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Cattaneo–Christov theory for a time-dependent magnetohydrodynamic Maxwell fluid flow through a stretching cylinder

This research work explores the thermal and mass transport phenomena for a time-dependent Maxwell fluid flow in the presence of Cattaneo–Christov concept. The Maxwell fluid flow is analyzed through a stretching cylinder and sheet. Brownian motion, non-uniform heat source/sink, thermophoresis, and variable thermal conductivity are operated in this study. A theoretical analysis of the modeled system of equations is explored with the help of HAM. Impacts of fixed constraints on velocity, thermal, and concentration functions are offered graphically. It is concluded that the velocity profile heightens quickly for Newtonian fluid equated to non-Newtonian fluid (Maxwell) via curvature parameter while the temperature and concentration distributions increase quickly for non-Newtonian fluid as equated to the Newtonian fluid via curvature parameter. The presence of Maxwell and magnetic parameters increases the size of the trapping bolus.


Introduction
Owing to the various uses throughout the fields of engineering and manufacturing equipment including insulation of nuclear reactors, heat exchangers, refrigerators, polymer processes, and plastics extrusion, research teams are now putting an excessive amount of focus on mass and heat transfer observation. The basic mathematical associations of Fourier's and Fick's principles are being used to explain the process of mass and heat transfer rate throughout a particular channel due to concentration and temperature variations, respectively. Computational and experimental investigations are achieved to study the characteristics of fluid flow and thermal transmission over the cylinder. Na and Pop 1 examined thermal performance in viscous flow field across a flowing cylinder in the trajectory and reverse of free flow. Kumari and Nath 2 studied a magnetohydrodynamic (MHD) time-dependent stagnation point viscous fluid flow. Toh et al. 3 introduce numerical analysis to the Newtonian fluid flow and thermal transmission system in micro-channels. According to this analysis, when the Reynolds number is low, the viscosity of liquid declines, and thermal energy increases. The analysis of Oldroyd-B nanofluid over an extending surface with Cattaneo-Christov theory (CCT) was presented by Khan et al. 4 The numerical investigation of incompressible Eyring-Powell nanofluid through a bi-directional extending sheet with CCT was offered by Wubshet. 5 The stagnation point of the time-dependent incompressible Maxwell fluid flow was introduced by Moshkin et al. 6 The special effects of Hall current on MHD flow of dusty fluid through a stretching surface were investigated by Gireesha et al. 7 The incompressible Maxwell fluid flow through a bidirectional extending surface with convective conditions was presented by Mahanthesh et al. 8 Consequences of Joule heating and Hall current viscous dissipative flows of nanofluid were Mahanthesh et al. 9,10 The Maxwell fluid flow with homogeneousheterogeneous reactions (HHR) and CCT between spiraling disks was presented by Ahmed et al. 11 The MHD and electrically conducting the mixed convective flow of Maxwell fluid with CCT over a spreading surface was evaluated by Saleem et al. 12 Mixed convection in MHD flow of nanofluid with nonlinear thermal conduction was investigated by Ellahi et al. 13 The Hall Effect on the convective flow of micropolar fluid with CCT was investigated by Shah et al. 14 In another article, Shah et al. 15 analyzed the MHD second-grade thin film flow with CCT over a time-dependent stretching sheet. The carbon nanotubes nanofluid between rotating disks with CCT was introduced by Bhattacharyya et al. 16 Researchers got interests in non-Newtonian fluids due to their diverse uses in the field of engineering to analyze the transportation thermal and solutal behaviors. To define the behaviors of various non-Newtonian fluids, researchers recommend different models. Furthermore, various numerical and analytical techniques are applied to solve highly nonlinear and complicated constitutive equations of different models. The buoyancy effect on MHD Oldroyd-B, Jeffrey, and Maxwell nanofluids through cone with variable properties were presented by Raju et al. 17 The Maxwell and Carreau nanofluids with different physical phenomena were offered by Hsiao. 18,19 Ahmed et al. 20 offered the Maxwell nanofluid flow with a thermal sink/source over a rotating disk. Shah et al. 21 presented the thermal conduction in a Casson ferrofluid through an extending surface. The MHD and viscous dissipative flow of micropolar nanofluid through an extending sheet was analyzed by Hsiao. 22 The MHD Maxwell fluid through an extending surface was considered by Abel et al. 23 The Oldroyd-B nanofluid flow with mass and thermal stratification conditions was inspected by Waqas et al. 24 The Oldroyd-B fluid flow with CCT and HHR under the impact of nonlinear thermal conduction was studied by Irfan et al. 25 The MHD Williamson nanofluid flow through a nonlinear extending plate was presented by Dawar et al. 26 The MHD non-Newtonian nanofluid flow through two dissimilar geometries with Joule heating was presented by Dawar et al. 27 The MHD Jeffrey fluid flow with Hall and ion slip influences inspected by Krishna. 28 Khader and Sharma. 29 analyzed the radiative flow of micropolar fluid with a magnetic effect over a stretching sheet. The MHD flow of Casson fluid with heating influence was presented by Tassaddiq et al. 30 Further related studies are mentioned in Abel et al., 31 Waqas et al., 32 Irfan et al., 33 Rajagopal, 34 and Alamri et al. 35 In light of the overhead declared studies, we anticipated the analytical study of time-dependent non-Newtonian Maxwell fluid with Cattaneo-Christov theory through an extending cylinder. The non-uniform heat source/sink, variable thermal conductivity, Brownian motion, and thermophoresis phenomena are operated. Section 2 represents the mathematical modeling of the proposed study. Section 3 deals with the analytical solution of the present model. Section 4 grants the results and physical discussion on the embedded parameter. Section 5 displays the concluding remarks of the present analysis.

