A fractional study of generalized Oldroyd-B fluid with ramped conditions via local & non-local kernels


 Convective flow is a self-sustained flow with the effect of the temperature gradient. The density is nonuniform due to the variation of temperature. The effect of the magnetic flux plays a major role in convective flow. The process of heat transfer is accompanied by mass transfer process; for instance condensation, evaporation and chemical process. Due to the applications of the heat and mass transfer combined effects in different field, the main aim of this paper is to do comprehensive analysis of heat and mass transfer of MHD unsteady Oldroyd-B fluid in the presence of ramped conditions. The new governing equations of MHD Oldroyd-B fluid have been fractionalized by means of singular and non-singular differentiable operators. In order to have an accurate physical significance of imposed conditions on the geometry of Oldroyd-B fluid, the ramped temperature, concentration and velocity are considered. The fractional solutions of temperature, concentration and velocity have been investigated by means of integral transform and inversion algorithm. The influence of physical parameters and flow is analyzed graphically via computational software (MATHCAD-15). The velocity profile decreases by increasing the Prandtl number. The existence of a Prandtl number may reflect the control of the thickness and enlargement of the thermal effect. The classical calculus is assumed as the instant rate of change of the output when the input level changes. Therefore it is not able to include the previous state of the system called the memory effect. Due to this reason, we applied the modern definition of fractional derivatives. Obtained generalized results are very important due to their vast applications in the field of engineering and applied sciences.


Introduction
Convective ow is a self-sustained ow that transfers heat energy into or out of the body by actual movement of uids particles that move energy with its mass. Thermal radiation and the e ect of magnetic ux plays an important role in convective ow. The di erent industrial problems and uid ow in the porous medium have achieved consideration in recent years. In the literature, di erent theories are made to see the phenomenon of heat transfer analysis. The convection heat transform between two heated cubes discusses by Mousazadeh et al. [1]. Sajad et al. [2] investigate the heat transfer and magnetic e ect on hybrid nano uids. Nazish et al. [3] analyze the in uence of heat and mass transform with a magnetic eld in the rate type uid model. The analysis of heat transfer mathematical model's subject to the slip boundary condition for the Maxwell uid discussed by Han et al. [4]. They explored the exact solutions using the e ect of relaxation time of the heat ux. Literature shows more interest in developed identical studies in [5][6][7].
Ramped heating plays a good role in real-life problems such as diagnoses of prognosis, analysis of heart function, and blood vessel system [8][9][10]. Moreover, Kundu [11] investigates the thermal therapy based on ramped heating to destroy the cancer cells on the human structure. Initially, convective viscous uid with ramped heating over vertical wall analyzes by Schertz [12] and Hayday [13]. The heat absorption ramped heating and thermal e ect near a moving wall discussed by Seth et al. [14]. Further authors [15] investigate the dynamical aspect of mass and heat transformation with Darcy's law, chemical reaction, and ther-mal conditions. Previously, there is less study which deals the parallel use of ramped heating with ramped velocity. It is complicated to apply these conditions, but they have broad signi cance as a physical aspect. Researchers investigated the ramped heating to investigate the ows of Newtonian and non-Newtonian [16][17][18][19]. In multiple subdivisions of emerging technologies, ramped wall velocity has found broad applications. For instance, in medical sciences, diagnoses of cardiovascular infections by means of treadmill testing (TT) or Ergo-meters is e cient employment of ramped velocity. Ramped velocity is a signi cant tool to recognize, determine medication, anticipate prognosis, and assess the working capability of blood vessels and heart.
The technique of fractional calculus has been used to formulate mathematical modeling in various technological development, engineering applications, and industrial sciences. Di erent valuable work has been discussed for modeling uid dynamics, signal processing, viscoelasticity, electrochemistry, and biological structure through fractional time derivatives [20]. This fractional di erential operator found useful conclusions for experts to treat cancer cells with a suitable amount of heat source and have compared the results to see the memory e ect of temperature function. As compared to classical models, the memory e ect is much stronger in fractional derivatives [21][22][23][24][25]. Over the last thirty years, Fractional derivative/calculus (FDs/FC) has captivated the numerous researchers after recognition of the fact that in comparison to the classical derivatives, FDs are more reliable operators to model the real-world physical phenomena. In dynamical problems, Fractional order models/ modeling are receiving a rapid popularity nowadays. The mathematical modeling of many physical and engineering models based on the idea of FC exhibits highly precise and accurate experimental results as compared to the models based on conventional calculus. For example, the fractional results of rate and di erential type's uids have a great resemblance with the results obtained experimentally. Tan [26], studied the generalized second grade uid and learn the analytic solution of time dependent Couette ow. Riaz et al. [27] investigate the optimal solution of unsteady generalized second grade uid via FD.
Riaz et al. [28] learn the view of newly FD operators of Maxwell uid in heat and mass transfer study. They consider the Maxwell uid and investigate the heat and mass transfer with the integer & non-integer order derivative. Some further investigation of uid ows and their properties equipped with FD establish in [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45]. Applications of combined impact of heat and mass transfer in engineering, applied sciences and FC, since they are connected to the historical data (memory e ect). Memory e ect in FC means the occurrence of process depends not only in the present state but also on the past history of the process. FC has ability to remember prior e ects of the input in order to calculate the current value of the output motivate us to investigate the time dependent natural convection ow of MHD Oldroyd-B uid.
The intent of this manuscript is to explore the analytical solution of MHD OBM with simultaneous use of ramped heating with ramped velocity. New de nitions of non-integer order derivatives C, CF and ABC implemented using Laplace integral transformation is used to gain the solution of velocity, temperature and concentration under impact of simultaneous use of ramped conditions. In Section (2), the dimensionless governing equations are developed. In Sections (3), (4) and (5), non-integer order derivatives with Laplace integral transform is used to nd the required solution of the concentration, temperature and velocity eld. In Section (6), the e ect of physical parameters is analyzed graphically. The concluding observation listed at the end.

