Unsteadiness Parameter Research Articles

Numerical study for bio-convection effects on MHD nano-fluid flow past a porous and extending wedge

We explored the insinuations of bio-convection and thermal radiation on nanofluid transportation across stretching permeable wedge with magnetic force. Appropriate similarity transformation variables are utilized to achieve ordinary differential equations. In order to tackle the non-linearity of these equations, numerical procedure based on shooting technique and Range Kutta method are harnessed on MATLAB platform. Computational and devour is carried out to evaluate the influence of controlling limitations on temperature, velocity, concentration of nanofluids and micro-organisms density. The growing strength of thermophoresis and Brownian motion enhance the fluid temperature. The profile volume fraction show decline against higher values of parameters which are Lewis number, unsteadiness and Brownian motion but opposite trend noted against higher value of Williamson and thermophoresis parameters. The skin friction values rise with the growing values of parameter of wedge angle for the moving wedge. The motile organism profile exhibits decrease against growing strength of Peclet number, bioconvection Lewis number, temperature difference and unsteady parameters while opposite behavior has been noted against wedge angle parameter.

Dual solutions of unsteady flow of copper-alumina/water based hybrid nanofluid with acute magnetic force and slip condition

Suspending particles of tiny solid in a fluid used to transport energy can enhance its thermal conductivity and heat transport properties. Our main goal of this examination is to study the radiative unsteady two-dimensional (2D) flow on a continuously diminishing, horizontal sheet. with suction for the hybrid water-based nanofluid and an aligned field of magnetic, including the combined suction, magnetic, and velocity slip conditions effect. The Tiwari & Das model of nanofluid equations is used, which takes into consideration the solid volume percentage. Equations of similarity are derived by employing the transformations of similarity, and the associated equations have been simplified numerically by employing the bvp4c method in MATLAB software for a variety of values of the nanoparticle volume fraction, the unsteadiness, and the wall mass suction in water. It is discovered that, within the given the unsteadiness parameter range, two solutions exist. Moreover, it is found that the fluid velocity slows down in 1st solution as volume fraction of copper nanoparticles rises but speeds up in the second solution at first before slowing down again. Using a temporal stability analysis, it is found that only one of the dual branches is stable over the long run, while the other is unstable.

Numerical analysis of the performance of a composite marine propeller blade subject to structural blade oscillations

High-fidelity Large Eddy Simulations (LES) are conducted over the oscillating marine propeller blades to investigate the effects of blade oscillation on the unsteady flow behaviour as well as to analyse the performance of the composite marine propeller blades. The accuracy of the results is first verified against the reference experimental results. A nonlinear frequency-domain solution or harmonic solution method is also applied to this study, based on a high-fidelity LES model, in addition to the traditional time-domain solution method to investigate the capabilities of the frequency-domain method in predicting the unsteady flow parameters, vortex generation, and hydrodynamic damping to determine the stability of the propeller blades. Results obtained reveal that the oscillation of the blade has a great impact on the vortex generation process which could impose implications on the performance and structural integrity of the composite marine propeller blades.

Inverse multiquadric kernel‐based neuro heuristic approach to analyze the unsteady MHD nanofluid flow via permeable elongating surface

AbstractIn this study, a novel neuro heuristic approach is designed to investigate the flow properties of magnetohydrodynamic (MHD) nanofluid along an exponentially extending sheet with a permeable medium with the impact of radiation as well as fluctuating heat source/sink. The designed scheme to handle the suggested problem is established through the well‐known biologically inspired neural networks (BINNs) by exploiting the inverse multiquadric kernel (IMQK), that is, BINNs‐IMQK which is quite a new approach. The partial differential equations (PDEs) which govern the fluidic flow are reformed into a nonlinear system of ordinary differential equations (ODEs) using the most fitted similarity transformations rules and numerically solved by varying the parametric values including unsteady parameter, Brownian motion parameter, suction/injection parameter, radiation parameter, Schmidt number together with Prandtl number to visualize the velocity, thermal gradient, and mass transfer in the suggested fluid problem. It is noticed that nanofluid temperature hikes by uplifting the value of the Brownian motion parameter but this effect is reversed in case of unsteady parameter. The obtained numerical results are verified through reference solution using the well‐known Adams method and the efficacy of the suggested solver is endorsed using a variety of statistical operators.

