# An Exact and Comparative Analysis of MHD Free Convection Flow of Water-Based Nanoparticles via CF Derivative

Authors: Aziz-Ur- Rehman; Muhammad Bilal Riaz; Syed Tauseef Saeed; Fahd Jarad; Hayder Natiq Jasim; Aytekin Enver
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Convective flow is a self-sustained flow with the effect of the temperature gradient. The density is nonuniform due to the variation in temperature. The effect of the magnetic flux plays a major role in convective flow. The process of heat transfer is accompanied by a mass transfer process, for instance, condensation, evaporation, and chemical process. Combination of water as base fluid and three types of nanoparticles named as copper, titanium dioxide, and aluminum oxide is taken into account. Due to the applications of the heat and mass transfer combined effects in different fields, the main aim of this paper is to do a comprehensive analysis of heat and mass transfer of MHD natural convection flow of water-based nano-particles in the presence of ramped conditions with Caputo–Fabrizio fractional time derivative. The exact fractional solutions of temperature, concentration, and velocity have been investigated by means of integral transform. 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. But, in the fractional calculus (FC), the rate of change is affected by all points of the considered interval to incorporate the previous history/memory effects of any system. Due to this reason, we applied the modern definition of fractional derivative. Here, the order of the fractional derivatives will be treated as an index of memory. The influence of physical parameters and flow is analyzed graphically via computational software (MATHCAD-15). Our results suggest that the incremental value of theM is observed for a decrease in the velocity field, which reflects to control resistive force.