Problem statement
We discussed the unsteady generalized OBF ow over an in nite plate. The ow representation and governing equations of OBF using geometry with appropriate conditions are analyzed in Figure 1 [33]: The dimensionless parameters are mentioned below: Applying (5) into (1) -(4), we get the set of dimensionless governing equations with corresponding conditions,

Solution of the temperature pro le . Caputo time derivatives
Fractional operators are quite exible for describing the behaviors of heat transfer of MHD Oldroyd-B model through the characterization of governing equations of the temperature (7) via Caputo-fractional operator (11) where C D ε τ is called Caputo-Fractional operator [29] and its inverse de ned below: Applying Eq. (12) to Eq. (10) with suitable condition on temperature, we have, The required solution of Eq. (13) is written as: We nd the unknown using (9) . Caputo-Fabrizio time derivatives where CF D ε τ is called Caputo-Fabrizio fractional operator [32] and its inverse de ned below: Applying Eq. (18) to Eq. (16) with suitable condition on temperature, we have, The required solution of Eq. (19) is written as: We nd the unknown using (9) .

Atangana-Baleanu time derivatives
Fractional operators are quite exible for describing the behaviors of heat transfer of MHD Oldroyd-B model through the characterization of governing equations of temperature (7) via ABC-fractional operator (23) of order ε. ABC where ABC D ε τ is called Atangana-Baleanu fractional operator [33,34] and its inverse de ned below: Applying Eq. (24) to Eq. (22) with suitable condition on temperature, The required solution of Eq. (25) is written as: We nd the unknown using (9)

Solution of the concentration pro le . Caputo time derivatives
Fractional operators are quite exible for describing the behaviors of heat transfer of MHD Oldroyd-B model through the characterization of governing equations of concentration (8) via Caputo-fractional operator, The required solution of Eq. (28) is written as: We nd the unknown using (9) .