Multiple solutions of unsteady flow of CNTs nanofluid over permeable shrinking surface with effects of dissipation and slip conditions

The objective of this paper is to analyze the unsteady incompressible flow of the viscous nanofluid on a contracting surface with viscous dissipation effects. Presented and contrasted are analyses of both multi-wall carbon nanotubes (MWNTs) and single-wall carbon nanotubes (SWNTs). As the common (or base) fluids, kerosene oil and water are utilized. In the existence of first-order thermal and velocity slip conditions, mathematical modeling and analysis are performed. Using the MATLAB software’s bvp4c solver tool, numerical solutions to the governing nonlinear modeled problems were obtained. This technique is particularly effective for developing many solutions to highly nonlinear differential equations. In addition, a comparison is done between this study and previously published works. The temperature, velocity, skin friction coefficient and heat-transfer rate have been explored for various significant factors included in the problem statements. In the unsteadiness parameter regime, dual solutions can be found. As the velocity slip parameter is increased, the flow slows down. In comparison to SWCNTs kerosene, MWCNTs kerosene oil has a greater velocity curve for the nanoparticles volume fraction. Increases in volume fraction decrease skin friction, whereas increases in the unsteadiness parameter speed up the drag force. Furthermore, as the Eckert number intensity increases, so do the temperature profiles in both solutions. Finally, the stability study revealed that the initial solution is robust, whereas the breakage in the second solution in the Nusselt number shows singularity, and thus the second solution is considered unstable.

Entropy production in unsteady MHD double diffusive and chemically reactive nanofluid flow past a rotating permeable plate using overlapping grid spectral collocation technique

The entropy production phenomenon due to the applied external magnetic fields, thermal dissipation, chemical reaction and the impact of rotating pervious plate on heat transfer, nanofluid friction and mass transfer has been numerically explored. The nanofluid flow is presumed to be unsteady, smooth, hydro-dynamically and electrically conducting. The well known Navier–Stokes time-dependent equations in cartesian coordinates for velocity, temperature and concentration are modelled and scaled down to a system of ODEs by applying suitable similarity transformations with unsteady dimensionless parameters. A new numerical scheme known as the overlapping grids spectral collocation method (OGSCM) is applied in finding the approximate solutions to the highly nonlinear ODEs. The results obtained are then used for solving the entropy equation and the rate of irreversibility is represented graphically. Several engineering variables, such as the coefficient of heat transfer, Nusselt number and Sherwood number, are also thoroughly examined. The OGSCM proved to be accurate, computationally efficient, time efficient and converges fast. We discovered that increased applied magnetic field, chemical reaction parameter, rotation parameter, and Schmidt parameter insert more energy into the system which reduces the thermal boundary layer and hence reducing the rate of entropy generation. Furthermore, the entropy generation rate is increased by increasing the Grashof number, Brinkman number, permeability, Prandtl number, and thermal radiation parameter.

Melting heat transfer analysis in magnetized bioconvection flow of sutterby nanoliquid conveying gyrotactic microorganisms

In biotechnology and biosensors bioconvection along with microorganisms play a important role. This article communicates a theoretic numerical analysis concerning the bioconvective Sutterby nanofluid flow over a stretchable wedge surface. Bioconvection is a remarkable occurrence of undercurrents fluid that is produced owing to the turning of microbes. It is considered for hydrodynamics unsteadiness and forms classified in interruption of inclined swimming microbes. Bioconvection is perceived practically in many uses for example pharmaceutical products, bio sensing applications, biomedical, bio-micro systems, biotechnology advancements and refining of mathematical models. Additionally, unsteady parameter influences are taken into account. Furthermore, no mass flux as well as heat sink/source consequences are measured in existing analysis. The similarity transformation are established for the non-linear PDEs of microorganism's field, nanofluid concentration, energy, momentum and mass for bioconvection flow of Sutterby nanofluid. Then, altered non-linear ODEs are resolved by utilizing the bvp4c technique. Moreover, nanofluids are declining in thermal and concentration fields and the greater number of Peclet number declines the field of microorganisms. Acquired numerical data displays that temperature field of nanofluid increases for more thermophoretic and unsteady parameters.

