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This study investigates the mixed convection on a vertically oriented exponentially stretching surface, considering the exponential distributions of velocity, concentration and temperature. The governing highly nonlinear partial differential equations (PDEs) are transmuted into highly nonlinear ordinary differential equations (ODEs) by considering appropriate similarity variables. The highly nonlinear differential equations are solved numerically using the Hermite wavelet method (HWM) in MATHEMATICA 12. The obtained results show excellent agreement with previous studies focusing on different special cases of the problem. A parametric analysis of the physical parameters is shown and illustrative numerical results are presented graphically. The analysis incorporates the radiation, chemical reaction and cross-diffusion effects. The interaction between radiation and chemical reaction is particularly relevant in high-temperature systems, such as combustion, industrial furnaces, photochemistry, polymerization and certain chemical reactions. The results indicate that an increase in radiation enhances the temperature profile in the momentum boundary layer. Higher reaction rate values lead to a decrease in the concentration profile, while an increase in the Dufour value results in increased velocity and temperature and decreased concentration.

This paper investigates the effects of radiation, internal heat source and magnetohydrodynamics (MHD) on the mixed convective boundary layer flow of a Casson nanofluid within a porous medium that is saturated and subject to an exponentially stretching sheet. The nanofluid model incorporates Brownian motion and thermophoresis, and the Darcy model is employed for the porous medium. By applying an appropriate similarity transformation, the nonlinear governing boundary layer equations are converted into a set of nonlinear coupled ordinary differential equations. These equations are then solved numerically using the Hermite wavelet method, with simulations conducted through the MATHEMATICA programming language. The analysis covers various aspects including temperature distribution, velocity, solute concentration and several engineering parameters such as skin friction coefficients, the Nusselt number (rate of heat transfer) and the Sherwood number (rate of mass transfer), all evaluated based on dimensionless physical parameters. The results indicate that elevated radiation intensifies temperatures and leads to thicker thermal boundary layers. As the Casson parameter increases, both the velocity and the momentum boundary layer become narrower. Additionally, a more pronounced chemical reaction rate reduces the thickness of the solutal boundary layer. The accuracy and reliability of the numerical Hermite wavelet method are validated through a comparative analysis with previous studies, demonstrating excellent concordance and confirming the robustness of the computational approach.

In this paper, the thermal analysis is performed on the three-dimensional flow of viscous nanofluid over a rotating disk. The disk rotation is assumed as of the form $(v\sim r^m)$ where $(m\ge 1)$, thus, there are different azimuthal velocities of the disk at different radii, and it increases with the increase in $m$. Also, the effect of nanoparticles on the thermal conductivity of fluid is investigated by thermophoresis and Brownian motion features. For the governing problem, suitable similarity transformations are employed that convert the produced partial differential equations to ordinary differential equations. The flow field and energy transport are presented graphically and discussed in detail for the torsional exponent $m$, $N_t$ and $N_b$, respectively. It is noted that the temperature and concentration profile rise for the thermophoresis parameter while the opposite trend showed for Brownian motion. Moreover, the power-law index parameter $m$ declines the flow field temperature and concentration profiles.

This paper depicts the fully developed natural convective flow on a conducting viscous fluid towards a nonlinearly stretching sheet. Furthermore, the porous dissipation, thermal radiation and heating parameter effects are implemented on both the vertical walls of the stretchy channel. To model the stretchy flow equations, the Cartesian coordinates’ system is utilized. Through the utilization of similarity variables, the nonlinear partial differential equations that describe the flow (mass, momentum and energy conservation) are converted into nonlinear ordinary differential equations. With the help of the MAPLE, a well-known fourth-order Runge–Kutta procedure is used to do a numerical evaluation of the stated nonlinear and non-dimensional set of equations. For each of the several nonlinear radiative parameters regulating the flow regime, the velocity and temperature distribution functions are determined, viz the nonlinear heating parameter ${\theta }_R$, Eckert number $Ec$, Prandtl number $Pr$, porosity variable $Pm$ and thermal radiation parameter $N_R$. Graphic representations are provided for every outcome. Furthermore, skin friction and Nusselt number are also computed to give an approximation of the surface shear stress and cooling rate, respectively. A remarkable compaction is obtained between computed numerical data and published results. It has been demonstrated that an increase in the value of the nonlinear parameter $Pm$ outcomes creates a reduction in the dimensionless translational velocity $g^{\prime }$ of both viscous and Newtonian fluids. Dimensionless temperature mostly upsurges with growth in nonlinear parameters $Ec$, $Pm$, ${\theta }_R$ and decreases with an intensification in convective parameters, $Pr$, $N_R$. There is a detailed discussion on the implications of all embedded stretching sheet variables on the flow. The flow regime is extremely useful in the technology of polymer processing as well as in the field of materials science.

