Analytical solutions for unsteady fractional maxwell nanofluid flow and mass transfer between two rigid spheres with periodic oscillations

In this paper, we develop axisymmetric squeeze flow and mass transfer of incompressible Maxwell nanofluid between two rigid spheres. The position of the boundary between the two rigid spheres varies periodically with time and space, and the unsteady magnetic field is considered. Analytical solutions are derived for velocity and concentration by using the method of Laplace integral variation. In addition, the radial velocity and concentration distributions are imaged and the relevant parameters are analyzed.


Introduction
Squeeze flow has been widely followed in the bioengineering, automotive engines, and food processing areas.In recent years, many related models have been developed [1][2][3][4][5] .Lang et al. reported a theoretical study of the transient squeezing flow of Newtonian fluid between the cylindrical gaps [1] .Taking non-Newtonian fluid into account, Gupta et al. [2] analyzed the squeezing flow of Casson nanofluid between two parallel disks.Squeeze flow between two disks is a simple but basic case, while squeezed flow between two spheres is an extended use of the disks concept.Xu et al. [3] studied squeeze flow between two arbitrary rigid spheres with interstitial Herschel-Bulkley fluid.Rodin [4] developed a new asymptotic solution to unify the classical solution for fluid squeezed by two plates with more recent solutions for fluid squeezed by either two spheres or a sphere and a plate.Particularly, to explore the effects of wall slip, the squeeze flow between two rigid spheres with a bi-viscosity fluid is examined by Zhou et al. [5] .However, most studies have only investigated the flow characteristics of squeeze flow and not their mass transfer characteristics.
At present, no researcher has derived an analytical solution for the fractional Maxwell nanofluid squeeze flow model under time and space-dependent boundaries.Inspired by the above studies, the squeezed flow and mass transfer of fractional Maxwell nanofluid between two rigid spheres are investigated considering oscillatory boundaries and an unsteady magnetic field.Analytical solutions for the velocity and concentration distributions are found by using the Laplace transform.

Mathematical formulation
The schematic of the problem investigated herein is shown in Figure 1.We consider two rigid spheres of radius R at an initial distance 0 h moving periodically and synchronously in the z-direction.In addition, we assume that the fractional Maxwell nanofluid is stationary in the sphere gap at the initial moment.At any instant, two rigid spheres move in opposite directions with the same magnitude of velocity.The film height is much smaller than the curvature radius, so we can use the Derjaguin approximation [6] .We assume that the height of the gap varies periodically in time and space: ( ) where  is the angular frequency, t is time, and 0 V is characteristic velocity.
The constitutive equation of the fractional Maxwell model is: where ,,     and S are the relaxation time, dynamic viscosity, deformation rate, and shear stress, respectively.Considering the influence of the unsteady magnetic field [7] , the Navier-stokes equations are given as: where the radial velocity is expressed in u, while the axial velocity is expressed in υ; The physical property parameters of the nanofluid are as follows [8] : where Initial along with boundary conditions considered for the presented problem are [9]  : where  is the degree of slippage; 21 , CC are the concentration of the upper and lower wall surfaces, respectively.
We introduce Equation (1) into Equation ( 2) and Equation (3), respectively.We introduce the following non-dimensional form of involved variables: /, hR , where  is the kinematic viscosity;  is the squeezing depth coefficient; Re is Reynolds number; 0 W is Womersley number; Ha is Hartmann number; and Sc is Schmidt number.Then the dimensionless governing equations can be obtained ("*" omitted for the sake of brevity): )

The analytical approach
In the case of a thin gap , Equation (10) implies that the vertical pressure gradient is negligible, the convective and the radial diffusive terms can be discarded in Equation (9).In narrow gaps, the radial concentration diffusion is negligible compared with the axial concentration diffusion, and the concentration diffusion process can be simplified to one-dimensional concentration diffusion in the axial direction in a cylindrical coordinate system.( ) ( ) ( ) =  [10]   , defining  as the format of ( ) We use the expressions of velocity in Equation ( 12), assuming that the pressure is equal to zero at 1 r = , and integrating the result concerning r and making a difference in z, then we get: ( ) ( ) ( ) Now we take the Laplace transform to Equation ( 14) and Equation ( 13), respectively, ( ) ( ) ( ) where F is the Laplace transform of F ,  is the Laplace transform of  , and s is the Laplace operator.
Equations ( 15) and ( 16) are solved analytically based on the initial and boundary conditions: where i is the imaginary number, and  , respectively.

Results and discussion
In this section, we present the influence of relevant parameters on the velocity distribution and concentration distribution.It is worth noting that the two rigid spheres perform a cyclic motion, with each cycle divided into four phases, and we will only give a short analysis of the first phase.The corresponding radial velocity distribution is shown in Figure 2. Figure 2(a) shows the velocity profiles with different volume fractions of nanoparticles that the velocity profiles decline with the decreasing volume fraction of nanoparticles.In Figure 2(b), we illustrate the radial velocity distributions for different Hartmann numbers, with the enlargement of the Ha, the velocity becomes smaller and sharper, and the magnetic field impedes the flow of the fluid to some extent.We can discover that the solution of concentration depends on the product of the Schmidt and Reynolds numbers.

Conclusion
This study investigates the fractional Maxwell fluid flow and mass transfer between two rigid spheres with periodic oscillations.Analytical solutions are derived by using the method of Laplace integral variation, and the findings are discussed by using diagrams.The novelty of this study is that the squeeze flow of fractional Maxwell nanofluid is investigated under an unsteady magnetic field, and the analytical solution for concentration is derived.
are the components of shear stress; nf  is the electrical conductivity of nanofluid; D is the coefficient of mass diffusion; and the unsteady magnetic field takes the form ( ) of the fluid and nanoparticles, respectively.
by solving the stream function, and the stream function  concerning r Then, we apply the inverse Laplace transform on Equation (17) and Equation (18), respectively.F and  are obtained 1 ln

Figure 2 .
Velocity distributions for different parameters.
Figure 3(a) compares the effect of three different ScRe on the velocity distribution, from which we find that the concentration variation increases as the ScRe increases, but shows an opposite trend in the upper and lower half axes of z.The influences of the squeezing depth on the concentration distributions are presented in Figure 3(b), and the concentration distribution crosses over at some points as the concentration varies more with the depth of the squeeze.

Figure 3 .
Figure 3. Temperature distributions for different parameters.