Problem formulation
The Maxwell fluid flow through a stretching cylinder of radius R is considered here. A magnetic field B = 0, 0, B 0 ð Þ of strength B 0 is applied normal to the fluid flow. The variable properties are taken into account to analyze the heat transmission of the fluid flow. Cattaneo-Christov theory, thermophoresis, and Brownian motion are also utilized. Assume that u and w are the components of velocity along zÀ and rÀ directions correspondingly. z is the axis of cylinder where r is normal to z as described in Figure 1. Furthermore, it is also assumed that u w t, z ð Þ= az 1Àat ð Þ is the unsteady stretching velocity of cylinder where a, g are positive constants T, T w , and T ' are the temperature, surface temperature, and free stream temperature. C, C w , and C ' are the concentration, surface concentration, and free stream concentration. According to all these assumptions, the leading equations are 34,35 : with boundary conditions 25 : where l is the relaxation time, n is the kinematic viscosity, r is the density, c p is the specific heat, l 1 and l 2 are the thermal and concentration relaxation times respectively, D B and D T are diffusion coefficients of Brownian motion and thermophoresis, K T ð Þ = k ' 1 + ec ð Þ is the variable thermal conductivity where k ' is the free stream conductivity, e is the small conductivity parameter, and c is the dimensionless temperature parameter, A Ã .0 and B Ã .0 are the heat generation parameters, A Ã \0 and B Ã \0 are the heat absorption parameters.
The similarity transformations are defined by: Using (6), (2)-(5) are reduced as: with transformed boundary conditions: Here, the unsteadiness parameter is indicated by S = a a À Á , magnetic parameter is demarcated by , curvature parameter is denoted by thermal and mass relaxation times parameters are rep- The skin friction, local Nusselt number, and Sherwood number are defined as: ffiffiffiffiffiffiffiffi