Effects of Flow Unsteadiness on the Transport of Bimodal Bed Material

The grain size distribution of the transported bed load was experimentally investigated under unsteady flow conditions with bimodal mixture of sand and gravel in a laboratory flume. Five various triangular hydrographs were generated. A clockwise behavior for the total bed load versus shear velocity was observed meaning that the bed load during rising limb was higher than that of falling limb. It was found that the percent finer at the plateau of bimodal sediment size distribution curve had higher values during the initial and final phases compared to those obtained during the peak time. At all plateaus, the percent finer values related to the hydrograph peak discharge were in the same order of magnitude with that of the bed material. The sand content of the transported bed material initially decreased, then maintained a constant value during a certain time interval and finally returned to its original value. The sand percent of the bed load decreased in the falling limb showing a counterclockwise loop and the duration of the hydrograph did not affect the results considerably. The greater the peak flow rate of the hydrograph, the greater was the hysteresis. The bimodality index was calculated for all transported sediment samples and it was revealed that its initial and final values were less than that of the bed material but it was approximately the same elsewhere. The 5% finer sediment amount was nearly equal during rising and falling limbs. It was revealed that D50 value of the bed load decreased in the rising limb showing a clockwise loop. The hysteresis was not considerably changed according to the hydrograph characteristics. The clockwise type hysteresis was also observed for the size group of D95. The lag increased as the peak flow rate increased. A strong relation was found between the dimensionless total bed load Wt* and the total work index Wk as well as Wk and the ratio WR/WF. The correlations between the dimensionless total bed load and the unsteadiness parameters P, and Pmod were very weak, whereas a high value of determination coefficient was obtained with the unsteadiness parameter Pgt, implying an appreciable interdependence.

Flow across moving plate at separated stagnation point: Features of corcione’s correlation with Thompson and Troian slip and melting heat

The dynamic features of nanofluids replicate multiple implications in the processes of cooling, environmental architecture, heat pumps, electromagnetic flow cytometry, energy generation, hyperthermia, and other related processes. The goal of this study is to investigate the Corcione correlation, which highlights the importance of fluid temperature, particle volume fraction, and particle size in increasing the nanofluid’s heat transfer rate and thermal conductivity considering the promising applications of nanofluids. In this case, we ponder about the flow of a nanofluid made up of and through a sliding plate, where the fluid experiences unstable separated stagnation point flow coupled with melting heat transfer. Nanoparticles with diameters of 28, 30, and 45 nm and concentrations of up to 4% by volume are all accounted for in the study. The Thomson and Troian slip conditions improve the mathematical model. In this work, we propose a novel mathematical model of a nanofluid and provide analogous solutions in the manner of ODEs. Estimated solutions to reduced ODEs are calculated using MATLAB's bvp4c method. Several graphics illustrate the impact of many elements. Over a given volume percentage, the Nusselt number and drag force coefficient are calculated for a range of nanoparticle sizes, and it is shown that both increase with increasing nanoparticle diameter (). Based on the findings, the skin friction coefficient and heat transfer rate both increase when stagnation strength parameters are raised. The thermal conductivity was also enhanced by the inclusion of the suction constraint. Curiously, improving heat-transfer performance is proportional to decreasing the unsteadiness parameter. The manufacturing and operations systems are two areas where this heavy engagement is particularly evident in furthering industrial progress. We may also infer that the incorporation of nanoparticles into the working fluid enhances heat transfer and that surface slip boosts draining control.

Unsteady MHD Casson Fluid Through a Porous Medium over an Inclined Plate in the Present of Chemical Reaction

The unsteady of MHD Casson fluid flow through an infinite inclined plate in the presence of chemical reaction is investigated. In this research, Casson fluid model is choose to be used in this study to characterize the non-Newtonian fluid behaviour. We are employed the Laplace transform technique with an appropriate boundary condition to convert the governing partial differential equations into ordinary differential equations. All the transformed equations are then solved numerically by using Mathematica. Finally, we were obtained the exact solution of momentum, energy and concentration of the cases. The results of flow features for different values of the governing parameters, unsteadiness parameter, Casson parameter are analysed in graphs and has been discussed in the result section. The results presented that fluid velocity rises with the increment of magnetic parameter due to the present of Lorentz force. Lorentz force helps in reducing heat for electronic system and radiators. Furthermore, the increasing of inclination angle and chemical reaction make the velocity increases. The result for Casson fluid explained that this fluid helps to lower the resistance of the yield stress so that the velocity become higher.