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## Body

## 1. Introduction

Convective flow is a self-sustained flow that transfers heat energy into or out of the body by actual movement of fluids particles that move energy with its mass. Thermal radiation and the effect of magnetic flux play an important role in convective flow. The different industrial problems and fluid flow in the porous medium have achieved consideration in recent years. In the literature, different theories are made to see the phenomenon of heat and mass transfer analysis. Radiation, convection, and conduction are three modes of heat transfer. Convection can be defined as heat transfer by the substance motion which may be air or water. It plays a central role in creating the weather clause on the plant. Nanoliquids are used to enhance the thermo-physical properties and the performance of conventional fluids in cooling and heating methods such as decrease the power of nuclear reactors, reduce temperature in vehicles radiators, controlled heat in different computer progressions, and handling thermal flows. In pharmaceutical industry, diagnoses and treatment of cancer are based on nanoliquid operators which comprise of different radiations. These noteworthy physical attributes of nanofluids and their implications are fascinating scientists and researchers. The term nanofluid is referred to the addition of some solid nanoparticles in regular fluid, sometimes known as base fluid. This idea was first introduced by Choi [1]. Nano-sized particles are used to raise the thermal conductivity of traditional fluids such as water and mineral oils. The creation of nanometer-sized particles involves carbides, carbon nanotubes, and metals. Nanocompounds have substantial applications in several procedures such as drugs delivery, water purification, bio diesel invention, and creation of carbon nanotubes, according to Sarli et al. [2]. Masuda et al. [3] presented higher thermo physical properties in nanofluids due to some nanosized particles, undoubtedly, huge difference in the structure of nanoparticles because it has different shape and size. Das et al. [4] studied that for different temperature range between 200 C to 500 C, the thermal conductivities of TiO2 and Al2O3 water-base fluids increased at most four times.The study of mass and thermal flows of incompressible, viscous nanofluids is highly significant because of essential applications of such flows in engineering, chemistry, and physics. Imposition of external magnetic field and placement of cavities filled with fluid and porous medium affect the flow of electrically insulated fluid in bearings, pumps, MHD motors, and generators. Such cavities can be portioned as horizontal and vertical cavities. Hamad et al. [5] observed the characteristics of fluid flow containing nanometer-sized particles over a vertical plate in the presence of the external magnetical field. The impacts of magnetic strength on some nanofluid flow are investigated by Das and Jana [6]. Turkyilmazoglu [7] discussed the mass with heat transmission of exact investigation of MHD flow of some fluids having nanometer-sized particles. The movement of nanofluid on a porous surface placement in a revolving system is examined by Sheikholeslami and Ganji [8]. Husanan et al. [9] presented the unsteady movement of nanofluids nested in a porous medium in the existence of an electromagnetic field. In [10–12], researchers studied and discussed the influence of exothermically and radiating heat for some microlevel fluid flow. Khan et al. [13] studied Casson-type nanofluid movement in the occurrence of thermo-radiation and heat consumption. In [14–18], some identical studies can be investigated. Ahmed and Dutta [19] first time floated the idea of effective velocity and temperature with ramped wall conditions at a similar time for mass transmission of Newtonian unsteady fluid flow transient through impulsively affecting long vertical plate. Generally, ramped velocity has a great advantage in the medical field especially diagnosed, and treatment for heart, blood, and cancer diseases is discussed and studied in [20–22]. Schetz [23], Hayday [24], and Malhotra et al. [25] are investigated nonuniform and time-dependent temperatures with ramped conditions. Kundu [26] highlighted important operability time-dependent conditions for temperature and also suggested five different forms of heating. Keolyar et al. [27] observed controlled heat conditions on unstable MHD radiated movement of some fluids with nanoparticle mixtures passing through a flat surface plate. Some researchers have discussed noteworthy facts of heat emission and mass transfer and elaborated under dissimilar physical phenomena in [28–30].The technique of fractional calculus has been used to formulate mathematical modeling in various technological development, engineering applications, and industrial sciences. Different valuable work has been discussed for modeling fluid dynamics, signal processing, viscoelasticity, electrochemistry, and biological structure through fractional time derivatives [31–33]. This fractional differential 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 effect of temperature function. As compared to classical models, the memory effect is much stronger in fractional derivatives [34–39]. Over the last thirty years, fractional derivative/calculus (FDs/FC) has captivated numerous researchers after recognition of the fact that in comparison to the classical derivatives, FDs are more reliable operators to model real-world physical phenomena. In dynamical problems, fractional-order model/modeling is receiving 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 differential type fluids have a great resemblance with the results obtained experimentally. For nonsingular kernel convective flow with ramped conditions, the temperature is studied by Riaz et al. [40]. Furthermore, the same author Riaz et al. [41] highlighted the heat effect on MHD Maxwell fluid by using local and nonlocal operators. Some other associated references dealing with fractional differential operators, MHD Maxwell fluid movement, heat emission, or fractional second-grade fluid are given in [42–54].Recently, Talha Anwar et al. [55] studied MHD-free convective flow of water base nanofluids having three types of nanoparticles named copper, titanium dioxide, and aluminum oxide [56], with the classical approach. They have not analyzed the behavior of fractional derivatives. The intent of this manuscript is to explore the exact and closed-form solution of MHD-free convective flow of water base nanofluids with the Caputo–Fabrizio fractional operator with simultaneous use of ramped heating with ramped velocity. Laplace integral transformation is used to gain the solutions of velocity, temperature, and concentration under the impact of ramped conditions. In Section 2, the dimensionless governing equations are developed. In Section 3, a noninteger order derivative with Laplace integral transform is used to find the required solution of the concentration, temperature, and velocity field. In Section 4, the effect of physical parameters is analyzed graphically. The concluding observation is listed at the end.


