Closed-form solution of oscillating Maxwell nano-fluid with heat and mass transfer

The primary goal of this article is to analyze the oscillating behavior of Maxwell Nano-fluid with regard to heat and mass transfer. Due to high thermal conductivity of engine oil is taken as a base fluid and graphene Nano-particles are introduced in it. Coupled partial differential equations are used to model the governing equations. To evaluate the given differential equations, certain dimensionless factors and Laplace transformations are used. The analytical solution is obtained for temperature, concentration and velocity. The temperature and concentration gradient are also finds to analyze the rate of heat and mass transfer. As a special case, the solution for Newtonian fluid is discussed. Finally, the behaviors of various physical factors are studied graphically for both sine and cosine oscillation and give physical meanings to the parameters.


Scientific Reports
| (2022) 12:12205 | https://doi.org/10.1038/s41598-022-16503-w www.nature.com/scientificreports/ To the best of author's knowledge no one has consider the oscillating Maxwell nanofluid with the heat and mass transfer. So, motivated by this we study this problem analytically. The aim of this work is to explore oscillating Maxwell nanofluid with heat and mass transfer. The suspension of graphene nanoparticles and engine oil (base fluid) is taken in consideration. The governing equation is solved through LT. The solution for temperature, concentration and velocity are calculated analytically. Temperature slope and concentration gradient in the form of Nusslet number and Sherwood number are also acquired. Finally, the influence of various embedded factors on temperature, concentration and velocity shows graphically as well as theoretically.

Problem statement
Let us assume Maxwell nanofluid passed on an infinite oscillating vertical plate with heat and mass transfer. ε is perpendicular to the plate while plate along x-axis. Both the fluid and the plate are initially at rest with ambient temperature T ∞ and ambient concentration C ∞ . After some time at t = 0 + the plate begins oscillation in its plane ( ε = 0 ) as indicated with velocity UH(t)e (iωt)i , where U is the amplitude, ω represents the frequency of the oscillation of the plate, H(t) is the unit step function and i is the unit vector in the vertical flow direction. Chemical reaction phenomenon is also incorporated to elaborate mass diffusion response. We suppose that the velocity, concentration and velocity is the function of ε and t. The governing equations is model in the following form. Figure 1 shows the geometry of the flow problem.
Here, ρ nf is the density of nanofluid, o represents the Maxwell fluid parameter, µ nf denotes the dynamic viscosity of nanofluid, g is the gravitational acceleration, (β t ) nf represents the coefficient of thermal expansion of nanofluid, c p denotes the specific heat, (β m ) nf represents the coefficient of mass expansion of nanofluid, K represents the chemical reaction parameter, M nf shows the diffusion species coefficient.
The corresponding initial conditions (ICs) and boundary conditions (BCs) are of the following form 49 : where T ∞ is the ambient temperature, ht represents the coefficient of heat transfer, C ∞ demonstrates the ambient concentration, and hc shows the coefficient of mass transfer. Using Rosseland approximations 9,31,33 and gaining the small temperature variation between the temperature T ∞ of the free stream and the fluids temperature T, exploring the Taylor theorem on T 4 about T ∞ and omitting the numbers of 2nd and higher order, we get (1) where φ * , κ * are respectively Stefan boltzman constants, is the mean absorption coefficient. Substituting (5) into (2) we obtain the following form The thermo-physical characteristics of nanoparticles were given by 35 ; The dimensionless parameters are given below; After substituting the above dimensionless parameters in Eqs. (1), (3) and (7) we get these governing dimensionless equations and dropping the • from the above dimensionless factors, where Here, θ denotes the dimensionless temperature, Pr shows Prandtl number, Gr t demonstrates thermal Grashof number, Gr m represents mass Grashof number, ϑ denotes volume fraction parameter and Sc represents Schmidt number.
The dimensionless ICs and BCs are as follow and skip • from the non-dimensional factors The thermo-physical property of graphene (nanoparticles) and engine oil (base fluid) are tabulated in table. 1

Problem solution
Temperature. Taking LT on Eq. (10) and also using the related ICs and BCs, we get the following transform form; The inverse LT of Eq. (13) has the following final form, Nusslet number. The Nusslet number, measure the rate of heat transfer at the plate can be acquired by differentiating Eq. (13) with respect to ε and using ε = 0 , we get the constant term. i.e.
This shows the heat is transfer due to purely conduction.