Control of magnetic dissipation and radiation on an unsteady stagnation point nanofluid flow: A numerical approach

This study investigates two-dimensional time-dependent stagnation point nanofluid flow over a permeable stretching or sinking sheet. The electrically conducting fluid allows for the interaction of a transverse magnetic field along with magnetic dissipation and thermal radiation to enrich the flow phenomena. The use of nanofluids is becoming increasingly important in a variety of industrial applications, as well as in engineering and biomedicine. The process of peristaltic pumping, the flow of blood in the artery, the drug delivery process, and besides that, the cooling of electronic devices for the better shape of the product at the final stage of production, the use of nanofluid, are all important. The transformed dimensionless governing equations are obtained by the implementation of various transformation rules, and the model is designed by using different thermophysical properties such as viscosity, conductivity, etc. Finally, the numerical technique RK-fourth-order method is deployed to solve the set of equations and the iteration process is continued up to the desired accuracy of 10[Formula: see text]. The validation of the current result is shown with the earlier investigation in particular cases, and further, the behavior of the physical parameters is presented through graphs. It is observed that the particle concentration has a greater impact in enhancing the velocity distribution for the variation of suction/injection. The behavior of the magnetic parameter vis-à-vis the unsteadiness parameter and the thermal buoyancy presents their greater contribution in enhancing the magnitude of nanofluid velocity.

Unsteady Prandtl nanofluidic flow through stretching sheet due to mixed convection, first-order slip, and chemical reaction

This article explores the unsteady flow, heat, and mass transport of a Prandtl nanofluid flow over a stretching sheet due to the existence of mixed convection, first-order slip, and chemical reaction. The flow characteristics are analysed for the interaction of thermal and solutal Biot numbers. The system of coupled non-linear leading partial differential equations (PDEs) is converted into non-linear ordinary differential equations (ODEs) with the facilitation of given similarity transformations. Furthermore, these equations are solved by applying a semi-analytical method termed the “Optimal Homotopy Analysis Method” (OHAM) using the computational software MATHEMATICA. The tables and Figures present the numerical computations of the pertinent parameters of the flow problems. The essential findings of this study are for the growing values of the Prandtl parameter, the velocity profile is enhanced, and the temperature profiles show the opposite behaviour. Unsteady parameters reduced the thickness of the heat and moving boundary layers. The temperature profiles are enhanced as the thermal Biot number grows, and similar characteristics can be observed in the solutal Biot number on concentration profiles.

Combined effects of Joule heating and binary chemical reaction of MHD Williamson nanofluid on Darcy–Forchheimer porous medium past unsteady stretching cylinder

In this study, the convective transport phenomena of Williamson nanofluid flow with variable thermal conductivity on Darcy–Forchheimer porous medium past unsteady stretching cylindrical sheet is presented. The conservation laws of mass, momentum, energy and concentration are used to formulate the governing boundary value problem of the flow regime. The flow is expressed with a couple of nonlinear partial differential equations. An appropriate similarity transformation and techniques are used to transform the couple of partial differential equations into a system of initial value problems. The system of initial value problems is then solved numerically using the Runge–Kutta fourth order with the shooting technique. The Python programming tool is used to carry out the computation. The influence of unsteady parameter, thermal and concentration Biot numbers, thermal conductivity parameter and other parameters on velocity, temperature and concentration profiles of the nanofluid are examined. The responses of the three transfer rates (i.e momentum, mass and heat) to many pertinent parameters are also investigated. Most flows in industries are in a controlled manner hence the cylindrical geometry is used to study the flow regime. The results of the study show that the convective mass transfer is initiated with a rise in thermal and concentration Biot numbers whereas the convective heat transfer is motivated and demotivated with a rise in thermal and concentration Biot numbers, respectively. Larger values of the unsteady parameter on the other hand creates substantial wall friction to the motion of the nanofluid. Further, the numerical method used is found in excellent agreement with some earlier works under common assumptions.

Effect of Magnetic field on Unsteady Mixed convection Micropolar Nanofluid flow in the Presence of Non-Uniform Heat Source/Sink.

: The idea of the current investigation is to study the two-dimensional unsteady mixed convection MHD flow of a micro-polar nanofluid with non-uniform heat source/sink. Effects of radiation, reaction, thermo-diffusion, and viscous dissipation are considerednanoparticles are suspended in water base fluid. The governing non-linear system of PDEs is transformed into a system of ODEs using the similarity transformation and HAM is employed for obtaining solutions. Effects of different physical parameter on momentum, microrotation, energy, and concentration profiles are discussed through graphs. From graphical results, it is concluded that the unsteady parameter and magnetic parameter tend to improve the motion of the fluid. It is also seen that microrotation is raised in the shrinking sheet and reduced in the stretching sheet with the increase in the unsteady parameter. The mass transfer process improved with thermal diffusion parameter while delayed with chemical reaction.