## 2. Mathematical Model

Let us assume that incompressible unsteady-free convective MHD fluid movement through a vertical plate is nested in a porous material with ramped temperature and energy transmission of a nanofluid. Both the fluid and plate have temperatureT∞, concentration C∞ at initial time τ=0. At τ>0, motion is started in the plate with velocity U0τ/τ0, also temperature raised to T∞+Tw−T∞τ/τ0 in the long vertical plate for 0<τ≤τ0. But, later on, the plate maintained constant temperature Tw, concentration Cw, and moving with uniform velocity U0 for τ>τ0. Suppose that the fluid flow is unidirectional and one dimensional with x-axis is assumed along the vertical plate, y-axis is considered in perpendicular direction to the plate, and plate is placed at y=0 but nanofluid movement to be constrained for y>0. Figure 1 provided the geometrical and physical interpretation of the considered model. Also, nanofluid is contained water as a base fluid and Cu,TiO2 and Al2O3 as nanoparticles. In the light of all aforementioned assumptions, the principal governing equations for a nanofluid can be expressed subject to the Boussinesq’s approximation are given as [19, 46, 55](1)ρnf∂uy,τ∂τ=μnf∂2uy,τ∂y2+μnfγ2k∗uy,τ+gρβTnfTy,τ−T∞+gρβCnfCy,τ−C∞−σnfβ02uy,τ,ρCpnf∂Ty,τ∂τ=Knf+16σrT∞33kr∂2Ty,τ∂y2−Q0Ty,τ+Q0T∞,∂Cy,τ∂τ=Dnf∂2Cy,τ∂τ2.Figure 1
Geometrical presentation of the stated problem.The corresponding initial and boundary conditions are stated as(2)uy,0=0,Ty,0=T∞,Cy,0=C∞,y≥0.(3)u0,τ=f1τ,T0,τ=f2τ,C0,τ=Cw.(4)f1τ=U0ττ0,0<τ≤τ0;U0,τ>τ0,f2τ=T∞+Tw−T∞ττ0,0<τ≤τ0;Tw,τ>τ0.(5)uy,τ⟶0,Ty,τ⟶∞,Cy,τ⟶∞asy⟶∞.The expressions for different properties of nanofluids then conventional fluids are density, specific heat capacity, coefficient of thermal expansion, electrical conductivity, thermal diffusion coefficient and dynamic viscosity are denoted byρnf, ρCpnf, ρβnf, σnf, Dnf, and μnf respectively and defined as(6)μnf=μw1−ϕ2.5,ρnf=ρw1−ϕ+ϕρnfρw,σnf=σw1+3ϕσ−1σ+2−ϕσ−1,σ=σnfσw,ρCpnf=ρCpw1−ϕ+ϕρCpnfρCpw,Dnf=1−ϕDf,ρβnf=ρβw1−ϕ+ϕρβnfρβw.Also, the effective role of thermal conductivity for nanoparticles is discussed as(7)KnfKw=Knp+2Kw−2Kw−KnfϕKnp+2Kw−Kw−Knfϕ.Subscripts used in the above equations arew, nf, and np which are represented as base fluid, nanofluid, and nanoparticles, respectively. For nondimensionalization, we considered the following set of new variables:(8)ζ=U0υwy,τ∗=U02υwτ,u∗=uu0,T∗=T−T∞Tw−T∞,Gr=gTw−T∞υβTwU03,C∗=C−C∞Cw−C∞,Gm=gC−C∞υβcwU0,M=σwβ02υwρU02,Nr=16σrT∞33kwkr,Pr=μCpwkw,1K=υw2γ2k∗u02,Sc=υDf,Q=Q0U02υρCpw.After employing the dimensionless quantities and removing the star notation, the following partial differential equations in the dimensionless form are derived as(9)π1∂uζ,τ∂τ=π4∂2uζ,τ∂ζ2+π2GrTζ,τ+π7GmTζ,τ−π3Muζ,τ−π4uζ,τK,(10)∂Tζ,τ∂τ=π5+Nrπ6Pr∂2Tζ,τ∂ζ2−Qπ6Tζ,τ,(11)∂Cζ,τ∂τ=b∂2Cζ,τ∂ζ2,where(12)π1=1−ϕ+ϕρnfρw,π2=1−ϕ+ϕρβTnfρβTw,π3=1+3ϕσ−1σ+2−ϕσ−1,π4=11−ϕ2.5,π5=Knp+2Kw−2Kw−KnfϕKnp+2Kw−Kw−Knfϕ,π6=1−ϕ+ϕρCpnfρCpw,π7=1−ϕ+ϕρβCnfρβCw,b=1−ϕSc,with stated conditions in the dimensionless form are(13)uζ,0=0,Tζ,0=0,Cζ,0=0,ζ≥0,u0,τ=Λτ,T0,τ=Λτ,0,τ=1whereΛτ=τ,0<τ≤1,1,τ>1,uζ,τ⟶0,Tζ,τ⟶0,Cζ,τ⟶0asζ⟶∞.


