Shape effect on MHD flow of time fractional Ferro-Brinkman type nanofluid with ramped heating

The colloidal suspension of nanometer-sized particles of Fe3O4 in traditional base fluids is referred to as Ferro-nanofluids. These fluids 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-nanofluids, the main objective of this study is to consider the MHD flow of water-based Ferro-nanofluid in the presence of thermal radiation, heat generation, and nanoparticle shape effect. The Caputo-Fabrizio time-fractional Brinkman type fluid model is utilized to demonstrate the proposed flow 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 figures to study the influence of pertinent flow parameters. The results revealed that the fractional parameter has a great impact on both temperature and velocity fields. In the case of ramped heating, both temperature and velocity fields 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 fields are reduced to classical form and validated with already published results.


Description of the problem
Assume heat transfer in MHD flow of time-fractional Ferro-Brinkman type nanofluid near an infinite vertical plate along the x-axis and y-axis is selected transverse to it. Magnetic nanoparticles of different shapes (blade, brick, spherical, and platelet) are dispersed into the water as base fluid to form Ferro-Brinkman type nanofluid. Thermal radiation, heat generation, and ramped wall heating are also considered. At t ≤ 0 , the fluid and plate are set in rest with u y, 0 = 0 and T y, 0 = T ∞ where T ∞ is the ambient temperature. Afterward at t = 0 + , the velocity and temperature fields are switched to u(0, t) = U 0 H(t) cos (ωt) and T(0, t) = T 0 + (T W − T ∞ )t/t 0 if 0 < t < t 0 or T(0, t) = T W if t > t 0 respectively. At this phase, the fluid starts flowing in x-direction as presented in Fig. 1. The Ferro-Brinkman type nanofluid experience magnetic force because the fluid is electrically www.nature.com/scientificreports/ conducting, thereby, an external magnetic field is employed normally to the flow direction. The governing equations of the proposed model are derived in the following section.

Mathematese formulation
In accordance with Rajagopal 30 and Fetecau, Fetecau 31 , the linear momentum equation for Brinkman type fluid can be written as where D Dt = ∂ ∂t + u ∂ ∂x + v ∂ ∂y + w ∂ ∂z refers to material time derivatives, V is the velocity vector, T represents the Cauchy stress tensor, ρF depicts the body forces, and I 0 exhibits the interaction force of porous medium which can be expressed as where α d is a positive coefficient of drag force which yields Eq. (1) to the following form In the case of Brinkman type fluid, the constitutive equation of Cauchy stress tensor is expressed by 32 where p is the scalar pressure, I is the identity tensor, µ is the dynamic viscosity, and A 1 is the Rivlin-Ericksen tensor determined by 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 flow is considered thereby, the velocity vector is defined as Bearing in mind, Eqs. (4) and (6), the ∇ · T is determined as whereas, the fluid flow 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 Based on Jaluria 33 , the body forces ρF for convection flow of electrically conducting Brinkman type fluid is given by where J × B is the Lorentz force, J is the current density, B = B 0 + b is the magnetic flux intensity, B 0 is applied magnetics filed acting in y-direction, b is the induced magnetic field and g = −g, 0, 0 is the gravitation acceleration. The Lorentz force can be defined by Maxwell's set of equations as 34 where µ m is the magnetic permeability and E is electric field intensity. The current density J is described by the generalized Ohm's law as 35 where σ is the electrical conductivity. Meanwhile, the magnetic Reynolds number is assumed small enough so that the induced field b is neglected compared to the applied field B 0 . Furthermore, it is assumed that there is no polarization and applied voltages thereby, the electric field E is ignored. Hence, Eq. (11) takes the following form (4) T = −pI + µA 1 , where β * = α d /ρ is the Brinkman type fluid parameter which corresponds to the drag force of highly non-Darcy's porous medium. The energy equation together with thermal radiation and heat generation is given by 37 The radiative heat flux q r in Eq. (22) is formulated by using via the Roseland approximation as 38 The T 4 is expanded along T ∞ by using the Taylor series as The temperature gradient is assumed to be small enough so, the higher-order terms are neglected which yield to The further simplification of Eq. (25) yield to the following The simplified form of T 4 from Eq. (26), is used in Eq. (23) which yield to (13)  For enhanced heat transfer, the Fe 3 O 4 has been dispersed into the water as base fluid to form water-Ferronanofluid. As refer to Khanafar et al. 39 and Tiwari and Das 40 Eqs. (21 and (29) can be written for Ferro-nanofluid flow as where ρ nf is the density, u y, t is the velocity, β * is the Brinkman type fluid parameter, µ nf is the dynamic velocity, σ nf is the electrical conductivity, B 0 is the uniform magnetic field, g gravitational acceleration, (β T ) nf is the thermal expansion, T y, t is the temperature, C p nf is the heat capacitance, k nf is the thermal conductivity, q r is the radiative heat flux and Q 0 is the heat generation. The corresponding initial and boundary conditions are given as The terms ρ nf , µ nf , σ nf , (β T ) nf , C p nf and k nf appeared in Eqs. (30) and (31) for the enhanced thermophysical properties nanofluid with different shapes (blade, brick, spherical, and platelet) nanoparticles defined as 41 . and where the subscript nf is used for nanofluid, f for base fluid water, and s for solid nanoparticles Fe 3 O 4 . Furthermore, in Eq. (35) a and b correspond to shape constant which affects the density factor of nanofluid and in Eq. (39) n is the experimental shape constituent. n can be evaluated as (27)