Concentration.
Applying LT on Eq. (11) and also utilizing the respective ICs and BCs, we acquire the following transform form; The Laplace inverse transform of Eq. (16)  Here,

Special cases
In the absence of nanoparticles, we obtained the solution of Fetecau et al. 33 . where

Numerical results and discussion
In order to see the physical meaning of the problem, we use the LT method to obtain the solution for temperature, concentration, velocity, rate of heat transfer and rate of mass transfer. These solutions have been studied graphically by giving numerical values to various embedded parameters like radiation factor, chemical reaction factor, thermal Grashof number, mass Grashof number, Maxwell fluid coefficient, Schmidt number, Prandtl number. The value of volume fraction parameter is taken 0.01. Figure 2 characterizes the concentration for variations of Schmidt number Sc and chemical reaction factor K. It is found that by increasing the value of Schmidt number Sc and chemical reaction factor K , the concentration of the nanofluid decreases. Physically, there is inverse relation between Schmidt number and mass diffusivity. As we enhance Schmidt number Sc , the mass diffusion is de-escalates. Thus, concentration profile decreases. Similarly, concentration profile decreases with the increasing estimation of chemical reaction factor K . This behavior is due to less fluid particles are produced as a product. In Fig. 3 the flow profile of Maxwell fluid is studied under the revamping of thermal Grashof number for both the sine and cosine oscillations. The velocity distribution for both sine and cosine oscillation is the growing function as we grow the value of thermal Grashof number Gr t . Physically, this characteristic is because of the viscous and thermal buoyancy forces in flow of fluid. The greater the value of Gr t shows the fluid is heated that bolsters the impact of thermal buoyancy forces because of the existence of convection currents. These currents get the value of great importance due to prevailing temperature slop and eventually cause the viscous forces to sink. As a result, the fluid's velocity enhances. Figure 4 displays the impact of mass Grashof number on velocity. It is also have same behavior like Fig. 3 i.e. the (38)  www.nature.com/scientificreports/ enhancement of Gr m enhances the velocity of the fluid. This is due to the enhancement in mass buoyancy force and buoyancy force enhances concentration gradient, which result enhances the velocity. Figure 5 portrays the behavior of radiation coefficient Rd for both sine and cosine oscillation. It characterize that the fluid's velocity accelerated with the greater value of Rd . Physically, rate of energy transfer explains this increase. As Rd increments, rate of energy transfer to the fluids grows which results to weak the bond between fluid particles. As a result these poorly associated particles collectively give much weaker viscosity to fluid motion and gradually fluid gets accelerated. Figure 6 shows the relationship between Schmidt number and velocity of the fluid. It is spotted that the increases in Schmidt number decelerate the fluid's velocity for both the oscillations. Physically,   www.nature.com/scientificreports/ as Schmidt number Sc increases, the molecular diffusivity reduces due to which velocity decreases. Figure 7 exhibits the effects of chemical reaction factor on velocity distribution for both the sine and cosine oscillation. Clearly Fig. 7 demonstrates the de-escalation in fluid velocity as we grow the value of chemical reaction factor. The variation of velocity distribution because of Maxwell fluid coefficient for both the oscillation is described in Fig. 8. It is realized the fluid flow is increasing function for greater value of . Physically, this observation is because of the retard in boundary layer thickness. Velocity shows the significant behavior in the main stream region and finally approaching to zero. Figure 9 shows the behavior of graphene nanoparticle on velocity profile. It can be seen that the velocity of nanofluid reduces for the growing value of volume fraction. This is because of increases the nanoparticles makes denser the fluid, so its velocity decelerates.    Figure 10 depicts the influence of volume fraction on temperature profile of nanofluid. It is observed that the temperature of nanofluid decelerates with accelerating the estimations of volume factor. Physically, this behavior is due to decrease of thermal conductivity on adding nanoparticles, which results decelerates the temperature of nanofluid. Figure 11 shows the increase in nanofluid's temperature with increasing radiation parameter. Since increase in at fixed value of and, decelerates the value of, therefore slop of radioactive heat flux increases which lead to grow   www.nature.com/scientificreports/ the radiative heat transfer rate and gradually the fluid's temperature increments, It means that thickness of energy boundary layer reduces and temperature is distributed more uniformly.
In order to authenticate our present solutions, Figs. 12 and 13, are presented. It can be observed if the volume fraction parameter ϑ are removed from temperature field and ϑ and Gr m are deleting from the velocity field of the current model, then the present solutions for temperature and velocity field are in excellent agreement with the velocity solution of ordinary Maxwell fluid model of Khan et al. 13 for both sine and cosine oscillations and the temperature solution of Fetecau et al. 33 .

Conclusion
The aspiration of this work was to evaluate the oscillating Maxwell nanofluid with heat and mass transfer. The analytical solution for temperature, concentration, and velocity were obtained through LT method. The rate of heat and mass transfer was also measured in the form of Nusslet number and Sherwood number. Finally, the effect of different physical factors were shown in discussion section graphically and theoretically for both sine and cosine oscillation. The solution of Newtonian fluid was also analyzed as a special case. Following are the key concepts of this work (Supplementary Information S1): • The concentration profile is decreasing function for Schmidt number Sc.
• Decrease occurs in concentration with increasing the estimation of chemical reaction factor K.
• The velocity of the nanofluid grows, when thermal Grashof number Gr t is accelerated.
• The nanofluid's velocity enhances with the enhancement of mass Grashof number Gr m .
• The velocity field is accelerated as we accelerate the estimation of Maxwell fluid parameter .
• Reduction occurs in velocity with higher value of Schmidt number Sc.
• The nanofluid's velocity is increasing function as we increase the value of radiation parameter Rd while decreasing function against the chemical reaction factor K. • Volume function also reduces the nanofluid's velocity.