## 3. Solution of the Problem

### 3.1. Exact Solution of Heat Profile

Applying the definition provided in equation (15) for CF on (6) and plugging equations (8), (9), and (10) yield(14)CFDταTζ,τ=π5+Nrπ6Pr∂2Tζ,τ∂ζ2−Qπ6Tζ,τ,where CFDτℑ is called Caputo–Fabrizio fractional operator [37], and its inverse defined is as follows:(15)CFDt℘fy,t=11−℘∫0texp−℘t−℘1−℘∂fy,τ∂τdτ,0<℘<1,(16)ℒCFDt℘fy,t=sℒfy,t−fy,01−℘s+℘,where ℘ is a fractional parameter. The above equation becomes after employing the Laplace transformation(17)∂2T¯ζ,p∂ζ2−βp1−αp+α+λT¯ζ,p=0,where(18)β=π6Prπ5+Nr,λ=QPrπ5+Nr.By using suitable conditions, the (17) has the solution in the form of(19)T¯ζ,p=1−e−pp2e−ζβp/1−αp+α+λ,=1−e−pp2e−ζb2p/p+η2+λ,=e−ζb2p/p+η2+λp2−e−pe−ζb2p/p+η2+λp2=Tr¯ζ,p−e−pTr¯ζ,p.After applying inverse Laplace transformation on the above equation, we get the solution(20)Tζ,τ=Trζ,τ−Trζ,τ0Gτ0,where(21)Trζ,τ=L−1e−ζλ+ψp/1+ϵpp2,=ψϵ∫0∞∫0τe−wψ+zϵerfcζ2zI02ϵψ−λϵwzdzdw+λϵ∫0∞∫0ν∫0τe−w1+λ+pϵerfcζ2zI02ϵψ−λϵqwdzdqdw,Gτ0 represent a standard Heaviside function with(22)τ0=τ−1,b2=β1−α,η2=α1−α,ψ=b2+λη2,ϵ=1η2.It is noticed that whenα⟶1, the results derived by CF in (14) for ramped wall temperature distribution are the same as those obtained by Talha Anwar et al. [55], which proves the validity of the derived results with the published literature.


### 3.2. Exact Solution of Mass Profile

Applying the definition provided in equation (15) for CF on equation (3) and plugging equations (3), (9), and (10) yield(23)DταCFCζ,τ=1−ϕSc∂2Cζ,τ∂ζ2.The above equation becomes after employing the Laplace transformation(24)∂2C¯ζ,p∂ζ2−bp1−αp+αC¯ζ,p=0.By using the equation, the (11) has the solution in the form(25)C¯ζ,p=1pe−ζbp/1−αp+α.Employing inverse Laplace transformation, the exact solution for concentration is written as(26)Cζ,τ=L−1e−ζbp/1−αp+αp,=1−2bπ∫0∞sinζ/1−αχχb+χ2e−α/1−αtχ2dχ.