Solutions of the problem
This section presents the exact solutions for the magnetic flow of time-fractional Ferro-Brinkman type nanofluid under the effect of a normal magnetic field. In this section, the problem modeled in Sect. 3 is first transformed to dimensionless form to diminish the units for simplification and reduction of number variables. For this purpose, the following dimensionless variables are implemented into Eqs. (30)-(33) after dropping the * sign for simplicity yield to the following form together with the following dimensionless conditions and 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, Pr eff is the effective Prandtl number, Q is the heat generation parameter, and φ 0 , φ 1 , φ 2 , φ 3 , φ 4 , φ 5 are constant terms. The Caputo-Fabrizio time-fractional derivative is used to transform Eqs. (41) and (42) to time-fractional for as The Caputo-Fabrizio time-fractional operator D α t (, .,) appeared in Eqs. (45) and (46)   where H(t − 1) is the Heaviside unit step function and the term θ Ramp y, t is defined by where Solutions for temperature filed with isothermal heating. In order to find exact solutions for isothermal heating, the boundary condition in the Laplace transform domain is given by The exact analytical solutions of Eq. (52) is obtained by using Eq. (59) as The final exact solution for isothermal heating is obtained after applying the inverse Laplace transform to Eq. (60) which yield to The exact solutions corresponding ramped and isothermal heating are respectively depicted in Eqs. (57) and (61) which satisfy the impose conditions in both cases. The exact solutions for the velocity field corresponding to ramped and the isothermal heating is presented in the following section.
Solution for velocity field. This section presents exact analytical solutions for the velocity field for both ramped and isothermal heating.
Solutions for velocity field with ramped heating. The Laplace transform is applied to Eq. (45) using initial condition from Eq. (43) yield to which takes the following form after simplification along with the transformed velocity boundary conditions The 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 field with ramped heating.
Solutions for velocity field with isothermal heating. Next, Eq. (45) is solved again for isothermal heating as For convenience in inverse Laplace transform, Eq. (74) can be written in more in suitable form as Gr.
(74) u Iso y, q = q q 2 + ω 2 e (77) u y, t = u C y, t + u 1 (t) * u 3(Iso) y, t − H(t − 1)u 3(Iso) y, t − 1 − u 1 y, t * θ y, t , Limiting solution for velocity field. The limiting solutions for the velocity field in case of ramped and isothermal heating is introduced in this subsection.
Limiting solution for velocity field with ramped heating. While taking into account Eqs.
(90)   45 and highlighted in Fig. 3. It is found that these solutions are alike which shows the correctness of the present results.

Results and Discussion
The exact analytical solutions (solutions for ramped and isothermal heating) for temperature and velocity fields are computed and displayed in numerous graphs to study the impact of pertinent parameters such as fractional parameter α , volume concentration φ , shape effect 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.

