Narrow Results

The unsteady Casson and Williamson nanofluid flow via a radiative cone in a magnetic surface with cross-diffusion and chemical processes is investigated numerically. The Casson and Williamson nanofluid models characterize the behavior of non-Newtonian fluids. We transform the nonlinear dimensional PDEs into dimensionally free PDEs by implementing dimensionless parameters. The governed partial differential equations are addressed by adopting the Crank–Nicolson implicit approach. A parametric analysis is used to determine the outcomes of skin friction, Nusselt and Sherwood numbers. Increased Brownian motion and the Soret effect promote the transfer of heat and mass. The thermal and mass transfer characteristic of the Casson nanofluid is higher than the Williamson nanofluid. Regarding momentum dispersion, the Williamson nanofluid clearly dominates the Casson nanofluid. Conical mixers are used in polymer blending and plastics manufacturing. The relevance of this mixing step in achieving specified product attributes shows how mass and heat transfer concepts can boost productivity and output in engineering industries.

Background and Objectives: This study is made to analyze the radiation effect in the flow of magnetohydrodynamic (MHD) Casson nanofluid when subjected to a magnetic field. The velocity slip over inclined nonlinear stretching surface in Forchheimer porous medium is taken into account. The Blood is considered as a base fluid and single-walled carbon nanotubes (SWCNTs) as nanoparticles in this study. The basic purpose of this study is to analyze the heat transfer and MHD effects on the Casson nanofluid which is nowhere found in previous studies and this laydown a pathway for the future researches.

Significance: Growing potential of Casson fluid by considering its applications to flow and energy transfer, the current analysis can be of great significance where working fluid used is non-Newtonian in nature.

Methodology: The mathematical model consisting of flow and heat equations is solved by using the Runge–Kutta fourth-order method along with shooting method in MATLAB using bvp4c solver.

Results: Graphical outputs of velocity and temperature fields are obtained for various values of magnetic parameter M, Prandtl number $\mathrm{\text{Pr}}$, Forchheimer parameter $F_s$, permeability parameter $K_p$ and concentration parameter $\varphi$. The numerical findings of coefficient of local skin friction and local Nusselt number are also tabulated. Casson fluid parameter in an increasing order impacted decreasingly on the skin friction of the fluid while magnetic number upgrade it along the sheet. The stability of fluid flow is effected by volumetric ratio of SWCNT’s nanoparticles. The boundary line temperature increases as radiation parameter rises.

This paper examines the influence of magnetized Casson nanofluid flow and heat transport phenomena towards a boundary layer flow over a nonlinear stretchable surface. The characteristics of the nanofluid are illustrated by considering Brownian motion and thermophoresis effects due to which the fluid is electrically conducting. The nonlinear Casson model is very useful to describe the fluid behavior and the flow curves of suspensions of pigments in lithographic varnishes intended for the preparation of printing inks. A uniform magnetic field, along with suction and chemical reaction are taken into account. Similarity transformations are employed to convert the PDEs into ODEs, and then solved numerically (Bvp4c) using MATLAB. This scheme consists of a finite difference scheme that implements three-stage Lobatto IIIa collocation formula which provides continuous solution upto fifth-order accuracy. Excellent correctness of the present results has been acquired which is compared with the previous one. The outcomes of various parameters on heat transfer rate, skin friction coefficient, nanoparticle concentration, Sherwood number, velocity and temperature profiles are demonstrated via tabular forms and pictorially. The most important fact is that an increase in the thermophoresis parameter, radiation and magnetic parameter boosts up the fluid temperature, resulting in an improvement in the thermal boundary layer.

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.