### 3.3. Exact Solution of Velocity Profile

Applying CF fractional time derivative operator as mentioned in equation (15) on (5) gives(27)ηCFDταuζ,τ=∂2uζ,τ∂ζ2+mGrTζ,τ+nGmCζ,τ−wuζ,τ.We employ Laplace transformation on the above equation and equations (8), (9), and (10) which gives(28)d2u¯ζ,pdζ2−ηp1−αp+α+wu¯ζ,p=−mGrT¯ζ,τ−nGmC¯ζ,τ,where(29)η=π1π4,w=Mπ3π4+1K,m=π2π4,n=π7π4.The result of the homogeneous part of velocity (28) is(30)uc¯ζ,p=c1eζη1p/p+η2+w+c2e−ζη1p/p+η2+w,and the particular solution is written as(31)up¯ζ,p=−mGr1−e−p/p2e−ζb2p/p+η2+λb2p/p+η2+λ−η1p/p+η2+w−nGm1/pe−ζb1p/p+η2b1p/p+η2−η1p/p+η2+w.So, the required solution of (28) has the following form:(32)u¯ζ,p=c1eζη1p/p+η2+w+c2e−ζη1p/p+η2+w−mGr1−e−p/p2e−ζb2p/p+η2+λb2p/p+η2+λ−η1p/p+η2+w−nGm1/pe−ζb1p/p+η2b1p/p+η2−η1p/p+η2+w.To find constantsc1 and c2 by using corresponding conditions, the solution is written as(33)u¯ζ,p=1−e−pp2e−ζη1p/p+η2+w+mGr1−e−pp+η2p2b3p+b4e−ζη1p/p+η2+w−e−ζb2p/p+η2+λ+nGmp+η2pb5p−b6e−ζη1p/p+η2+w−e−ζb1p/p+η2,where(34)η1=η1−α,η2=α1−α,b1=b1−α,b2=β1−α,b3=b2−η1+λ−w,b4=λ−wη2,b5=b1−η1−w,b6=wη2.Equation (33) can be written in a more precise form as(35)u¯ζ,p=Φζ,p−e−pΦζ,p+mGrb3Φζ,p−e−pΦζ,p−T¯ζ,p+mGrη4b3δζ,pΦζ,p−e−pδζ,pΦζ,p−δζ,pT¯ζ,p+nGmb5μζ,p−C¯ζ,p+nGmη6b5φζ,pμζ,p−φζ,pC¯ζ,p.After applying inverse Laplace transformation on the above equation, we get the solution(36)uζ,τ=Φζ,τ−Φζ,τ0Gτ0+mGrb3Φζ,τ−Φζ,τ0Gτ0−Tζ,τ+mGrη4b3Υζ,τ−Υζ,τ0Gτ0−δ∗Tτ+nGmb5μζ,τ−Cζ,τ+nGmη6b5φ∗μτ−φ∗Cτ,where(37)Φζ,τ=L−1e−ζw+γp/1+ϵpp2,=γϵ∫0∞∫0τe−sγ+zϵerfcζ2zI02ϵγ−wϵszdzds+wϵ∫0∞∫0ν∫0τe−s1+w+pϵerfcζ2zI02ϵγ−wϵqsdzdqds,and Gτ0 represent a standard Heaviside function with(38)τ0=τ−1,b2=β1−α,η2=α1−α,ψ=b2+λη2,γ=η1+wη2ϵ=1η2,η3=−b4b3,η4=η2+η3,η5=b6b5,η6=η2+η5,δζ,τ=L−11p−η3=eη3τ,φζ,τ=L−11p−η5=eη5τ,Υζ,τ=δ∗Φτ,μζ,τ=L−1e−ζw+γp/1+ϵpp,=e−ζw−ζγ2π∫0∞∫0τ1τe−ϵτ+ζ2/4v+wvI12γvτdτdv.It is detected that whenα⟶1 and Gm=0, the fractionalized model CF in equation (3) for ramped wall velocity profile is converted into integer order model as calculated by Talha Anwar et al. [55]. Also, we achieved the same results as Seth et al. [29] for ramped wall temperature with ϕ=0 and Gm=0. Moreover, we derived similar solutions as obtained by Chandran et al. [28] when porosity parameter 1/K⟶0, Gm=0, and magnetic parameter M⟶0. This authenticates the current acquired results with the published literature.


## 3.1. Exact Solution of Heat Profile

Applying the definition provided in equation (15) for CF on (6) and plugging equations (8), (9), and (10) yield(14)CFDταTζ,τ=π5+Nrπ6Pr∂2Tζ,τ∂ζ2−Qπ6Tζ,τ,where CFDτℑ is called Caputo–Fabrizio fractional operator [37], and its inverse defined is as follows:(15)CFDt℘fy,t=11−℘∫0texp−℘t−℘1−℘∂fy,τ∂τdτ,0<℘<1,(16)ℒCFDt℘fy,t=sℒfy,t−fy,01−℘s+℘,where ℘ is a fractional parameter. The above equation becomes after employing the Laplace transformation(17)∂2T¯ζ,p∂ζ2−βp1−αp+α+λT¯ζ,p=0,where(18)β=π6Prπ5+Nr,λ=QPrπ5+Nr.By using suitable conditions, the (17) has the solution in the form of(19)T¯ζ,p=1−e−pp2e−ζβp/1−αp+α+λ,=1−e−pp2e−ζb2p/p+η2+λ,=e−ζb2p/p+η2+λp2−e−pe−ζb2p/p+η2+λp2=Tr¯ζ,p−e−pTr¯ζ,p.After applying inverse Laplace transformation on the above equation, we get the solution(20)Tζ,τ=Trζ,τ−Trζ,τ0Gτ0,where(21)Trζ,τ=L−1e−ζλ+ψp/1+ϵpp2,=ψϵ∫0∞∫0τe−wψ+zϵerfcζ2zI02ϵψ−λϵwzdzdw+λϵ∫0∞∫0ν∫0τe−w1+λ+pϵerfcζ2zI02ϵψ−λϵqwdzdqdw,Gτ0 represent a standard Heaviside function with(22)τ0=τ−1,b2=β1−α,η2=α1−α,ψ=b2+λη2,ϵ=1η2.It is noticed that whenα⟶1, the results derived by CF in (14) for ramped wall temperature distribution are the same as those obtained by Talha Anwar et al. [55], which proves the validity of the derived results with the published literature.


