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In this paper, we apply the extended direct algebraic method to examine the soliton solutions as well as hyperbolic and trigonometric functions solutions of convective–diffusive Cahn–Hilliard equation describing the dynamic of separation phase for ternary iron alloys of (Fe ‐ Cr ‐ Mo) and (Fe ‐ X ‐ Cu). The outcomes reveal that our technique is very dynamic and straightforward. It is observed that the obtained exact solutions of this model are new in the literature. Moreover, various 2D and 3D graphs of the obtained solutions are presented to examine the physical understanding of the obtained results. [ABSTRACT FROM AUTHOR]

An endeavor is consummate to study the steady mixed convection solute transfer in two‐dimensional viscous fluid. A joule heating incompressible flow along an inclined stretching cylinder and double stratification impact on Jeffrey Nanofluid is scrutinized. Acquired nanomaterial pattern comprises the phenomena of thermophoresis and Brownian motion. By applying rule of approximation transformation, the nonlinear PDEs converted into ODEs. Shooting via Newton Raphson technique is used to solve the ODE'S with boundary conditions into initial conditions, also applying Runge–Kutta–Fehlberg technique of fourth order for numerical purpose. Computation (−1/2Rex0.5Cfx)$(-1/2 Re_{x}^{0.5}C_{f_x })$ for skin friction, (−Nrθ′(0)$-N_{r}\theta ^{^{\prime }}(0)$) for Nusselt, and (−ϕ′(0))$(-\phi ^{^{\prime }}(0))$ for Sherwood number are fetched in table format by using mathematical software Matlab. The results reveal that M (0, 0.3, 0.6, 0.9), γ (0.1, 0.3, 0.7, 1.3), and α (0,π/4,π/3,5π/4)$(0, \pi /4, \pi /3, 5\pi /4)$ will increase (−1/2Rex0.5Cfx)$(-1/2 Re_{x}^{0.5}C_{f_x })$ upto 7.51, 8.55, and 10.27%, respectively. The behavior of Nt$N_t$ on −Nrθ′(0)$-N_{r}\theta ^{^{\prime }}(0)$ and (−ϕ′(0))$(-\phi ^{^{\prime }}(0))$ falling off upto 11.20 and 28.38%, respectively with the increase of Nt$N_t$, but in Nb$N_b$ shows opposite reaction on Sherwood number. It is culminated that (f′(η))$(f^{\prime }(\eta))$ for velocity augments with the value of Deborah number of four different sizes (0.8, 1.1, 1.4, and 1.7) upto 14.28, 16.6, and 20% while situation is reversed in case of magnetic factor (0, 0.3, 0.6, 0.9) upto 20, 28.12, and 39.13%. Effect of θ(η)$\theta (\eta)$ for temperature enhanced with the value of Brownian motion on (0.0, 0.2, 0.4, and 0.6), about 16.66, 20, and 25%, respectively. The maximum effect percentage with radiation factor on θ(η)$\theta (\eta)$ is 39 and minimum percentage is 25.71. Eckert number on (− 4, 0, 4, 8) and thermophoresis on (−0.2,−0.1,0.1,0.2)$(-0.2, -0.1, 0.1, 0.2)$ diminished with pile‐up Prandtl number on (1.2, 1.5, 1.8, and 2.1). Impact of (ϕ(η))$(\phi (\eta))$ for concentration and local Sherwood number showed reverse situation on Schmidt number. [ABSTRACT FROM AUTHOR]

In this work, we prove Ostrowski–Mercer inequalities for differentiable harmonically convex functions. It is also shown that the newly proved inequalities can be converted into some existing inequalities. Furthermore, it is provided that how the newly discovered inequalities can be applied to special means of real numbers. [ABSTRACT FROM AUTHOR]

This article disputes the study of convective flow for improved nanofluid along with an erect heated plate via Prabhakar‐like energy transport. The governing equations for this mathematical model are obtained by Prabhakar fractional derivative. To attain the generalized results for dimensionless velocity profile and temperature profile, a scheme of Laplace transform is applied. By applying the conditions of nanofluid flow, we develop the constantly accelerated, variables accelerated, and non‐uniform accelerated solution of the model. Prabhakar fractional derivative for improved nanofluid based on generalized Fourier's thermal flux is determined for heat transfer. Different structures of graphs are performed for ordinary fractional parameters. As a result, it is found that temperature of Ag − H2O is higher than Cu − H2O and TiO2 − H2O nanoparticles and the reverse trend can be found for velocity. Furthermore, temperature and velocity can be enhanced by increasing the values of fractional parameters. [ABSTRACT FROM AUTHOR]

