq. 19). In the absence of heat generation, → the solutions presented in Eqs. (83) and (86) from the present study and Eq. (19) from Nandkeolyar et al.45, for t 0.5, 1.5 are computed and displayed in Fig. 2. Tis fgure clearly indicates that the solutions are identical = which validates the present solutions for the Temperature feld.
Limiting solution for velocity feld. Te limiting solutions for the velocity feld in case of ramped and isothermal heating is introduced in this subsection.
Limiting solution for velocity feld with ramped heating. While taking into account Eqs. (50), (62) is reduced to classical form by applying limα 1 to which yield to → d2u y, q φ β φ φ φ θ 0 qu y, q u y,0 u q, t 1 2 2Mu q, t 3Gr q, t . (87) − + = dy − + Te analytical solution of Eq. (87) is determined by using Eq. (80) and boundary conditions from Eq. (64) as
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q q q Gr0 1 e− y√a q a Gr0 1 e− y√a q Q u y, q − e− 6 + 7 − e− 0 + 1 . = q2 ω2 + a q a q2 − a q a q2 (88) 8 9 8 9 + + + where as a0 a6, a9 a7 Q1. = − = + By using partial fraction, Eq. (88) can be reduced to the following form
1 y√a q a 1 y√a q a u y, q e− 6 + 7 e− 6 + 7 = 2 q iω + 2 q iω + − Gr 1 1 0 y√a6q a7 q y√a6q a7 e− + e− e− + + a q a q2 − q2 (89) 8 + 9 Gr0 1 y√a q Q q 1 y√a q Q e− 0 + 1 e− e− 0 + 1 . − a q a q2 − q2 8 + 9 In order to fnd the inverse Laplace transform, Eq. (89) can be written in a more suitable form as
1 a7 1 a7 u y, q � y√a6, , iω, q � y√a6, , iω, q = 2 a − + 2 a 6 6 a7 q a7 F q � y√a6, , q e− � y√a6, , q + a − a (90) 6 6 Q1 q Q1 F q � y√a0, , q e− � y√a0, , q , − a − a 0 0 where
a7 1 y√a6q a7 � y√a6, , iω, q e− + , (91) a6 − = q iω + a7 1 y√a6q a7 � y√a6, , iω, q e− + , (92) a6 = q iω − 1 a7 y√a6q a7 y√a6, , q e− + , a = q2 (93) 6
Gr0 F q , (94) = a8q a9 + y√a , Q1 , q and the function 0 a0 is previously defned in Eq. (82). Applying the inverse Laplace transform to Eq. (90) which yield to
1 a7 1 a7 u y, t � y√a6, , iω, t � y√a6, , iω, t = 2 a − + 2 a 6 6 a7 q a7 F q � y√a6, , t e− � y√a6, , t 1 H(t 1) + ∗ a − a − − (95) 6 6 Q1 Q1 F q � y√a0, , t � y√a0, , t 1 H(t 1) , − ∗ a − a − − 0 0 where
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a7 y√a6 iω y√a6 a7 e− a6 − erfc iω t iωt � 2√t − a − a7 e � �� 6 � � � y√a6, , iω, t , a6 − = 2 a7 � � y√a6 iω y√a6 a7 e a6 − erfc iω t � � 2√t + � a6 − � � � a7 y√a6 iω y√a6 a7 e− + a6 erfc iω t iωt � 2√t − + a a7 e � �� 6 � � � y√a6, , iω, t , a6 = 2 a7 (96) � � y√a6 iω y√a6 a7 e a6 − erfc iω t � 2√t + + a � �� 6 � � y√a a ya √a ey√a7 erfc 6 7 √t t 6 6 a7 1 2√t + a6 + 2a7 � y√a6, , t � �� � , a6 = 2 y√a y√a6 a7 ya6√a6 � � e− 7 erfc √t t + 2√t − a − 2a � 6 �� 7 � Gr0 a9 t F q e− a8 , = a8 � � y√a , Q1 , t and the function 0 a0 is already presented in Eq. (84) . Limiting solution for velocity feld with isothermal heating. In order to reduce Eq. (62) to classical form for the velocity feld with isothermal heating, Eq. (85) is used which yield to
q 1 Gr0 y√a q a 1 Gr0 y√a q Q u y, q e− 6 + 7 e− 0 + 1 , = q2 ω2 + q a q a − q a q a (97) 8 9 8 9 + + + For the convenience in the inverse Laplace transform, Eq. (97) is reduced to the following form
