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## Applied Mathematics for Engineering Problems in Biomechanics and Robotics 2021

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Research Article | Open Access

Volume 2021 |Article ID 5557708 | https://doi.org/10.1155/2021/5557708

Haroon Ur Rasheed, Saeed Islam, Zeeshan Khan, Sayer O. Alharbi, Waqar A Khan, Waris Khan, Ilyas Khan, "Thermal Radiation Effects on Unsteady Stagnation Point Nanofluid Flow in View of Convective Boundary Conditions", Mathematical Problems in Engineering, vol. 2021, Article ID 5557708, 13 pages, 2021. https://doi.org/10.1155/2021/5557708

# Thermal Radiation Effects on Unsteady Stagnation Point Nanofluid Flow in View of Convective Boundary Conditions

Revised15 Apr 2021
Accepted12 Jun 2021
Published21 Jun 2021

#### Abstract

The present communication particularizes nonlinear convective non-Newtonian stagnation point flow and heat transference effects in stretchable flow of nanofluid. Magnetohydromagnetic steady viscous flow of nanofluid is examined. Heat transfer attributes of nanofluids are addressed via a numerical algorithm. Conductivity and diffusivity characteristics of fluid are depending on temperature and concentration and furthermore, on mass conservation, momentum, energy, and concentration yield partial differential equations (PDEs). The boundary layer flow concept pioneered by Prandtl has been employed to simplify the nonlinear constitutive flow laws which are then changed to ordinary differential equations. A built-in bvp4c algorithm in Mathematica software yields convergent outcomes of nonlinear (ODEs) systems. A comprehensive analysis has been made elucidating the physical significance of various governing parameters effects presented graphically. Additionally, the flow nature was confirmed versus streamlines.

#### 2. Mathematical Formulation

Here, we formulate unsteady non-Newtonian stagnation point stretchable flow of incompressible viscous nanofluid along with convection boundary conditions. Lower surface of sheet heated convectively fluid having temperature and heat transportation transfer coefficient and the ambient fluid having temperature and concentration , respectively. The stretchable sheet velocity at the surface is defined by the relation , where is the fixed positive constant. Magnetic field source is adequate with strength positioned in normal to stretchable surface for magnetization. The influence of induced magnetic properties is small with respect to the applied one and is neglected. Furthermore, there is a thermal equilibrium to base fluid and suspended nanoparticles. Flow diagram and coordinate axes are shown in Figure 1. Boundary layer governs flow expressions byMass conservation expression:Momentum expression:Energy expression:Concentration of nanoparticles:Subject to flow conditions are as follows:Herein, velocity components are in coordinates axes, along and normal to flow directionThe kinematics viscosity is , inside boundary layer temperature is , nanoparticles effective heat capacity is , thermal capacity of fluid is , fluid density is , , and is the ambient temperature away from stretchable surface.Introducing dimensionless parameters,Here, the stream function is defined byEquation (1) is justified automatically, and equations (2)–(4) take the following forms by employing equation (6):Transformed conditions:Variables seeming in constitutive flow laws (8)–(11) are defined as follows: the magnetic parameter; Lewis number; Prandtl number; velocity ratio parameter; Brownian motion parameter; thermophoresis parameter; and Biot number, respectively.Expressions of physical quantities are defined byBy substituting (13) in (12), we have

#### 3. Numerical Solution and Convergence Analysis

For numerical solution, the 1st order differential equations are found from equations (8)–(10) by introducing the following transformations relations. Let the transformations variables be defined by

Following are the first-order differential equations generated:

The initial boundary conditions are

A suitable step size is selected for the entire numerical computation work. A possible convergence limit has been adopted as . Furthermore, the boundary conditions given in (11) value at infinity is chosen as ; after selecting the appropriate value for , all numerical solutions converge to it and are satisfied correctly. Finally, residual error analysis has been evaluated. Dwindles in error perceive for higher order deformations as shown in Figure 2. Furthermore, the algorithm of bvp4c routine calculation is shown in Figure 3.