Atangana-Baleanu time derivatives
The required solution of Eq. (34) is written as: We nd the unknown using (9)

Solution of the velocity pro le . Caputo time derivatives
We apply Eq. (12) for the solutions of the Eq. (6), The solution of homogeneous part of (37) is: The general solution can be given as

. Caputo-Fabrizio time derivatives
We apply Eq. (18) for the solutions of the Eq.(6), The complementary solution of (41) is: The general solution can be given as

Validations of results
a) If we neglect Gm = and (ε, ϖ) → , then the required results are identical which obtained by [33]. b) If we neglect Gm = , then the required results are identical which obtained by [19]. c) If we neglect = and r = , then required results are identical which obtained by [16].

Results and discussion
This part is devoted for physical interpretation of heat and mass transfer is executed on the motion of Oldroyd-B uid near a porous surface. The impact of thermal radiation, magnetic eld, and ramped conditions are also analyzed via Fractional derivative to obtain a solution via inversion algorithm. The impact of physical parameters such as Pr,M, Gm, Gr,K,ε,ϖ, and r on energy, concentration and velocity pro le are discussed using graphs. The time e ect on all fractional derivative operators and classical model analyzed in gure (2). It is clearly show that  for altered time the behavior of velocity pro le are same. The resultant velocity of ABC model is huge with respect      to other fractional models as well as classical model. Figure (3) analyzes the behavior depends on Speci c heat and conductivity of Pr . The velocity decreases as increase in the value of Pr . The lower Pr enhance the thermal conductivity and increase the boundary layer. Figure (4) investigate the domination of M on velocity components. The magnetic eld increases as the velocity decreases. By enhancing the value of M, the Lorentz force also increases. Due to this force the uid ow on the boundary layer is slow down. It is perceive that the behavior of uid pro le for ABC model is e ective as compared to other models. Thermal and isothermal conditions represent the domination of Gr shown in gures (5). Physically, Gr shows the relation between thermal forces to viscous force. For variation of time, the behavior of velocities is unique. The inuence of Gm is illustrated in gure (6). It is notice that resultant velocity increase with enhance of in all fractional operators. It is also show that velocity increase with increase of time. The velocity eld of ABC is huge as compared to Caputo and Caputo-Fabrizio. C, CF and ABC models analyzed the in uence of fractional parameters ε andϑ on velocity via graphs as shown in gures (7) and (8). With large value of, the velocity pro le also enhances. The behavior of velocity eld is same for variation of time. Further memory e ect of ABC is good as compared to other operators. Figure (8) represent the behavior of velocity pro le for another fractional parameterε. The behavior of ε is reverse toϑ. With increase inε, velocity eld reduce for variation of time. Di erent physical properties are more e ective to discuss in ABC model due to its non-local kernel. The in uence of is illustrated in gure (9). It is notice that resultant velocity decrease with enhance of in all fractional operators due to thickness of boundary layer. It is also show that velocity increase with increase of time. Figures (10) show the e ect of retardation time r. The behavior of and r are reversible. Enhance r, the resultant velocity enhance with variation of time. The in uence of all physical parameters on velocity pro le using ABC model is more e ective as compared to other models.

Conclusions
The comprehensive analysis of the time fractional Analysis and heat transfer of a Oldroyd-B uid in the presence of magnetic eld with ramped conditions via has been investigated. To obtained the solution by using Laplace transformation, to compare the results between C, CF and ABC. To demonstrated in several graphs to analyze the e ects of all parameters. The following major ndings of this study are given below: i) It is observed that the behaviors of uid velocity for relaxation and retardation pro les are opposite to each other.
ii) Velocity behaves as a decreasing function for M and Pr.
iii) Increasing the worth of Gr and Gm, the velocity pro le also enhances. iv) Velocity eld for the ABC fractional operator is higher than the CF and Caputo fractional operator. v) As the fractional parameter approaches to 1, then fractional models convert into classical model. vi) Ramping of the enclosing wall is a salient technique to control the temperature and velocity of the uid.