1 y√a q a 1 y√a q a u y, q e− 6 + 7 e− 6 + 7 = 2 q iω + 2 q iω + − (98) Gr0 1 1 y√a6q a7 Gr 0 1 1 y√a0q Q1 a9 e− + a9 e− + , + a8 q q − a8 q q + a8 + a8 which can be written in a further simplifed form as
1 a7 1 a7 u y, q � y√a6, , iω, q � y√a6, , iω, q = 2 a6 − + 2 a6 (99) Gr0 1 a7 a9 Gr0 1 Q1 a9 � y√a6, , , q � y√a0, , , q , + a q a a − a q a a 8 6 8 8 0 8 where a a 1 y√a , 7 , 9 , q e y√a6q a7 , 6 a9 − + (100) a6 a8 = q + a8
Q a 1 y√a , 1 , 9 , q e y√a0q Q1 , 0 a9 − + (101) a0 a8 = q + a8
a7 a7 and functions � y√a6, , iω, q and � y√a6, , iω, q are given in Eqs. (91) and (92) respectively. Next, a6 − a6 taking the inverse Laplace transform of Eq. (99), the exact solution for the classical velocity feld in case of iso- thermal heating is given by
1 a7 1 a7 u y, t � y√a6, , iω, t � y√a6, , iω, t = 2 a − + 2 a 6 6 t t (102) Gr0 a7 a9 Gr0 Q1 a9 � y√a6, , , τ dτ � y√a0, , , τ dτ, + a8 a6 a8 − a8 a0 a8 0 0 Besides this, the viscosity of Brinkman and thermal conductivity of Maxwell’s from 41 are set in Eqs. (95) and (102). Afer making β 0 and Q 0 , in Eqs. (95) and (102) with TiO nanoparticles are computed for → → 2 t 0.5, 1.5 together with Eq. (20) from Nandkeolyar et al.45 and highlighted in Fig. 3. It is found that these solu- = tions are alike which shows the correctness of the present results.
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Figure 3. Comparison of present solutions presented in Eqs. (95), (102) and Nandkeolyar et al.45, Eq. (20) for t = 0.5, 1.5
Figure 4. Consequence of α on θ y, t when φ = 0.04,Nr = 0.5 , Q = 0.5 in case of cylindrical shape nanoparticles.
Results and Discussion Te exact analytical solutions (solutions for ramped and isothermal heating) for temperature and velocity felds are computed and displayed in numerous graphs to study the impact of pertinent parameters such as fractional parameter α , volume concentration φ , shape efect of nanoparticles, thermal radiation Nr , heat generation Q , Brinkman parameter β , magnetic parameter M , and thermal Grashof number Gr . In order to provide a clear understanding, ramped and isothermal solutions are simultaneously plotted in Figs. 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16. It is essential to underline that these graphs satisfy all the initial and boundary conditions. Moreover, for ramped heating time is chosen in 0 < t < 1 and for isothermal heating, it is selected as t > 0 . Precisely, for ramped heating time is chosen t 0.5 and for isothermal heating, it is taken t 1.5. = = Efects of fow parameters on both velocity and temperature felds. Figures 4 and 5 present the impact of α on temperature and velocity felds. In the case of isothermal heating, both the temperature and veloc- ity feld increase with increasing α near the heated plate. But this trend reverses at a certain point away from the plate. Physically this fact can be justifed as α is increasing the thickness of thermal and momentum boundary
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Figure 5. Consequence of α on u y, t when φ = 0.04 , β = 0.5 , M = 0.5 , Gr = 5 , Nr = 0.5 , Q = 0.5 and ωt = 0.15 in case of cylindrical shape nanoparticles.