This paper concerns with the numerical investigation on a boundary layer flow of nanolubricant/liquid flow over a convectively heated rotating disk. Following that, the research was supported by the inclusion of the convection process as well as the influence of heterogeneous and homogeneous reactions on the flow of the nanolubricant/liquid in concern. A comparative analysis is made in terms Zinc Oxide (ZnO)-Society of Automotive Engineers 50 nanolubricant and Zinc Oxide-Kerosene using two different models. Transforming the governing nonlinear equations yields the nonlinear ODE set. Numerical investigations use Runge–Kutta method. Flow, temperature and concentration field controlling factors are also explored numerically. The heat transport and shear stress and characteristics are evaluated for important factors. Results reveals that, ZnO-SAE50 nanolubricant shows augmented heat transport than ZnO-Kerosene nanoliquid for increased values of Q. But reverse trend is seen for increased Biot number values. Nanolubricant shows improved heat transport rate than ZnO-Kerosene nanoliquid for augmented values of heat source and porosity parameters.

The theme of the current effort is to theoretically analyze the entropy generation and heat transfer aspects of Casson nanofluid flow triggered by rotating porous disc with the presence of magnetic dipole, nonlinear thermal radiation, viscous dissipation and Joule heating. The modeling of the nanofluid can be described with the combination of Brownian motion and thermophoresis by incorporating the passive control boundaries, and the governing PDEs are transformed into a set of highly nonlinear ODEs. The resulting equations are then solved analytically using HAM technique. The present results are compared with previously published results, which are in excellent agreement. The effect of pertinent nondimensional parameters on the entropy generation, hydrodynamic, heat and mass transport aspects is discussed via graphical illustrations. Both radial and tangential velocities are affected by accelerating the values of Hartmann number and porosity parameter. The temperature profile is upsurged by improving the radiation and thermal ratio parameter. Increasing the Casson parameter and Brinkman number leads to improved entropy generation rate. Moreover, skin friction, heat and mass transfer rates are examined with the help of the tables. It is believed that this study can be utilized as coolants by numerous automotive and engineering industries, namely the electronic devices, electrical motor, spin coating, fabrication of spacecraft, thermal insulation, nuclear reactors, etc.

This study investigates the effects of complete slip conditions on the peristaltic pumping of a Casson nanofluid with suction and injection in a vertical due to the crucial role that nano liquids play in a variety of technological and medical fields, particularly in peristalsis, a mechanism that transports liquids. The Casson fluid belongs to a class of non-Newtonian fluids that, through a particular stress threshold magnitude, exhibit elastic solid behavior before changing to liquid behavior. These fluids have several uses in engineering, food preparation, drilling and other fields. After establishing the governing conservation equations, the resulting flow model is effectively simulated using the realistic assumptions of a long wavelength and a low Reynolds number. The temperature distributions, velocity, pressure rate per wavelength and nanoparticle concentration of the resulting flow problem have been solved analytically. The effects of all physical factors on temperature, velocity, concentration fields, pressure rate, frictional force and pressure gradient are graphically examined using Wolfram MATHEMATICA software. There are a variety of biofluids that cannot be classified as liquids. For example, blood contains WBC, RBC and plasma. It is essential to model biofluids (blood) as nanofluids given the physical properties of these biofluids. According to reports, one of the finest yield stress models is the Casson model, and blood exhibits a similar behavior. We took these facts into consideration when thinking about Casson nanofluid flow in a vertical layer under peristalsis. Additionally, the suction and injection mechanisms can be used to represent the exchange of carbon dioxide in bold. In order to understand how blood flows through small blood vessels, this model must be examined. The obtained results show that the Newtonian case and those found in the literature have a very good agreement. Since the liquid moves faster and more effectively when the value is increased, it becomes clear that this increases the strength of the velocity.  In other words, nanoperistaltic pumps can maintain a pressure differential that increases or decreases at all operating flow rates with an increasing thermophoresis effect. Furthermore, it is obvious that the pressure reduction in a Casson fluid is greater than in a Newtonian fluid.

