Narrow Results

In this paper, we thoroughly examine the influences of slip effects and stagnation point flows in the context of an upper-convected non-Newtonian Maxwell nanofluid interacting with a stretching sheet. The existence of a heat generation, transverse magnetic field, and thermal radiation induces a flow resulting from a linearly stretched sheet. The application of the shooting method involves deriving nonlinear ordinary differential equations from the governing partial differential equations, followed by their solution. The effects of dimensionless governing parameters, including velocity ratio, Brownian motion parameter, thermophoresis parameter, velocity slip parameter, Lewis numbers, solutal slip parameter, Maxwell parameter, magnetic number, Eckert number, thermal slip parameter, chemical reactions parameter, and heat source parameter, are examined. The outcomes are illustrated and discussed through graphical representations, showcasing their impact on the velocity field, as well as heat and mass transfer characteristics. Tabular data are generated to display numerical values for physical parameters, including the skin-friction coefficient, local Sherwood number, and the reduced local Nusselt number. The findings suggest that an increase in the velocity slip parameter results in a reduction of both the local Sherwood number and the local Nusselt number. Furthermore, an increase in the strength of the magnetic field leads to a decrease in velocity profiles while simultaneously elevating temperature and concentration profiles.

In this paper, He’s homotopy perturbation method (HPM) is used, which is an approximate analytical method for solving numerically the problem of Newtonian fluid flow past a porous exponentially stretching sheet with Joule heating and convective boundary condition. The major feature of HPM is that it does not need the small parameters in the equations and hence the determination of classical perturbation can be discarded. Due to the complete efficiency of the HPM, it becomes practically well suited for use in this field of study. Also, the obtained solutions for both the velocity and temperature field are graphically sketched. The results reveal that the proposed method is very effective, convenient, and quite accurate to systems of nonlinear differential equations. Results of this study shed light on the accuracy and efficiency of the HPM in solving these types of nonlinear boundary layer equations.

In this work, an approximate analytical solution for the problem of non-Newtonian Casson fluid flow past a porous exponentially stretching sheet with Joule heating and convective boundary condition is obtained using a relatively new technique; He’s homotopy perturbation (HPM). The major feature of HPM is that it does not need the small parameters in the equations, and hence the determination of classical perturbation can be discarded. Due to the complete efficiency of the HPM, it becomes practically well suited for use in this field of study. Also, the obtained solutions for both the velocity and temperature field are graphically sketched. The results reveal that the method is very effective, convenient and quite accurate to systems of nonlinear equations.

The analysis of Boundary Layer Flows (BLFs) is of huge significance due to its large-scale applications most likely in aerodynamics, manufacturing of vehicles front faces and many other advanced research areas. Therefore, this study is organized by exercising significant important engineering parameters like surface convection, radiative flux, MHD and thermal and mass diffusion. The developed model embedded the concrete effects of velocity slip, diffusion convection, thermal radiations, magnetic field, chemical reaction and thermo-mass gradients. Later, the numerical treatment is made via RK scheme and the model efficaciously is tackled with desired accuracy. Then the results are decorated over the region (wedge and sheet) and deeply examined. It is observed that slippery surface of wedge and sheet $(S_1=0.1,0.3,0.5,0.7,0.9)$ enlarges the fluid movement rapidly. The directed magnetic field ($M=0.0,0.1,0.2,0.3,0.4)$ controlled the fluid motion which can be beneficial from industrial view point. Further, temperature and concentration distributions upsurge by strengthening convective heat ($B_i=0.1,0.2,0.3,0.4,0.5)$ and diffusion convection processes. Moreover, skin friction and Nusselt number improved against the parameters under consideration.

This research is focused on the examination of an unsteady flow of an electromagnetic nanofluid close to a stagnation point over an expanded sheet kept horizontally. Buongiorno’s nanofluid model is revised with the combined influence of the externally applied electric and magnetic fluxes. Moreover, the underneath surface offers multiple slips into the nanofluid flow. The leading partial differential equations (PDE) are renovated to the nonlinear ordinary differential equations (ODE) with the assistance of similarity transformations. Thus, the outcomes are received numerically by using the RK-6 with Nachtsheim–Swigert shooting technique. The enlistment of the outcomes for the momentum, energy and concentration profiles along with the skin-friction coefficient $(C_{fx}^{\ast })$, Nusselt number $({\text{Nu}}_x^{\ast })$ and Sherwood number $({\text{Sh}}_x^{\ast })$ for several parametric values are presented in a graphical and tabular form and discussed in detail. The variation of streamlines with respect to the unsteadiness parameter is also recorded. Statistical inspection reveals that the flow parameters are highly correlated with the wall shear stress, wall heat and mass fluxes. Findings indicate that the escalation of electric flux tries to intensify the hydrodynamic boundary layer meanwhile the magnetic flux assists to stabilize the growth by reducing it for both the steady and unsteady flow patterns. Influence of velocity slip parameter $\xi$ from 0.0 to 1.5 causes the reduction in ${\text{Nu}}_x^{\ast }$ by 16.98% for steady flow while 60.27% for time-dependent flow case. Moreover, we expect that these theoretical findings are very much helpful for several engineering and industrial applications such as polymer sheet productions, manufacturing automobile machines, cooling microelectronic chips, etc.