This article aims to investigate free convection of a Casson fluid past a vertical plate embedded in porous medium with invariant wall temperature. It is assumed that the fluid can conduct electricity and it is flowing across a porous medium. The partial differential equations governing the model are made dimensionless by using dimensionless parameters. The Laplace transform method is applied to get analytical results. Furthermore, the hybrid fractional model is developed and the exact solutions for momentum and energy equations are acquired. The obtained results are compared with classical ones and the effect of hybrid fractional parameters are analyzed graphically by using MathCad software. Skin friction and heat transfer rate Nu is analyzed for small and large times and for hybrid fractional parameter β . We also have seen the increasing velocity profiles for buoyancy parameter Gr, whereas temperature of the fluid decreases for PrPr. The rate of heat transfer (Nu) and skin friction (Cf ) can be minimized by increasing the values of β . Furthermore, the constant proportional Caputo derivative model exhibits more decay in velocity in comparison with classical model given in Khalid et al. Therefore, the constant proportional Caputo differential model demonstrates better memory function than the classical one. Moreover, the obtained results are identical to already published results of Khalid et al. and Imran et al. [ABSTRACT FROM AUTHOR]

The single‐walled carbon nanotubes are the solid tiny materials in nature with efficient thermal conductivity and has many useful applications in biological domain specifically in the formation of biological nanofluid. The physical effects of single‐wall carbon nanotube (SWCNT) are considered in this study for free convection slippage flow of fractional viscous fluid by considering the thermal radiation, heat generation, chemical reaction of order first, and Newtonian heating through a porous medium. The fractional flow model of viscous fluid with SWCNT is constructed by inserting most recent approach of non‐integer derivative (constant proportional Caputo fractional derivative). Non‐dimensional boundary layer equations which govern the flow are solved analytically via integral transform, namely, the Laplace transform, and explicit transformed results for field variables, temperature, concentration, and velocity are established. Thermal and mass fluxes are also obtained numerically and presented in tabular form. The parametric study is carried out by plotting some graphs of thermal, concentration, and momentum profiles. In the light of paramedic study, it is noted that from the plots of field variables with slip condition are higher than that with the no‐slip condition. The constant proportional Caputo (CPC) fractional derivative is a best choice to get more stable temperature, concentration, and velocity fields than that of ordinary fields. More precisely, fractional parameter controls the boundary thickness of temperature, concentration, and velocity fields. [ABSTRACT FROM AUTHOR]

The present study is aimed to analyze the unsteady micropolar nanofluid flow passing over an oscillating infinite vertical plate. The flow is affected by thermal radiation and Newtonian heating. Single‐walled carbon nanotubes (SWCNTs) are added to enrich the thermal properties of the micropolar fluid. Kerosene is taken as the base liquid to enhance heat transfer. By using dimensional analysis, the governing equations for temperature, velocity, and microrotation are reduced to dimensionless form and after that, these equations have been solved by applying Laplace transform method to get the exact solutions. Finally, we have presented the effects of material and flow parameters and illustrated graphically by the Mathcad software. We found that microrotation, temperature, and velocity are decreasing functions of Prandtl number but have shown increasing behavior for Grashof number. Furthermore, we found that SWCNTs‐water‐based nanofluid has a comparatively higher heat transfer rate than SWCNTs‐kerosene and SWCNTs‐engine oil‐based nanofluids. [ABSTRACT FROM AUTHOR]

In this article, micropolar nanofluid boundary layer flow over a slanted stretching surface with Soret and Dufour effect is studied. The inclined stretching surface in this study is considered permeable and linear. In this problem, the Buongiorno model is considered for thermal efficiencies of fluid flow in the existence of Brownian movement and thermophoresis properties. The nonlinear problem for Micropolar Nanofluid flow over the slanted channel is developed to think about the heat and mass exchange phenomenon by incorporating portent flow factors to strengthened boundary layers. In this study, nonlinear partial differential equations are converted to nonlinear ordinary differential equations by utilizing appropriate similarity transformations then elucidated the numerical outcomes by the Keller-Box technique. An examination of the set-up results is performed with accessible outcomes and perceived in a good settlement without involved impacts. Numerical and graphical outcomes are additionally displayed in tables and charts. [ABSTRACT FROM AUTHOR]