Inquisitive researchers have studied the movement of peristaltic nanofluids due to the enlightening impact of nanoliquids in various technological and therapeutic fields, particularly in the fluids transport mechanisms known as peristalsis. The Casson fluid belongs to a group of non-Newtonian fluids that, according to a particular stress threshold, exhibit elastic solid behavior before changing to liquid behavior. These fluids are known as viscoelastic fluids and have several uses in engineering, food preparation, drilling and other fields. The Casson nanofluid model is used in this investigation. In order to better understand this, this study examines the peristaltic motion of a Casson nanofluid in a vertical layer with suction/injection. The resulting flow model is successfully simulated under the realistic assumptions of long wavelength and low Reynolds number after obtaining the governing conservation equations. By using workable transformations, the derived partial differential equations are mathematically converted into a dimensionless form. Analytical solutions have been found for the resulting flow problem’s temperature distribution, velocity, pressure rate per wavelength and concentration of nanoparticles. Using Wolfram Mathematica software, the impacts of all physical characteristics on temperature, velocity, concentration fields, pressure rate, frictional force and pressure gradient are graphically studied. The influences of thermophoresis parameter Nt raise the temperature and diminish fluid concentration. By raising the suction and injection parameter k values, the velocity in the directions of x and y is decreased. The pressure rate enhances by raising the Reynolds number and diminishes by enhancing the Grashof number.

This paper presents an investigation of magnetohydrodynamics (MHD) Casson nanofluid flow along a stretchable surface through a permeable medium. The modeling of the physical phenomena is considered with impact of thermal radiation, heat generation, slip conditions and suction. Transformations of the governing set of mathematical equations for the physical model are carried out into nonlinear ordinary differential equations (ODEs) with appropriate similarity variables. The nonlinear ODE solutions are carried out using the optimal homotopy analysis technique (OHAM), and the findings are presented for determining the influences of the emerging important parameters. The results indicate that velocity field increases in respect of porosity parameter, Casson fluid parameter and magnetic parameter while it declines for enhancing velocity slip and suction parameters. The temperature profile shows rising behavior for heat source, Prandtl number, thermophoresis, radiation and Brownian motion parameters while it declines for enhancing thermal slip parameter. Moreover, the concentration profile enhances for rise in Brownian motion parameter while it reduces for Schmidt number and nanoparticle parameter. We also showed the accuracy of the present results by indicating that skin friction values for varied magnetic parameters agree with earlier findings in the literature.

This research examines the analytical investigations for radiating Casson nanofluid models with unsteady convecting flow, which are valued by the order of Caputo fractional derivative (CFD). Ethylene glycol (EG) is used as a base fluid, and graphene oxide (GO) and carbon nanotubes (CNTs) are used as nanoparticles. The problem’s leading PDEs are nondimensionalized by applying the proper nondimensional variables. The solutions to the dimensionless governing equations are found by using the Fourier sine and Laplace transformation techniques together. For an enormous study of the problem, graphical illustrations and tables are developed by using MATLAB software programming with the help of Euler inversion. We examine the impact on the fractional heat and momentum equation of the $\alpha$, Gr, Pr, $\varphi$, R, $\beta$, oscillations. Using the properties of the fluid, important discoveries were made that indicated a number of elements for a number of flow parameters as well as fractional parameters. The thermal profiles are increased for $R,\varphi$ decreased for $\alpha$ at $\tau =0.4$ and $\tau =1.4$. The velocity profiles are increased for R and Gr decreased for $\beta$ and $\varphi$ at $\tau =0.4$ and 1.4. Different shapes of nanoparticles are performed for ordinary fractional parameters, which are increased for temperature as well as velocity.