This paper addresses the effects of applied magnetic field and partial slip effects in peristalsis of water-based nanofluids in an asymmetric flow configuration. Analysis is carried out using silver and copper nanoparticles. Viscous dissipation, mixed convection, Ohmic heating and heat generation/absorption are considered. Mathematical modeling is done employing lubrication approximations. Resulting coupled system is solved numerically. Physical quantities like axial velocity, pressure gradient, temperature and heat transfer rate are graphically analyzed. Comparison between the silver–water and copper–water nanofluids is presented and analyzed. Results show that the maximum velocity, temperature and heat transfer rate at the wall in silver–water nanofluid are comparatively greater than that of copper–water nanofluid. It is also observed that addition of nanoparticles results in a decrease in the velocity and temperature of fluid. However, the heat transfer rate at the wall is enhanced through addition of nanoparticles.

In nanotechnology, the nanofluids are decomposition of base materials and nanoparticles where the nanoparticles are immersed in base liquid. The utilization of such nanoparticles into base liquids can significantly enhance the thermal features of resulting materials which involve applications in various industrial and technological processes. While studying the rheological features of non-Newtonian fluids, the constant viscosity assumptions are followed in many investigations. However, by considering the viscosity as a temperature-dependent is quite useful to improve the heating processes along with nanoparticles. Keeping such motivations in mind, this investigation reports the temperature-dependent viscosity and variable heat-dependent conductivity in bioconvection flow of couple stress nanoparticles encountered by a moving surface. The famous Reynolds exponential viscosity model is used to deploy the relations for temperature-dependent viscosity. Moreover, the activation energy and higher order slip (Wu’s slip) are also elaborated to make this investigation more novel and unique. The emerging flow equations for governing flow problem are formulated which are altered into non-dimensional forms. The numerical simulations with applications of Runge–Kutta fourth–order algorithm are focused to obtain the desired solution. Before analyzing the significant physical features of various parameters, the confirmation of solution is done by comparing the results with already reported investigations as limiting cases. The results are graphically elaborated with relevant physical consequences. Various plots for velocity, temperature, concentration, wall shear stress, local Nusselt number, local Sherwood number and motile density numbers are prepared.

The progress in new inventions in the modern technological world demands outstanding heat transport. Unfortunately, common solvents are unable to produce such desired amount of heat which compelled the scientists and researchers towards the development of new heat transfer fluids (Nanofluids). Therefore, the study of C₈H${}_{18}$ with hybridization of [(MnZnFe₂O₄–NiZnFe₂O${}_4)$]${}_{\mathrm{\text{hnps}}}$ under novel effects of thermal radiations and convective heat conditions over a slippery permeable surface is organized. The modified thermophysical correlations for hybrid nanofluids were used and successfully achieved the modified heat transfer model. After numerical investigation, the results were plotted under varying parameters and provided for comprehensive discussion. The results revealed faster fluid motion and $G^{\prime }(\eta )$ is dominant. The velocities drop significantly due to the permeability of the surface. Further, thermal radiations potentially boost heat transfer by providing extra energy to fluid particles. The temperature coefficient ${\beta }_w$ due to nonlinear thermal radiations also indicated faster heat transport.

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 study of electro-osmosis, peristalsis and heat transfer with numerous slips, such as velocity slip, thermal slip and concentration slip, may be used to construct biomimetic thermal pumping systems at the microscale of interest in physiological transport phenomena. A mathematical model has been developed to investigate magnetohydro-dynamics non-Newtonian (Carreau fluid) flow induced by the forces to produce a pressure gradient. The walls of the microchannels erode as they expand. The Poisson and Nernst–Planck equations are used to model electro-osmotic processes. This procedure results in Boltzmann circulation of the electric potential across the electric double layer. The governing equations are simplified by approximations such as a low Reynolds number and a long wavelength. The ND Solver in Mathematica simulates and compares simplified coupled nonlinear governing equations. We investigate novel physical parameters affecting flow, heat transfer and pumping. Additionally, a fundamental peristaltic pumping phenomenon known as trapping is graphically provided and briefly discussed. The model’s findings show that the velocity increases as the electric field intensifies, implying that electro-osmosis may improve peristaltic flow.