## 3.2. Exact Solution of Mass Profile

Applying the definition provided in equation (15) for CF on equation (3) and plugging equations (3), (9), and (10) yield(23)DταCFCζ,τ=1−ϕSc∂2Cζ,τ∂ζ2.The above equation becomes after employing the Laplace transformation(24)∂2C¯ζ,p∂ζ2−bp1−αp+αC¯ζ,p=0.By using the equation, the (11) has the solution in the form(25)C¯ζ,p=1pe−ζbp/1−αp+α.Employing inverse Laplace transformation, the exact solution for concentration is written as(26)Cζ,τ=L−1e−ζbp/1−αp+αp,=1−2bπ∫0∞sinζ/1−αχχb+χ2e−α/1−αtχ2dχ.


## 3.3. Exact Solution of Velocity Profile

Applying CF fractional time derivative operator as mentioned in equation (15) on (5) gives(27)ηCFDταuζ,τ=∂2uζ,τ∂ζ2+mGrTζ,τ+nGmCζ,τ−wuζ,τ.We employ Laplace transformation on the above equation and equations (8), (9), and (10) which gives(28)d2u¯ζ,pdζ2−ηp1−αp+α+wu¯ζ,p=−mGrT¯ζ,τ−nGmC¯ζ,τ,where(29)η=π1π4,w=Mπ3π4+1K,m=π2π4,n=π7π4.The result of the homogeneous part of velocity (28) is(30)uc¯ζ,p=c1eζη1p/p+η2+w+c2e−ζη1p/p+η2+w,and the particular solution is written as(31)up¯ζ,p=−mGr1−e−p/p2e−ζb2p/p+η2+λb2p/p+η2+λ−η1p/p+η2+w−nGm1/pe−ζb1p/p+η2b1p/p+η2−η1p/p+η2+w.So, the required solution of (28) has the following form:(32)u¯ζ,p=c1eζη1p/p+η2+w+c2e−ζη1p/p+η2+w−mGr1−e−p/p2e−ζb2p/p+η2+λb2p/p+η2+λ−η1p/p+η2+w−nGm1/pe−ζb1p/p+η2b1p/p+η2−η1p/p+η2+w.To find constantsc1 and c2 by using corresponding conditions, the solution is written as(33)u¯ζ,p=1−e−pp2e−ζη1p/p+η2+w+mGr1−e−pp+η2p2b3p+b4e−ζη1p/p+η2+w−e−ζb2p/p+η2+λ+nGmp+η2pb5p−b6e−ζη1p/p+η2+w−e−ζb1p/p+η2,where(34)η1=η1−α,η2=α1−α,b1=b1−α,b2=β1−α,b3=b2−η1+λ−w,b4=λ−wη2,b5=b1−η1−w,b6=wη2.Equation (33) can be written in a more precise form as(35)u¯ζ,p=Φζ,p−e−pΦζ,p+mGrb3Φζ,p−e−pΦζ,p−T¯ζ,p+mGrη4b3δζ,pΦζ,p−e−pδζ,pΦζ,p−δζ,pT¯ζ,p+nGmb5μζ,p−C¯ζ,p+nGmη6b5φζ,pμζ,p−φζ,pC¯ζ,p.After applying inverse Laplace transformation on the above equation, we get the solution(36)uζ,τ=Φζ,τ−Φζ,τ0Gτ0+mGrb3Φζ,τ−Φζ,τ0Gτ0−Tζ,τ+mGrη4b3Υζ,τ−Υζ,τ0Gτ0−δ∗Tτ+nGmb5μζ,τ−Cζ,τ+nGmη6b5φ∗μτ−φ∗Cτ,where(37)Φζ,τ=L−1e−ζw+γp/1+ϵpp2,=γϵ∫0∞∫0τe−sγ+zϵerfcζ2zI02ϵγ−wϵszdzds+wϵ∫0∞∫0ν∫0τe−s1+w+pϵerfcζ2zI02ϵγ−wϵqsdzdqds,and Gτ0 represent a standard Heaviside function with(38)τ0=τ−1,b2=β1−α,η2=α1−α,ψ=b2+λη2,γ=η1+wη2ϵ=1η2,η3=−b4b3,η4=η2+η3,η5=b6b5,η6=η2+η5,δζ,τ=L−11p−η3=eη3τ,φζ,τ=L−11p−η5=eη5τ,Υζ,τ=δ∗Φτ,μζ,τ=L−1e−ζw+γp/1+ϵpp,=e−ζw−ζγ2π∫0∞∫0τ1τe−ϵτ+ζ2/4v+wvI12γvτdτdv.It is detected that whenα⟶1 and Gm=0, the fractionalized model CF in equation (3) for ramped wall velocity profile is converted into integer order model as calculated by Talha Anwar et al. [55]. Also, we achieved the same results as Seth et al. [29] for ramped wall temperature with ϕ=0 and Gm=0. Moreover, we derived similar solutions as obtained by Chandran et al. [28] when porosity parameter 1/K⟶0, Gm=0, and magnetic parameter M⟶0. This authenticates the current acquired results with the published literature.