The transport phenomena of Casson nanofluid flow between two parallel disks subject to convective boundary conditions are analyzed in this paper. The mathematical model incorporates the impact of thermophoresis and Brownian motion since the Buongiorno’s nanoliquid model is adopted to characterize the nanoliquid’s transport features. The appropriate similarity transformations are applied to obtain the resulting nondimensional ordinary differential equations from the basic governing equations. The resulting ordinary differential equations and the associated boundary conditions are solved analytically by adopting the homotopy perturbation technique. Further, a statistical experiment is conducted to identify notable flow parameters which cause significant impact on the heat transfer rate. The characteristics of critical pertinent parameters on the flow field are graphically manifested. It is worth noting that the Casson nanofluid velocity escalates by augmenting the magnetic field parameter in the case of injection near the disks. Nanoparticle concentration is considerably diminished with an increment in thermophoresis parameter. In the cases of equal and unequal Biot numbers, the heat transfer rate is promoted with higher values of the Brownian motion parameter. Among the Casson fluid parameter, squeezing parameter and magnetic field parameter, the heat transfer rate discloses the highest positive sensitivity with the lowest value of the Casson fluid parameter.

This paper examines the effect of rotation on thermal instability under Hele-Shaw cell saturated by Casson nanofluid using both linear and nonlinear ways. The nanofluid model incorporates Brownian and thermophoresis diffusion. While conducting an analysis of nonlinear stability numerically using the truncated Fourier series method, analysis of linear stability is performed analytically using the normal mode methodology. The outcomes are all displayed graphically. The results demonstrate that the rotation has dual effect on Hele-Shaw parameter as well as Casson parameter, for higher value of rotation it has destabilizing effect and it stabilizing  the system for lower values of rotation. Lewis number and concentration Rayleigh number promote the onset of convective motion within the system. On the other hand, rotation stabilize the system. Understanding the behavior of heat and mass transportation, the concentration of nanoparticles and fluid phase, utilize the Nusselt number when Nusselt numbers are assessed as a function of time, it is found that the variation of the rotation, Hele-Shaw and Casson parameter has a major influence on the heat and mass transfer. Both steady and unsteady weakly nonlinear analyses are performed to understand the heat transport in the system. It is concluded that the Casson nanofluid parameter has both stabilizing and destabilizing impact depending upon the rate of rotation and therefore this work can be possibly utilized in both places, where heat removal and heat conservation are required.

A computational mechanism has been formulated to scientifically analyze the phenomenon of combined convection in the occurrence of magnetohydrodynamic flow behavior of Casson nanofluid. This study emphasizes the trajectory of the nanofluid along a stretching sheet with nonlinear permeability, while considering additional physical effects such as reversible chemical reaction, thermal radiation, energy source/sink, suction and viscous dissipation. In this study, we have additionally integrated the nanofluid paradigm proposed by Buongiorno, which encompasses the repercussion of thermophoresis along with Brownian motion. The utilization of appropriate similarity renovations serves the purpose of recasting the dominant multivariate differential equations to a collection of nonlinear differential equations in single variable. The Runge–Kutta shooting mechanism is implemented for the intention of numerically determining the unknown boundary conditions. The solution to the dominant equations was obtained by employing the fourth-order Runge–Kutta technique and to ensure the dependability of the outcomes, the bvp4c method was employed. The graphical and tabulated observations are presented herein to facilitate a comprehensive analysis of the underlying physical characteristics inherent in the problem at hand. The utilization of the Casson parameter has been found to be advantageous in the reduction of shear stress rate, along with the enhancement of mass and energy transfer rates. Additionally, the application of suction has been observed to be beneficial in the enhancement of Sherwood number and energy transfer rate. Within the scope of the context of the esterification process, the purpose of this proposal is to evaluate the impact that numerous arithmetical values exert on the temperature field, the velocity profile and the volumetric concentration. An in-depth graphical analysis that takes into account the relevant physical consequences is used to evaluate the most important factors that relate to the physical quantities that describe the contours of temperature, velocity and concentration. In addition to being accompanied by their respective explanations, the tabular portrayal of the local Sherwood number, shear stress rate along with local Nusselt number all feature in this paper. There is a clear distinction that can be made between irreversible and reversible flows in the evaluation of the local Sherwood number, rate of shear stress and local Nusselt number. This differentiation is brought about by taking into analysis thermophoresis parameter, Brownian motion parameter and suction parameter. The results that were produced from the theoretical simulations have significant repercussions for a variety of fields that are related to energy engineering. The findings derived from the analysis indicate a negative correlation between the Brownian motion parameter and both irreversible and reversible flow in terms of rate of energy transfer and shear stress rate. It is imperative to highlight that reversible flow holds greater significance in comparison to irreversible flow.