HAM solution
The linear operators and initial guesses are defined as: x with where < 1 À < 7 are called arbitrary constants. Further detail of HAM can be found in Liao. 36238 Results and discussion   greater curvature parameter which reduces the impact of boundary in the Maxwell fluid flow. Thus, the velocity profile heightens. A similar impact of curvature parameter g on temperature and concentration profiles is depicted. Furthermore, the influence of curvature parameter on Newtonian and non-Newtonian (Maxwell) fluid flows is compared. The increasing impression of g on x 0 j ð Þ is greater for Newtonian fluid equated to non-Newtonian (Maxwell) fluid while a contrary conduct is observed on thermal and concentration profiles. Here, the Maxwell parameter b plays an important role for Newtonian and non-Newtonian fluids. Physically, with the increasing Maxwell parameter, the fluid behaves like a solid which consequently increases the non-Newtonian fluid velocity while diminishes the thermal and mass transport. Thus, the greater impact on x 0 j ð Þ is depicted for Newtonian fluid associated to non-Newtonian (Maxwell) fluid while this behavior is opposite for c j ð Þ and F j ð Þ of Newtonian and non-Newtonian (Maxwell) fluids. Figures 5 to 7 display the variation in x 0 j ð Þ, c j ð Þ, and F j ð Þ due to unsteadiness parameter S. x 0 j ð Þ reduces while c j ð Þ and F j ð Þ increase with the higher S. Physically, S has direct relation with positive constant a. The increasing a heightens the unsteadiness parameter which consequently increases the stretching rate of the sheet and cylinder. Thus, the velocity reduces with higher unsteadiness parameter. However, this impact is reverse on thermal and concentration profiles. The stretching rate of the sheet and cylinder increase the thermal and mass profiles of the fluid flow. Figure 8      In the existence of heat generation parameters, the thermal boundary layer gains additional energy which leads the temperature to escalate. Thus, the temperature profile rises with higher heat generation. However, the absorption parameters absorb the heat energy from the boundary layer which result a gradual decrease in the fluid flow temperature. From these figures, we have seen a rapid increase for the case of heat generation while this impact is slow for the case of absorption. Figure 12 shows the variation in c j ð Þ due to Prandtl number Pr. The higher Pr diminishes the temperature of the fluid flow. Physically, the higher Pr declines the thermal diffusivity of the fluid which     consequently reduces the fluid temperature which demonstrates thinning in the boundary layer. Thus, the thermal profile declines with higher Prandtl number. In addition, due to the increase in thermal conductivity there can be overrun throughout the thermal boundary layer at sometimes. This consequence can be mitigated by using a heat sink to regulate the temperature. Figures 13 and 14 show the variation in c j ð Þ and F j ð Þ due to thermal b t and solutal b c relaxation times factors respectively. Here, c j ð Þ and F j ð Þ reduce with higher values of b t and b c respectively. In CCT, the instantaneous transmission of thermal waver control by relaxation time parameters requires additional time for the transportation of mass and heat. Thus, the thermal and mass profiles reduce with greater values of b t and b c . Figure 15 shows the change in concentration profile via Lewis number Le. The concentration profile reduces with greater Le. Actually, the higher values of Le reduce the thermal diffusivity and increase the Brownian diffusivity of the fluid which results reduction in F j ð Þ. Thus, the increasing Le moderates F j ð Þ.  ð Þ whereas the higher Pr increases Àc 0 0 ð Þ. Furthermore, these effects are greater for cylinder as compared to the sheet. Table 3 displays the numerical estimations of ÀF 0 0 ð Þ for cylinder and sheet via different embedded factors when b t = b c = 0. The higher b, S, and e reduce ÀF 0 0 ð Þ whereas the higher Le increases ÀF 0 0 ð Þ. Furthermore, these effects are greater for cylinder as compared to the sheet.

Conclusion
The MHD flow of Maxwell fluid with variable thermal conductivity and non-uniform heat source/sink through a stretching cylinder is analyzed here. A magnetic field is applied normal to the fluid flow. The Cattaneo-Christov theory is taken for heat and mass transmission utilization. Final remarks are recorded underneath:     Table 3. Numerical values of ÀF 0 0 ð Þ for cylinder and sheet An increasing conduct is testified for velocity, thermal, and mass profiles via curvature parameter. The increasing impression of curvature parameter on velocity profile is higher for Newtonian fluid (Maxwell) as compared to non-Newtonian fluid while a contrary conduct is observed on thermal and concentration profiles. The velocity profile reduces while the thermal and concentration profiles increase with the higher unsteadiness parameter. The increasing Maxwell and magnetic parameters declines the velocity profile. The greater heat generation and heat absorption heighten the thermal profile while the escalating Prandtl number and thermal relaxation time factor reduce the thermal profile. The higher mass relaxation and Lewis number reduce the concentration profile. The presence of Maxwell and magnetic parameters increase the size of trapping bolus.

Author contributions
Saeed Islam: Conceptualization, methodology, formal analysis, and software, resources. Abdullah Dawar: Conceptualization, investigation, writing original draft, preparation, methodology, and software. Zahir Shah: Writing review and editing, software, visualization, writing review and editing, and validation. Adnan Tariq: Writing review and editing, software, visualization, writing review and editing, and validation.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.