Narrow Results

To model several engineering and physical models, the approach of the fractional derivative is highly anticipated. As compared to the ordinary derivatives, the fractional derivatives with more flexibility can estimate the data due to the involvement of the fractional-order derivatives. Due to these advantages of the fractional approach, this study communicates with the determination of the fractional-based exact outcomes of an oscillatory rectangular duct problem of a generalized second-grade fluid. The approach of the fractional operator is involved in the relationship of the constitutive equations. For cosine oscillation of the rectangular duct, exact results of the magnetized unsteady flow problem are evaluated through the technique of Laplace transform with double finite Fourier sine transform. This study concludes that the velocity field exhibits escalating behavior relative to the improved fractional parameter. Moreover, the magnetic parameter with increasing values declines the flow field while the accelerating values of the fluid parameter enhance the velocity field.

In recent years, nanotechnologies have been widely used in several fields regarding their rapid developments which create a lot of prospects for researchers and engineers. More specifically, replacement of conventional liquids with nanoliquids is considered as an innovative solution to heat transfer problems. This research work reports mixed convective phenomenon of viscoelastic fluids with variable stretched surface. This study prospects the applications in biotechnology and irrigation systems. Characteristics of thermophoresis, nonlinear thermal radiation and Brownian moment are examined. Significant influence of no mass flux relation and variable thicked surface on second grade is studied in this work. Moreover, heat-mass transportation phenomenon is explored through thermophoresis, nonlinear thermal radiation and Brownian diffusion. Transformation procedure is adopted in order to achieve required ODEs. HAM procedure is adopted to obtain the series solution. Outcomes of physical parameters for velocity, temperature and concentration fields are discussed. Here velocity profile of second-grade fluid is detected to be a declining function K (local second-grade parameter). Higher $N_b$ (Brownian moment parameter) intensifies the temperature profile while concentration of viscoelastic liquid dwindles against $N_b$.

Sequel to the significance of the dynamics of second-grade fluid in the industry, nothing is known on the increasing inclined magnetic force, Newtonian heating, slip flow and Prabhakar-like fractional kind of Newtonian heating. In this study, the novel modified and efficient fractional approach, namely, the Prabhakar fractional technique is applied on an unsteady flow of second-grade liquid flowing on an inclined oscillating plate. The flowing fluid is moving under the effects of an applied inclined magnetic field, furthermore, the Newtonian heating effect and slip on the boundaries of the wall are also considered in the boundary conditions. After nondimensionalizing the governing equations, the solution of coupled equations is attained with the help of integral transform i.e., Laplace transformation method, and some numerical approaches are also applied for the Laplace inverse. The results convey that the momentum field decrease with increase in fractional constraints. The thermal change exhibits a declining tendency for increased values of fractional parameters. Furthermore, in comparison of velocity field attained by our applied fractional system with the existing literature, the curves of both Stehfest and Tzou’s methods overlay with each other which signifies the validity of our attained outcomes of momentum profile.

The flow of viscoelastic fluid owing to stretching surfaces (SSs) has become well-known among the demanding study fields of fluid mechanics due to its numerous practical applications. As a result, the major goal of this work is to scrutinize the three-dimensional (3D) unsteady flow of second-grade nanofluid. A magnetic field influences the fluid when an indefinitely long, straight wire delivers an electric current. Thermal radiation is taken into consideration while studying heat transmission characteristics. The effective thermal conductivity of nanofluid flow is scrutinized by adopting Koo–Kleinstreuer–Li (KKL) nanofluid model. Through suitable transformations, relevant flow expressions are converted to ordinary differential equations (ODEs). The numerical approach Runge–Kutta–fourth–fifth Fehlberg’s order method (RKF-45) with shooting technique is used to solve the generated ODEs. Further, it was noted that the CPU time to estimate the solutions of flow fields was less than (2.23 s). The outcome reveals that the rise in values of the time-dependent parameter positively impacts both velocity and thermal profiles. The improved heat transport is seen for augmented values of radiation and ferromagnetic interaction parameters.

The main purpose of this study is to scrutinize the axisymmetric flow of second-grade nanofluid with variable viscosity near a stagnation point under the influence of the Cattaneo–Christov double diffusion model. A Riga plate is assumed to be the source of the three-dimensional flow. The consequences of the anisotropic slip conditions and the Buongiorno model are included in the formulation of the constituting equations of the fluid flow. The highly nonlinear ordinary differential equations are developed with the executions of the relevant similarity transformations on the problem’s governing equations. For the asymptotic analysis, the appropriate expansions are implemented on the nonlinear system of ordinary differential equations. The bvp4c technique of the MATLAB package is implemented on the nonlinear ordinary equations for the numerical solutions. In order to explore the impact of the parameter slip factor on the pattern of velocities, temperature, and concentration profiles, several graphs are plotted. From this study, we conclude that the normalized velocities boost up with the amplification in the slip factor parameter but the temperature profile exhibits a declining behavior. The concentration field exhibits the accelerating behavior relative to the slip parameter.