This research captures nonlinear thermo-solutal buoyancy (i.e., nonlinear mixed convection) impact in nanofluid flow based on magnetized Casson model. The generalized porosity concept (i.e., Darcy–Forchheimer relationship) is employed by considering incompressible liquid that saturates the porous space. Effects of thermophoresis, Robin conditions, thermo-solutal stratifications and Brownian diffusion are accounted. Consideration of transpiration phenomenon captures suction/injection aspects. Fluid mechanics basic laws are depleted to simplify the governing rheological expressions. A transformation procedure is then employed to convert the nonlinear governing partial systems into differential systems. Homotopy methodology is used to obtain analytical solutions and convergence is ensured. Graphical and tabular outcomes are presented to address the importance of emerging variables.

This work is motivated by the importance of vertical vibration in nanofluids suspension, because of its wide applications in heat exchangers from one fluid to another, controlling the mixing in microvolumes, pharmaceutical industry, and engineering. The current study analyzes the effects of high-frequency vertical vibration on a horizontal Casson nanofluid layer heated from below saturated between rigid-rigid, rigid-free, and free-free boundary conditions. Field equations are derived by using time-average method, which describes the vibrational thermo-convection. These field equations are solved by utilizing linear stability analysis and Galerkin technique to obtain the thermal stability curve. By using this curve, the impacts of vertical vibration and physical governing factors on temperature profile are discussed. High-frequency vertical vibration is observed to have a stabilizing effect on Casson nanofluid suspension, thereby reducing thermo-convective heat transfer from one fluid to another. Stability threshold of vibrated suspension is increased approximately as $5.748\%$ and $81.937\%$ when $R_v\to 50$ and $R_v\to 100$, respectively. The significance of Brownian motion and thermophoretic force on convective heat transfer with vertical vibration is discussed. The resisting nature of Casson nanofluid on vibrational convective heat transfer is also tabulated.

This study addresses the applications of non-uniform heat source/sink and viscous dissipation in the magnetohydrodynamic (MHD) flow of Casson nanoparticles toward a porous stretchable sheet. The free convective flow for the inclusion of thermal buoyancy along with dissipative heat encourages the flow phenomena. The governing equations of the flow model are presented in terms of partial differential equations and then transformed into a system of ordinary differential equations using similarity transformations. These systems of equations are solved numerically by using the Runge–Kutta fourth-order method with a very efficient shooting technique. The effects of various parameters such as the Casson parameter, elastic parameter, porosity parameter, Prandtl number, non-uniform heat source/sink constant, Eckert number, skin fraction and Nusselt number on the flow area are computed and represented graphically. The present results reflect that the velocity of nanoparticles declined effectively with the porosity parameter and nanoparticle volume fraction. The temperature profile is increased with the elastic parameter and heat source parameter while decreasing with the Eckert number and Casson fluid parameter. Moreover, it is observed that when the Hartmann number is maximum, a retardation in wall shear force against nanoparticles volume fraction is marked.