Figure 6. Consequence of φ on θ y, t when α = 0.5,Nr = 0.5 , Q = 0.5 in case of cylindrical shape nanoparticles.
layers gradually increasing and became thickest near the plate at α 1 . However, away from the plate, the ther- = mal and momentum boundary layers behave oppositely. In the case of ramped heating, the trend of temperature and velocity profle is straight forward. Increasing the value of α , decreasing the temperature and velocity felds. Tis is because the thickness of thermal and momentum boundary layers is inversely related to α in case of ramped heating. So, when α is increased, thermal and momentum boundary layers are gradually decreased as a result the temperature and velocity felds decreased. Additionally, as in Figs. 6 and 7 it can be observed that the temperature and velocity feld are signifcantly afected by φ . It is found that the temperature fled increases with increasing values of φ for both ramped and isothermal heating. It can be clearly seen from Eq. (39) that an incre- ment in φ corresponds to the enhancement in thermal conductivity of the nanofuid as a result the temperature profle increases. Furthermore, it can be seen in Fig. 4 that the velocity feld decrease with increasing values of φ for both ramped and isothermal heating. Tis is due to the dynamic viscosity of nanofuid presented in Eq. (35). Te dynamic viscosity is directly related to the volume concentration of nanoparticles. Increasing values of φ(0 <φ 0.04) leads to an increase in viscosity of the nanofuid and the fuid became thick. Hence, an increase ≤ in viscosity resists to nanofuid fow. Figures 8 and 9 depict the comparison of temperature and velocity felds for diferent shapes of nanoparticles. It is noticed from Fig. 8 that the temperature feld for blade shape nanoparticles is higher followed by platelet,
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Figure 7. Consequence of φ on u y, t when α = 0.5 , β = 0.5 , M = 0.5 , Gr = 5 , Nr = 0.5 , Q = 0.5 and ωt = 0.15 in case of cylindrical shape nanoparticles.
Figure 8. Consequence of diferent shapes of nanoparticles on θ y, t when α = 0.5 , φ = 0.04 , Nr = 0.5 , Q = 0.5
spherical, and brick shaped nanoparticles due to the shape factor n involving in Eq. (39). Besides this, the velocity profle is higher for brick shape nanoparticles fowed by the blade, spherical and platelet shaped nanoparticles which depend on the values of shape constants a and b involving in Eq. (35). Meanwhile, the behavior of tem- perature and velocity felds for the thermal radiation parameter Nr is studied in Figs. 10 and 11. As expected, an increase in Nr results of an increase in both the temperature and velocity feld as Nr indicates the proportional contribution of conduction heat transfer to the thermal radiation. Hence, the temperature feld signifying an increasing trend. Furthermore, increasing Nr twist the rate of heat transfer to the nanofuid as a result the attrac- tive forces holding the nanofuid molecule weaken as a result, decreasing the viscosity which accelerates the fuid velocity. Variations in temperature and velocity felds due to heat generation Q are depicted in Figs. 12 and 13, where Q is selected arbitrary 0.2, 0.5, 0.8, and 1.0. It is observed that both the temperature and velocity felds for ramped and isothermal heating increasing for increasing values of Q because the existence of heat generation causes an increment in the energy level due to which the thickness of thermal and momentum boundary grow at the oscillating boundary as a result the temperature and velocity feld increases.
The impact of fow parameters which efect only the velocity feld. Te infuence of the Brinkman type fuid parameter β on the velocity profles for isothermal and ramped heating is displayed in Fig. 14. β is
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Figure 9. Consequence of diferent shapes of nanoparticles on u y, t when α = 0.5 , φ = 0.04 , β = 0.5 , M = 0.5 , Gr = 5 , Nr = 0.5 , Q = 0.5 and ωt = 0.15.
Figure 10. Consequence of Nr on θ y, t when α = 0.5 , φ = 0.04 , Q = 0.5 in case of cylindrical shape nanoparticles.
the magnitude of the drag force of a highly non-Darcy’s porous medium. Te velocity felds for both isothermal and ramped heating decelerated with an increment in β because of a strong drag force. Hence, increment in β increase the drag forces which decelerate the velocity feld. Meanwhile, the efect of the magnetic parameter M is illustrated in Fig. 15 on the ramped and isothermal velocity felds. It is revealed that the isothermal velocity is higher than the ramped velocity. Te isothermal and ramped velocity felds decelerated together for greater values of M due to the applied magnetic feld which results in the presence of intense Lorentz force. Tis force works as a dragging force exhibits persistent resistance to the nanofuid fow. Ultimately, the isothermal and ramped velocity fled dropped. But away from the plate, the Lorentz force became poor and nanofuid comes to rest. Besides this, the infuence of thermal Grashof number Gr is highlighted in Fig. 16 for both ramped and
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Figure 11. Consequence of Nr on u y, t when α = 0.5 , φ = 0.04 , β = 0.5 , M = 0.5 , Gr = 5 , Q = 0.5 and ωt = 0.15 in case of cylindrical shape nanoparticles.