Effects of flow parameters on both velocity and temperature fields. Figures 4 and 5 present the
impact of α on temperature and velocity fields. In the case of isothermal heating, both the temperature and velocity field 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 justified as α is increasing the thickness of thermal and momentum boundary www.nature.com/scientificreports/ layers gradually increasing and became thickest near the plate at α = 1 . However, away from the plate, the thermal and momentum boundary layers behave oppositely. In the case of ramped heating, the trend of temperature and velocity profile is straight forward. Increasing the value of α , decreasing the temperature and velocity fields. This 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 fields decreased. Additionally, as in Figs. 6 and 7 it can be observed that the temperature and velocity field are significantly affected by φ . It is found that the temperature filed increases with increasing values of φ for both ramped and isothermal heating. It can be clearly seen from Eq. (39) that an increment in φ corresponds to the enhancement in thermal conductivity of the nanofluid as a result the temperature profile increases. Furthermore, it can be seen in Fig. 4 that the velocity field decrease with increasing values of φ for both ramped and isothermal heating. This is due to the dynamic viscosity of nanofluid presented in Eq. (35). The 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 nanofluid and the fluid became thick. Hence, an increase in viscosity resists to nanofluid flow. Figures 8 and 9 depict the comparison of temperature and velocity fields for different shapes of nanoparticles. It is noticed from Fig. 8 that the temperature field for blade shape nanoparticles is higher followed by platelet,  www.nature.com/scientificreports/ spherical, and brick shaped nanoparticles due to the shape factor n involving in Eq. (39). Besides this, the velocity profile is higher for brick shape nanoparticles flowed 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 temperature and velocity fields 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 field as Nr indicates the proportional contribution of conduction heat transfer to the thermal radiation. Hence, the temperature field signifying an increasing trend. Furthermore, increasing Nr twist the rate of heat transfer to the nanofluid as a result the attractive forces holding the nanofluid molecule weaken as a result, decreasing the viscosity which accelerates the fluid velocity. Variations in temperature and velocity fields 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 fields 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 field increases.
The impact of flow parameters which effect only the velocity field. The influence of the Brinkman type fluid parameter β on the velocity profiles for isothermal and ramped heating is displayed in Fig. 14. β is  www.nature.com/scientificreports/ the magnitude of the drag force of a highly non-Darcy's porous medium. The velocity fields 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 field. Meanwhile, the effect of the magnetic parameter M is illustrated in Fig. 15 on the ramped and isothermal velocity fields. It is revealed that the isothermal velocity is higher than the ramped velocity. The isothermal and ramped velocity fields decelerated together for greater values of M due to the applied magnetic field which results in the presence of intense Lorentz force. This force works as a dragging force exhibits persistent resistance to the nanofluid flow. Ultimately, the isothermal and ramped velocity filed dropped. But away from the plate, the Lorentz force became poor and nanofluid comes to rest. Besides this, the influence of thermal Grashof number Gr is highlighted in Fig. 16 for both ramped and  In the proposed problem, the convection flow of nanofluid driven by thermal buoyancy force is considered. As a result, it has a tendency to increase the velocity field in both ramped and isothermal heating cases.

Conclusion
This manuscript has been considered the MHD flow of Ferro-nanofluid near a vertical plate in the presence of thermal radiation, heat generation, and the shape effect of the nanoparticle. The oscillating boundary conditions together with isothermal and ramped heating have been taken at the solid boundary. The flow phenomenon has been modeled in the form of time-fractional Caputo-Fabrizio fractional derivatives. The model has been solved for the exact analytical solutions via the Laplace transform method. The obtained solutions for temperature and velocity field have been simultaneously plotted for ramped and isothermal heating. The results have been revealed that the temperature field for blade shape nanoparticles is higher followed by platelet, spherical and brick shaped nanoparticles due to shape factor n whereas the velocity profile is higher for brick shape nanoparticles flowed by the blade, spherical and platelet shaped nanoparticles which depend on the values of shape constants a and b . Moreover, the temperature and velocity fields increase with increasing values of α near the plate in case of isothermal heating. But away from the plate, this effect reverses. Besides this the temperature field increase with increasing φ . However, the velocity filed behaves opposite to this for φ . Meanwhile, the temperature and velocity fields increase with increasing Nr and Q . Finally, it has been noticed that the velocity field decreases for