The aim of this research is to develop a fractional supported thermal model for studying the features of modified hybrid nanofluid endorsed by uniformly accelerating plate. The novel impact of this work is observing the comparative thermal enhancement of water base fluid by utilizing four types of nanoparticles. The silver, copper, aluminum oxide and titanium oxide nanomaterials are utilized to present the comparative thermal aspect of modified hybrid nanofluid model. Moreover, the inclined direction of magnetic impact is treated. The second-grade nonlinear model is used to explore the base fluid properties. The fractional model is first attained into dimensionless form. The fractional computations with employing the Prabhakar fractional mathematical definitions are reported. The motivations for suggesting the Prabhakar algorithm are justified as this fractional algorithm contains modern definitions without any restriction of singularities. The verification of model is accomplished after simulating the comparison task with already performed studies. The physical dynamic and thermal enhancement of transportation phenomenon is performed for specific range of flow parameters like $0.1\le \alpha \le 0.7,0.1\le \beta \le 0.7,$$0.1\le \gamma \le 0.7,0.7\le \mathrm{\text{Pr}}\le 4.5,0.01\le \phi \le 0.04,0.8\le \mathrm{\text{Sc}}\le 2.6,1.7\le \mathrm{\text{Gr}}\le 3.7,1.5\le \mathrm{\text{Gm}}\le 3.2$ and $\frac{\pi }{2}{\le 𝜃}_1\le \frac{\pi }{6}$ Based on the computational model, it is concluded that the thermal transportation phenomenon is more impressive for water-based titanium oxide nanoparticles. The temperature profile rises due to factional parameter for both copper–water- and sliver — water-based hybrid nanofluid suspension.

The exponentially stretching curved surface is considered for second-grade fluid flow. The slip effects are applied at the assumed surface. The Brownian motion, chemical reaction, and thermophoresis effects are studied to determine the influence of the second-grade fluid. The boundary layer approximation is applied to the governing equation which is reduced in terms of partial differential equations. These partial differential equations become the dimensionless form after using the appropriate transformations. Further, the numerical approach is imposed on the dimensionless system to calculate the problem-related results. Graphs and tables are used to show how involving physical parameters have an effect. The velocity profile and skin friction have been increased due to increasing values of curvature parameters. The velocity function declined due to an increment in the second-grade fluid parameter. It means that the thickness of the boundary layer shown as decline when the second-grade fluid parameter rises while the second-grade fluid parameter has an inverse relation with viscosity. The Prandtl number increased which increased the temperature function. Physically, the thickness of the thermal boundary is controlled by the Prandtl number, and temperature increases by increasing the value of the Prandtl number.

This research is focused on studying the behavior of bubbles in one-dimensional nozzle flows. The study delves into the complex flow of cavitating bubbles within geometries of varying cross-sectional areas, using a second-grade fluid. The cavitation phenomenon poses a potential threat, as it can harm surfaces upon bubble collapse or interaction with adjacent boundaries. Thus, the proposed model unveils the intricate behavior of cavitating bubbles in response to nozzle shape and fluid properties. The governing equations are first modeled using the Rayleigh–Plesset equation along with continuity and momentum equations for bubbly liquids. They are then simplified and nondimensionalized to solve by means of the RK method. The influence of several nondimensional parameters, including the Reynolds number, fluid parameter, cavitation number and the number of bubbles, on the flow behavior of second-grade fluid through various channels is illustrated using graphs. To validate the results, the Newtonian formulation was applied in the current setting, and it was found that the obtained results matched the available literature. Also, critical values of void fractions for the radius of the bubble are presented. The non-Newtonian properties of the fluid are observed a significant outcome for the reduction of cavitation damage and noise.

In this work, weakly nonlinear wave equation in mono-dispersed, isothermal, bubbly, slightly compressible viscoelastic liquid is derived, using reduction perturbation method. The constitutive equation is based on the second-grade fluid model. Viscosity, relaxation, retardation and surface tension are considered under isothermal condition, which modifies the Rayleigh–Plesset equation for compressible liquid. A kink traveling wave solution is obtained using the tangent hyperbolic method combined with the Riccati equation. Graphical representation of the solution is given and analyzed with different parameter values.

We are examining a mathematical model of second-grade Casson fluid flow over a Riga surface. This analysis takes into account the effects of heat generation on a second-grade Casson fluid. Additionally, we have discussed the impacts of Soret and Dufour in the presence of thermal radiation. The study also includes an analysis of the influences of thermal and concentration slip. By considering these factors in the fluid flow, we have constructed a set of partial differential equations. Furthermore, these differential equations have been transformed into dimensionless ordinary differential equations. We have presented the influences of relevant parameters through tables and graphs. Velocity curve reveals declining behavior due to larger values of magnetic field factor. A high magnetic field coefficient means a strong magnetic field. As a result, the force acting on the charged particle increases, causing it to slow down or change direction. This change in force can cause the particle’s velocity to decrease.

The numerical methods such as the artificial neural networks with greater probability and nonlinear configurations are more suitable for estimation and modeling of the problem parameters. The numerical methods are easy to use in applications as these methods do not require costly and time-consuming tests like the experimental study. In this study, we use the Levenberg–Marquardt-based backpropagation Process (LMP) to create a computing paradigm that makes use of the strength of artificial neural networks (ANN), known as (ANN-LMP). Here we use the ANN-LMP to obtain the solution of the second-grade fluid in a rotating frame in a porous material with the impact of a transverse magnetic field. The 1000 data set points in the interval $[0,1]$ are used for the network training to determine the effect of various physical parameters of the flow problem under consideration. The experiment is executed of six scenarios with different physical paramaters. ANN-LMP is used for evaluating the mean square errors (MSE), training (TR), validation (VL), testing (TT), performance (PF) and fitting (FT) of the data. The problem has been verified by error histograms (EH) and regression (RG) measurements, which show high consistency with observed solutions with accuracy ranging from E-5 to E-8. Characteristics of various concerned parameters on the velocity and temperature profiles are studied.