Figure 12. Consequence of Q on θ y, t when α = 0.5 , φ = 0.04 , Nr = 0.5 in case of cylindrical shape nanoparticles.
isothermal heating. It is demonstrated in this fgure that the velocity feld increases with increasing Gr . Te Gr shows the proportional strength of the buoyancy force to the viscous force. thereby, an increase in Gr leads to an increase in thermal buoyancy force. In the proposed problem, the convection fow of nanofuid driven by thermal buoyancy force is considered. As a result, it has a tendency to increase the velocity feld in both ramped and isothermal heating cases.
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Figure 13. Consequence of Q on u y, t when α = 0.5 , φ = 0.04 , β = 0.5 , M = 0.5 , Gr = 5 , Nr = 0.5 and ωt = 0.15 in case of cylindrical shape nanoparticles.
Figure 14. Consequence of β on u y, t when α = 0.5 , φ = 0.04 , M = 0.5 , Gr = 5 , Nr = 0.5 , and ωt = 0.15 in case of cylindrical shape nanoparticles.
Conclusion Tis manuscript has been considered the MHD fow of Ferro-nanofuid near a vertical plate in the presence of thermal radiation, heat generation, and the shape efect of the nanoparticle. Te oscillating boundary conditions together with isothermal and ramped heating have been taken at the solid boundary. Te fow phenomenon has been modeled in the form of time-fractional Caputo-Fabrizio fractional derivatives. Te model has been solved for the exact analytical solutions via the Laplace transform method. Te obtained solutions for temperature and velocity feld have been simultaneously plotted for ramped and isothermal heating. Te results have been revealed that the temperature feld for blade shape nanoparticles is higher followed by platelet, spherical and brick shaped nanoparticles due to shape factor n whereas the velocity profle is higher for brick shape nanoparticles fowed by the blade, spherical and platelet shaped nanoparticles which depend on the values of shape constants a and b . Moreover, the temperature and velocity felds increase with increasing values of α near the plate in case of isothermal heating. But away from the plate, this efect reverses. Besides this the temperature feld increase with increasing φ . However, the velocity fled behaves opposite to this for φ . Meanwhile, the temperature and velocity felds increase with increasing Nr and Q . Finally, it has been noticed that the velocity feld decreases for
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Figure 15. Consequence of M on u y, t when α = 0.5 , φ = 0.04 , β = 0.5 , Gr = 5 , Nr = 0.5 , and ωt = 0.15 in case of cylindrical shape nanoparticles.
Figure 16. Consequence of Gr on u y, t when α = 0.5 , φ = 0.04 , β = 0.5 , M = 0.5 , Nr = 0.5 , and ωt = 0.15 for spherical s case.
increasing β and M whereas it increases with increasing Gr . Furthermore, this work can be extended in the future by developing a non-linear model with fractional derivatives. Meanwhile, the fractional boundary layer fow can be taken in a channel and cylindrical tubes with new fractional operators (Supplementary information S1).
Received: 8 August 2020; Accepted: 20 November 2020
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Scientifc Reports | (2021) 11:3725 | https://doi.org/10.1038/s41598-020-78421-z 21 Vol.:(0123456789) www.nature.com/scientificreports/
Acknowledgements Te authors would like to acknowledge Ministry of higher Education (MOHE) and Research Management Centre-UTM, Universiti Teknologi Malaysia (UTM) for the fnancial support through vote numbers 5F004, 07G70, 07G72, 07G76, 07G77, 08G33 and 5F278.for this research. Author contributions I.K. formulated the problem. A.Q.M transformed the problem into a dimensionless form. M.S. solved the problem and plotted the graphs. S.S. discussed the results. M.S and I.K. wrote the manuscript. S.S and A.Q.M. proofread the manuscript.
Competing interests Te authors declare no competing interests. Additional information Supplementary Information Te online version contains supplementary material available at https://doi. org/10.1038/s41598-020-78421-z. Correspondence and requests for materials should be addressed to I.K. or S.S. Reprints and permissions information is available at www.nature.com/reprints. Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afliations. Open Access Tis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. Te images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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