This research intends to investigate the entropy generation on the magnetized double diffusive heat and mass transfer flow of the Casson nanofluid under the influence of an inclined magnetic field in a porous medium. Additionally, the combined impact of heat absorption, chemical reaction, Brownian diffusion, source/sink, and thermophoresis phenomena is also taken care of. The fluid flow involves convective boundary conditions for both temperature and concentration instead of a constant value at the surface. The flow-regulating system involved nonlinear PDEs that are turned into nonlinear systems of ODEs by using scaling variables and then solved this system numerically in Matlab using the bvp4c strategy, which is a collocation technique based on the Lobatto 3-stage FDM algorithm. Graphical representations illustrate the behavior of fluid velocity, entropy generation, concentration, and temperature in response to changes in flow parameters. Physical quantities like skin friction coefficient, Nusselt number, and Sherwood number have been investigated using 2D and 3D plots. Here, we concluded that the inclined magnetic field decimates the flow velocity gradually and greater values of the magnetic field lead to an increased rate of entropy generation. Furthermore, it has been noted that the temperature profile improves as the Brownian motion of particles increases, and the distribution of energy also enhances with larger values of the thermophoresis. The obtained key findings are discussed in a physical manner using graphic representation.

This paper presents a two-dimensional unsteady laminar boundary layer mixed convection flow heat and mass transfer along a vertical plate filled with Casson nanofluid located in a porous quiescent medium that contains both nanoparticles and gyrotactic microorganisms. This permeable vertical plate is assumed to be moving in the same direction as the free stream velocity. The flow is subject to a variable heat flux, a zero nanoparticle flux and a constant density of motile microorganisms on the surface. The free stream velocity is time-dependent resulting in a non-similar solution. The transport equations are solved using the bivariate spectral quasilinearization method. A grid independence test for the validity of the result is given. The significance of the inclusion of motile microorganisms to heat transfer processes is discussed. We show, inter alia, that introducing motile microorganisms into the flow reduces the skin friction coefficient and that the random motion of the nanoparticles improves the rate of transfer of the motile microorganisms.

Bio-convection is an important phenomenon which is described by hydrodynamic instability and pattern in suspension of biased swimming microorganisms. This hydrodynamics instability arises due to the coupling force between the motion of the micoorganisms and fluid flow. It becomes more significant when nanoparticles are immersed in the base fluid with non-Newtonian rheology. This study presents the bio-convection for a viscoelastic Casson nanofluid flow over a stretching sheet. The Cattaneo–Christov double diffusion, induced magnetic field, thermal radiation, heat generation, viscous dissipation and chemical reaction are taken into account. The boundary condition is enriched with the suction / injection and melting phenomena at the surface. Highly coupled nonlinear governing equations are simplified into a system of coupled ordinary differential equation by using proper similarity transformation. The spectral quasi-linearization method (SQLM) is used to solve the transformed governing equations numerically. Good agreement is observed with the numerical data investigated in the previous outstanding works. It is observed that the density of the motile microorganisms depends on Peclet number and bio-convective Lewis number. Bio-convection Rayleigh number increases the possibility of bio-convection in the system which results in the enhancement of temperature. It is also examined that temperature and concentration profiles increase with the Eckert number and thermophoresis parameter.

This research work presents the numerical solution of the Darcy–Forchheimer flow of Casson nanofluidic model (DFFCNFM) on stretching sheets with Newtonian heating by utilizing a combination of nonlinear input–output neural networks with backpropagation of Levenberg–Marquardt computational approach. The presented study investigates the impact of electroosmosis forces on the boundary layer of the Casson nanofluid, focusing on viscous and Joule dissipations. A dataset for DFFCNFM is generated for the different events with backward differentiation formula (BDF) by varying Casson fluid parameter ($\beta )$, permeability parameter (Da), electric parameter ($E_1)$, Reynolds number (Re) relative to stretching velocity, magnetic field ($M)$, Eckert number (Ec), and Prandtl number (Pr). The artificial intelligence-inspired technique via nonlinear input–output neural networks with backpropagation of Levenberg–Marquardt is utilized from the generated dataset for DFFCNFM to find the approximate solutions. The satisfactory performance levels, as indicated by the mean square error (MSE), have been attained consistently with magnitude around 10${}^{-12}$–10${}^{-14}$ for all scenarios of DFFCNFM. The precision and performance validation is effectively established by the negligible MSE, close proximity to the unit value of regression metric, and the distribution of instances in error-histograms.