## 4. Results and Discussion

In order to achieve the goal of having a comprehensive understanding of the physical mechanism of current problem completely, a parametric analysis is performed, and computed solutions are revealed with the assistance of graphs. In this section, the influence of many pertinent parameters such asϕ,Gr,M,K,Gm,Nr,Sc, and fractional parameter α an obtained solution of water base nanoparticles with ramped conditions subject to the application of different fractional operators CF is studied by using MARHCAD-15 software.It is observed that temperature, concentration, and velocity profile are controlled by the fractional parameterα. Figures 2 and 3 illustrate for different time levels the effect of fractional parameter α and Sc on concentration. For small and large values of time t and using different values of fractional parameter α, it is observed from Figure 2 that the behavior of concentration is increasing for a small time and decreasing for a large time. Also, Figure 3 illustrates the behavior of concentration for Sc at two different values of time, and it is observed that concentration is decreasing as the value of Sc is increasing. Figure 4 displays the temperature decreasing for a small time and increasing for a large time, using various values of the fractional parameter α. It is detected that increasing the values of radiation parameter Nr elevates the temperature distribution for a small and large time in Figure 5. The impact of nanoparticle solid volume fraction ϕ is analyzed on temperature distribution for varying times in Figure 6. A raise in the solid volume fraction parameter elevates the temperature contour. Indeed, an increase in the value of ϕ increased fluid density and cause to increase in the thickness of the boundary layer which helps to improve the thermal conductivity and surface heat transfer rate is also increased.Figure 2
Concentration behavior for different values ofα when ϕ=0.1, Sc = 3.Figure 3
Concentration behavior of Sc for two distinct values of t whenϕ=0.1,α=0.3.Figure 4
Temperature behavior for different values ofα when Pr = 6.2, ϕ=0.1, Q = 0.4, Nr = 0.5.Figure 5
Temperature behavior for two distinct time levels of t for Nr when Pr = 6.2,α=0.3, Q = 0.4.Figure 6
Temperature behavior for two different values ofϕ when Pr = 6.2, α=0.3,Q = 0.5, Nr = 0.5.Figure7 displays the graphs of nanofluid velocity for different values of Gr with various values of fractional parameter α. It is noticed that the velocity profile is elevated corresponding to rising values of Gr. It is observed that the velocity and Gr have direct relation. Generally, Gr deals with buoyancy and viscous forces, and the strength of the viscous force is decreased due to a raise in Gr. Consequently, nanofluid velocity is elevated when it is close to an oscillating plate, and then, the nanofluid flow is apart from the plate, forces become delicate, and gradually, the motion of fluid approaches zero. In Figure 8, four different graphs are drawn for dissimilar values of Gm and numerous values of the fractional parameter α. The mass Grashof number is the ratio of mass buoyancy force to viscous force, which causes free convection. It is noted that the fluid velocity is increasing if we increase the value of Gm.Figure 7
Velocity behavior for four different values of Gr via ABC when Pr = 6.2,ϕ=0.1, Gm = 0.25, Sc = 3, M = 2, Q = 0.5, Nr = 0.5, K = 0.5.Figure 8
Velocity behavior for four distinct values of Gm whenϕ=0.3, t = 5, M = 2, Q = 0.5, Gr = 5, Sc = 3, K = 2, Pr = 6.2, Nr = 0.5, K = 0.5.In Figure9, the graphs of nanofluid velocity for different values of Sc are displayed, with various values of fractional parameter α. It is observed that the velocity profile is decreased corresponding to increasing values of Sc. In Figure 10, it is analyzed that velocity is elevated for different values of K using various values of fractional parameter α. So, Figure 10 shows the relationship between dimensionless velocity and permeability parameter K which highlights the fact that nanofluid velocity is accelerated correspondingly to increase the values of K for CF fractional operator varying fractional parameter α.Figure 9
Velocity behavior for four different values of Sc whenϕ=0.3, t = 5, M = 2, Gm = 0.25, Gr = 5, Q = 0.5, Nr = 0.5, Pr = 6.2, K = 0.5.Figure 10
Velocity behavior for two different values of K when Pr = 6.2,ϕ=0.1, t = 5, M = 7, Gm = 0.25, Sc = 3, Q = 0.4, Nr = 0.5, Gr = 15.In Figure11, it is analyzed that four different graphs are traced for fixed time, but different values of M and various values of fractional parameter α via CF the velocity is decreased. In Figure 12, the impact of fractional parameter is investigated on nanofluid velocity at two different levels of time. For small and large values of time t and using different values of fractional parameter α, it is observed that the behavior of velocity is decreasing for a small time and increasing for greater values of the time. This rapid decay in velocity is due to an increase in momentum boundary layer for increasing α. It is inspected that temperature and velocity profile is controlled by a time-fractional parameter. It is observed that the variation of fractional parameter α on the nanofluids velocity profile and comparison between the CF factional model with an ordinary model is discussed in Figure 13. The velocity profile is reduced by increasing the value of the fractional parameter α. In Figure 14, it is noticed that when α⟶1, the CF fractional model becomes an ordinary model.Figure 11
Velocity comparison between CF and ordinary model when Pr = 6.2,ϕ=0.1, t = 0.35, Gm = 0.25, Sc = 3, K = 2, Q = 0.5, Nr = 0.5, Gr = 5.Figure 12
Velocity behavior for different values ofα when Pr = 6.2, ϕ=0.1, t = 0.35,M = 2, Gm = 0.25, Sc = 3, Q = 0.5, Nr = 0.5, K = 3, Gr = 5.Figure 13
Velocity comparison between CF and ordinary model when Pr = 6.2,ϕ=0.1, t = 0.35, M = 2, Gm = 0.25, Sc = 3, Q = 0.5, Nr = 0.5, Gr = 5.Figure 14
Velocity comparison with [55].