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 aim of this work is to present the magnetized flow of Casson nanomaterials confined due to porous space with stability framework. The slip mechanism for thermal concentration diffusion has been elaborated. The shrinking surface with exponential velocity induced the flow. The new block method is imposed for the simulation process. The resulting systems of ODEs of the third and second orders are solved jointly using the block method, which is appropriate for dealing with the different orders of the system of ODEs. From a physical standpoint, graphs of different profiles for increasing values of the various applied parameters have been drawn and discussed in detail. To satisfy the infinite boundary conditions, we assigned numerical values such that all profiles converge asymptotically at $\eta \to \infty$. Furthermore, numerical results from the block method show that velocity profile declines with rising Casson and porous parameter values, as expected. It is noted that the heat transfer rate enhanced with the thermal slip parameter. A lower thermal profile due to larger Casson fluid parameter is observed.

In this paper, a non-isothermal study of the calendering processes is presented using Carreau–Yasuda model along with nonlinear slip condition introduced at the upper roll surface. The flow equations for the problem are developed and converted into dimensionless form with the help of dimensionless variables and then finally simplified by a well-known lubrication approximation theory. The final equations are solved numerically using “bvp4c” to find stream function and velocity profiles, while the hybrid numerical method which is the combination of shooting and finite difference methods is used to solve the energy equation. Graphs show the impact of the concerned material parameters on various quantities of interest. The pressure distribution decreases with the increasing values of the slip parameter and Weissenberg number. The mechanical variables show an increasing trend with the increasing values of the slip parameter and Weissenberg number. The temperature distribution increases with an increase in the Brinkman number, while temperature shows declining trend near the roll surface with the increasing values of the slip parameter. The force separating the two rollers, total power input into both rolls, increase with the increasing values of the Weissenberg number and slip parameters. The results show that the Newtonian model predicts higher pressure in the nip zone than the Carreau–Yasuda model. It is interesting to note that for the case of shear thinning, the Carreau–Yasuda model predicts 30% less pressure in the nip region when compared to the Newtonian model.

A fractional technique is used to evaluate the temperature, mass, and velocity flow of single and double wall CNTs over a vertical plate. Slip boundary conditions and applied magnetic force are addressed. Human blood is used to examine how base fluid behaves. Applying the proper dimensionless variables results in the dimensionless formulation of initial and boundary conditions related to the governed dimensional concentration, momentum, and energy equations. The Laplace transform technique is used to resolve the dimensionless governing partial differential equations and get the solutions. The constant proportional Caputo (CPC) time-fractional derivative is a unique class of fractional model used in the simulation technique. The fundamental definitions are used to support the said model first. Using MWCNTs and SWCNTs in comparison to the flow characteristics, a thermal and mass study is given. The heat and mass transfer processes for single-walled carbon nanotubes (SWCNTs) have been shown to typically be progressive. The momentum profile decreases as the fractional variables rise. Multi-walled carbon nanotubes (MWCNTs) show more progressive velocity control as a result of the magnetic parameter. Graphs demonstrate the influence of embedding factors on the velocity, energy, and concentration profiles.

In this paper, the effect of heat and mass transfer on particle–fluid suspension due to peristaltic motion is examined with of slip effects. The governing equations of fluid phase and particulate phase for Casson fluid model with embedded particles are interpreted under the approximation of long wavelength and neglecting the inertial forces. The obtained coupled resulting partial differential equations are solved analytically and an exact form of solutions are conferred. The impact of various sundry parameters are plotted and discussed for velocity, temperature and concentration distribution for both fluid and particle phase. Numerical solution is evaluated for pressure rise along the whole channel. The present analysis reveals various interesting behavior that warrant further analysis on various Newtonian and non-Newtonian fluids. In the present flow problem, the influence of slip represents opposite attitude on the walls of the channel whereas due to the impact of particle volume fractions, the velocity of the fluid diminishes along the whole length of the channel.

Present study examines the mixed convective peristaltic transport of Cu–H₂O nanofluid with velocity slip and convective boundary conditions. Analysis is performed using the two-phase model of the nanofluid. Viscous dissipation and heat generation/absorption effects are also taken into account. Problem is formulated using the long wavelength and low Reynolds number approach. Numerical solutions for the pressure rise per wavelength, pressure gradient, axial velocity, temperature and heat transfer rate at the boundary are obtained and studied through graphs. Results show that the area of peristaltic pumping decreases with an increase in the nanoparticles volume fraction. Increase in the velocity slip parameter shows an increase of the pressure gradient in the occluded part of the channel. Further, addition of copper nanoparticles reduces both the axial velocity and temperature of the base fluid. Temperature of the nanofluid also decreases sufficiently for an increase in the value of Biot number.