Due to high sensitivity of fluid particles to temperature changes, the thermophysical characteristics of fluids are known to exhibit significant variations when exposed to temperature rise, which can impact the overall efficiency of conveying fluid substances in manufacturing processes across different surfaces and media. Industrial and mechanical processes such as aerodynamic extrusion in plastic sheets, fiber technology and biological dynamics as transpiration and muscle cramps exhibit different surface dynamics. The Reiner–Philippoff (RP) fluid model exemplifies shear-thinning, shear-thickening and Newtonian behavior. This study investigates the flow of a RP fluid over a stretching and shrinking surface with nonlinear thermal convection and variable thermophysical properties. The mathematical models describing the dynamical system are transformed through a similar group of transformations. Flow variables such as the fluid’s velocity, temperature and solute concentration were numerically obtained through the spectral local linearization technique (SLLM). Subject to the validation and accuracy of the numerical results, the effects of pertinent parameters on the flow were investigated. The findings in this study reveal that the effects of the flow variables on stretching and shrinking sheets are not diametrically opposite. Moreover, the radiation negatively impacts the flow variables over a stretching sheet compared to the positive effect experienced at some point in flow over a shrinking sheet.

The analysis of non-Newtonian fluid flow over an oscillating surface often involves numerical simulations and experimental investigations. Computational fluid dynamics method including finite difference or finite element techniques can be used to crack the governing equations of the fluid flow. In this work, we used the Crank–Nicolson numerical technique to analyze the numerical behavior of unsteady boundary layer flow of Casson fluid with natural convection past an oscillating vertical plate. The temperature-dependent viscosity is assumed for the flow analysis. The impact of chemical reaction and heat generation coefficient is used to examine the mass and heat transferal rates. The investigation of non-Newtonian fluid flow over an oscillating surface is crucial for a wide range of industrial, biomedical, and scientific applications. The governing model of equations occurs in the form of nondimensional PDEs and then we use the dimensionless variables in order to achieve the dimensional PDEs. These equations are numerically solved by using the Crank–Nicolson technique. The Crank–Nicolson scheme is used because it has the ability to provide accurate and stable solutions and make it a valuable numerical technique in various scientific and engineering disciplines. The findings indicate the significance of numerous parameters on the mass, velocity and energy regions. The numerical outcomes of skin friction are observed due to fluid parameter, viscosity parameter, Grashof numbers of heat and solutal rates.

The focus of this paper is on physics of coherent structures in boundary layer flow in near-wall region of a vibrating plate. A dynamical model is developed based on Galerkin projection of the governing equation of the wall layer flow onto a subspace spanned by the orthogonal divergence-free Fourier basis functions. The interactive physics of the coherent structures with the wall vibration is studied with the established model truncated at any order. The compared results show that the prevailing coherent structures in the layer flow near a vibrating wall region are captured.

Our previous studies in quiescent air environment [Z. J. Zhao et al., AIAA J.53(5) (2015) 1336; J. G. Zheng et al., Phys. Fluids26(3) (2014) 036102] reveal experimentally and numerically that the shock wave generated by the nanosecond pulsed plasma is fundamentally a microblast wave. The shock-induced burst perturbations (overpressure and induced velocity) are found to be restricted to a very narrow region (about 1 mm) behind the shock front and last only for a few microseconds. These results indicate that the pulsed nanosecond dielectric barrier discharge (DBD) plasma actuator has stronger local effects in time and spatial domain. In this paper, we further investigate the effects of pulsed plasma on the boundary layer flow over a flat plate. The present investigation reveals that the nanosecond pulsed plasma actuator generates intense perturbations and tends to promote the laminar boundary over a flat plate to turbulent flow. The heat effect after the pulsed plasma discharge was observed in the external flow, lasting a few milliseconds for a single pulse and reaching a quasi-stable state for multi-pulses.

This study looks at the numerical results of magnetohydrodynamic liquid flowing across a stretching sheet at its stagnation point while also experiencing chemical reaction, viscous dissipation, thermal radiation and variable magnetic field. The fundamental partial differential equations that regulate physical phenomena are converted into nonlinear ordinary differential equations by using the appropriate similarity transformations. Later, resolved by numerically using Mathematica. The effects of various non-dimensional parameters like velocity ratio parameter ($\lambda$), porosity (k), magnetic parameter (M), radiation parameter (R), chemical reaction parameter ($\gamma$), etc. on the velocity, temperature and concentration profiles as well as on the local skin-friction and the local Nusselt number are discussed in detail and displayed through graphs. The numerical evaluation of the physical quantities was provided in tabular form for the various values of the relevant stream parameters. It is important to note that magnetic field interaction increases concentration distribution and fluid temperature while decreasing velocities at all domain flow locations.