## 5. Conclusion

The comprehensive analysis of the time fractional derivative with heat and mass transfer to evaluate the physical effects of application of simultaneous ramped wall velocity and ramped wall temperature condition on unsteady, MHD convection flow of some nanofluids. In addition, heat injection/consumption and heat radiative flux are also inculcated in the model. However, in this work, exact and closed-form solutions are derived by employing the Laplace transformation. For physical significance, the graphs of various system parameters are demonstrated. The following major findings of this study are given as follows:(i)
The temperature profile increases due to enlargement in volume fractionϕ and by increasing the value of Nr also noted that decay in concentration profile by enhancing the values of Sc.(ii)
Nanofluid velocity is a decreasing function of magnetic parameterM.(iii)
It is observed that the effect ofα for small and large time on temperature, concentration, and velocity profile is quite opposite.(iv)
For the greater value of K, velocity profile is elevated for varying fractional parameterα for fractional operator applied on nanofluid velocity also noted that decay in velocity for growing values of the Sc.(v)
Velocity profile enhances by increasing the values ofGr and Gm for a fixed time and different values of α.(vi)
CF time-fractional operator converges to ordinary model whenα⟶1.(vii)
In the future, we solved this fractionalized problem through a numerical approach with different techniques. Also, we solved this viscous model via other conditions.



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*Source: 9977188-2022-06-16.xml*