#### 4. Results and Discussion

The constative flow laws of nonlinear differential systems consisting of (8)–(10) subjected to specified boundary are solved numerically by employing the bvp4c algorithm. The attributes of affective pertinent flow factors on velocity field, thermal field, and nanoparticles volume are evaluated graphically and discussed in Figures 425.

Figures 4 and 5 depict variations in velocity subjected to velocity ratio parameter . These figures unveil enhancement in boundary layer thickness for higher , perceive when , and diminish the hydromagnetic boundary layer thickness for lower values of when . In reality, stretching velocity is less than the free stream velocity, due to which the ratio of stretching velocity to free stream velocity is less than unity; consequently, decreasing force vanishes, and clearly, profile is augmented. Attributes of are shown in Figure 6. Herein, we noticed is lower subjected to increment in magnetic field. Such scenario found due to higher implies strong resistance force came into existence called Lorentz force. In consequence, dwindles in velocity profile perceived. The contribution of the Prandtl number on thermal field is evaluated as shown in Figure 7. The thermal diffusivity diminishes when boosts through larger values. Hence, decays. Consequently, thermal boundary layer thickness escalates. Figure 8 explains variations in thermal field curves for the thermophoresis parameter. This figure unveils enhancement in when is enlarged. In fact, nanoparticles attain extra heat subject to the factor. In consequence, upsurges. The temperature field curves for the Biot number also called convective heating are shown in Figure 9. One can clearly perceive that is a growing function of . In reality, heat transference at surface boosts through larger . Consequently, upsurges. Hence, thermal boundary layer thickness escalates. Variations in temperature field against are interprets as shown in Figure 10. Clearly, larger velocity ratio factor diminishes thermal boundary layer thickness.

Impact of concentration profile against the parameter is portrayed as shown in Figure 11. This figure witnesses enhancement in . Clearly, is an augmenting function of the Brownian motion parameter. In reality, one can observe heat transference boosts through larger . Consequently, augments. Figure 12 discloses contribution of the parameter on profile. Clearly, dwindles subject to increment in . Figure 13 shows the effects of Prandtl number on concentration gradient. As the influence of the parameter increases, nanoparticles diffuse outside which reduced the concentration at the surface. The contribution of Lewis number on the concentration profile is shown in Figure 14. An increasing value of the parameter shows a poor diffusion coefficient as a result of small penetration depth observed for and displays decreasing trends in the concentration profile. The contribution of Boit number parameter via is shown in Figure 15. Clearly, larger estimations correspond to upsurges in the profile. Figure 16 unveils velocity ratio factor influences on . It is viewed that the concentration profile is a decreasing function of . Such scenario is noticed because augmentation in yields reduction in fluid thermal conductivity. Consequently, vanishes. Figures 17 and 18 emphasize , and influences on . Here, skin friction diminishes for are augmented. However, opposite appearances noticed for enlarged. The attributes of , and interpreted on are shown in Figures 1921. These figures confirm that Nusselt number upsurges subject to augmented in , and parameters. Figures 2225 explain the impression of magnetic and velocity ratio parameters on streamlines.

#### 5. Conclusions

Here, we formulated hydrodynamics stagnation point nanofluid flow over a surface with the magnetic effect and convective boundary condition. Furthermore, heat transportation features subject to viscous dissipation has been discussed. We witnessed the following are the noteworthy points from aforementioned investigation:We found lower when velocity ratio factor incremented and upsurges for a higher magnetic field valueAn augmentation in and yields an opposite report for the solutal boundary layerLarger velocity parameter and Boit number yields escalatesAn increment in thermal boundary is noticed for higher and diminishes for larger valuesBoth parameters and yield higher thermal fieldThe solutal boundary layer thickness is near stretching sheet negative and zero elsewhere

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under the project number (RGP-2019-6).

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