The major objective of this research is to create a novel mathematical model for the flow of an electro-osmotic boundary layer in a micropolar Williamson nanofluid. This development is achieved by considering the influence of electro-osmotic force on an incompressible micropolar Williamson nanofluid through a Darcian flow (Darcy model) when combined with a binary chemical reaction and the energy of activation over a linearly stretching sheet. The constituent parts of the energy equation include heat radiation, thermal and mass transport, along with joule-based heating and dissipation of viscous fluids. The problem is computationally analyzed using an equation set of nonlinear partial differential equations (PDEs), which are then similarly converted into a system of ordinary differentiation equations (ODEs). The obtained ODEs system is solved numerically using the built-in command (Parametric ND Solve) via MATHEMATICA software. A series of figures are used to demonstrate numerically and graphically the influence of physical characteristics on the fluid behavior. The behavior of flow is obtained by studying the streamlines around the plate in contour and three-dimensional form. In the case of non-Newtonian fluid, the results indicated that the boundary layer velocity is slower compared with the Newtonian fluid case. It is also mentioned that the graphic representation of the results shows that increasing the radiation parameter enhances thermal distribution, which means that the rate of heat transfer improved. The main findings revealed that the upward trend in the Williamson number diminishes the nanofluid flow while enhancing the skin friction coefficient. Also, it is discovered that as $m_e$ rises, the fluid’s velocity distributions in the boundary layer rise. Consequently, this type of research’s significance stems from its potential uses in biomedical engineering since it could be used to dewater liquids and solids from infected human tissues.

Micropolar fluids provide a more accurate description of fluid behavior at small scales, where classical continuum models may fail to capture complicated microscale interactions. These interactions become significant in situations involving non-Newtonian fluids, complex geometries, and materials with internal structures. Unlike traditional fluids, micropolar fluids consider the effects of micro-rotations and micro-deformations. The objective of this exploration is to investigate the characteristics of flow in a micropolar-Casson fluid that is doubly stratified, and is induced by a stretching sheet. Additionally, the study investigates the transfer of both thermal energy and mass species over a two-dimensional porous medium. By utilizing suitable similarity transformations, a system of nonlinear ordinary differential equations (ODEs) is derived by transforming the governing partial differential equations (PDEs) that describe the physics of the problem. The combination of a fourth-order Runge–Kutta integration scheme and the Shooting method is utilized to solve these equations. Various graphs and tables are utilized to illustrate the impact of physical parameters on the dimensionless quantities. Our findings exhibit strong conformity with the existing results in literature for specific scenarios. Furthermore, the outcomes highlight that the velocity and micro-rotation are enhanced by the material parameter.

Magnetohydrodynamic (MHD) principles may be used to study the flow of arterial blood under the action of an applied magnetic field. Such studies are of potential value in the treatment of cardiovascular disorders that may be associated with accelerated circulation. With an aim to providing a generalized model for studying the flow of blood in an electromagnetic field environment, a numerical model is developed here, by treating blood as a non-Newtonian fluid, the motion of which is taken to be governed by Walter's B-fluid model. The channel flow characteristics of the fluid are studied here, when the channel is porous and is subjected to an external magnetic field. Using the similarity transformation and boundary layer approximations, the associated nonlinear partial differential equations of the problem are reduced to nonlinear ordinary differential equations. These are solved numerically by developing a finite difference scheme. The study provides useful estimates for the influence of Reynolds number Re, Hartmann number M, and viscoelastic parameter K₁ on the flow characteristics. It bears the potential to explore some important information about the hemodynamical flow of blood in an artery when it is under the action of an external magnetic field.

In this paper, the boundary-layer natural convection flow on a permeable vertical plate with thermal radiation and mass transfer is studied when the plate moves in its own plane. A uniform temperature with uniform species concentration at the plate is affected and the fluid is considered to be a gray, absorbing–emitting. A viscous flow model is presented using boundary-layer theory comprising the momentum, energy, and concentration equations, which is solved analytically by means of an excellent method called homotopy analysis method (HAM). First, a comparison between HAM results and those obtained by means of a higher-order numerical method, namely differential quadrature method (DQM), is done. Close agreement of two sets of results indicates the accuracy of the HAM. The velocity, temperature, and concentration distributions are displayed graphically, and a parametric study is performed in which the effect of various parameters on the skin friction, the local Nusselt number (Nn), and the local Sherwood number (Mu) are investigated.