Numerical Analysis of Magnetohydrodynamic and Dissipated Hybrid Casson Nanofluid Flow Over an Unsteady Stretchable Rotating Disk with Cattaneo-Christov Heat Flux Model

The study scrutinized MHD and dissipated (SWCNTs-Fe3O4)/C2H6O2 hybrid Casson nanofluids flow over an unsteady stretchable rotating disk with a Cattaneo-Christov heat flux model. By means of proper similarity conversion, the boundary layer flow governing PDEs was changed into systems of dimensionless coupled nonlinear ordinary differential equations. Subsequently, the consequent nonlinear momentum and energy equations with their boundary conditions were worked out numerically employing the spectral quasilinearization method (SQLM). The convergence, stability, and accuracy of the SQLM were established as a computationally efficient method to solve a coupled system of boundary layer problems. It is specified that 5% of SWCNTs, 20% of Fe3O4, and 75% of C2H6O2 being taken for the preparation of (SWCNTs−Fe3O4)/C2H6O2 hybrid nanofluid with shape factor n1 = n2 = 3, and the values of the parameters used are fixed to M = 5, S = 0.5, β = 5, κ = 0.5, Ec = 2, Λ = 2, Pr = 7.3, α = 0.5, δ = 0. The effects of more perceptible parameters on velocity and thermal flow fields were considered and scrutinized carefully via graphs and tables. The results disclose that the momentum and thermal boundary layer thickness markedly declined with more value of the unsteady parameter. The local heat transfer rate improves nearly by 14% as 0.2 volume of Fe3O4 nanoparticles dispersed in 0.05 volume of SWCNTs and 0.75 volume of C2H6O2 nanofluid, hence, in realistic uses adding more values of nanoparticles in the hybrid nanofluids is useful to progress the heating process. The study is novel since to the best of the author’s knowledge, no paper has been published so far on the unsteady flow of (SWNT-Fe3O4)-Ethylene glycol hybrid Casson nanofluid with the effects of the Cattaneo-Christov heat flux model. As well, the model used for the thermophysical properties of the hybrid nanofluid is a new approach. Generally, hybrid nanofluids of (SWCNTs-Fe3O4)/C2H6O2 show better flow distributions with good stability of thermal properties than their mono counterparts.

Stability Analysis of Mass Transfer Effects on the Unsteady Stagnation-Point Flow Past a Permeable Sheet

This study involves the problem of mass transfer effects on unsteady stagnation-point flow through a permeable stretching/shrinking sheet. Similarity transformations in the form of nonlinear partial differential equations to ordinary differential equations were used to reduce the basic governing equations. The resulting equations were then solved numerically using the bvp4c function and shooting method. The programming codes for bvp4c function were built using Matlab software, whereas the programming codes for shooting method were built using Maple software. The effects of the Schmidt number, stretching/shrinking, suction/injection, unsteadiness and the reaction rate parameter on the flow and mass transfer characteristics were numerically analysed. The boundary layer thickness and velocity gradient are also influenced by these parameters. Dual solutions were figured out for the problems involving stretching/shrinking sheet at the stagnation-point. To determine the stability of the solutions obtained, stability analysis was performed by solving the linear eigenvalue problems using the bvp4c function. The analysis showed that for the first solution, there was an initial decay of disturbance, while the second and third solutions were opposite, where an initial growth of disturbance were detected for both. This result indicated that the first solution was more stable and physically realizable compared to others.

Chemically reactive and thin film flow analysis of cross nano-liquid over a moving surface

Vapour generalized liquids hydrodynamics plays a pivotal role in the absorption columns or design of distillation using structured packing since they indomitable the fluid dynamics restrictions of the columns and control rates of mass balance which is the main purpose of this particular work. Particularly, a first attempt is made regarding this work to explore an investigation for time-dependent thin film flow of Cross nano-liquid over a moving surface. Additionally, some physical aspects during this illustration in terms of heat absorption/generation, thermal radiation, magnetic field, non-constant thermal conductivity, and activation energy with chemical reaction are studied. Some of the suitable similarity variables are introduced to convert the leading problem in the form of ordinary differential equations and then the transformed problem is approximated by one of the built-in collocation methods by using MATLAB. Very interesting results in terms of monotonic reduction in the film thickness for various uplifting values of unsteadiness and magnetic parameters and vice versa for the Wessenberg number are found. Moreover, temperature and concentration are concluded as enhancement functions of variable thermal conductivity and activation energy, respectively. The last comparative work is presented to validate the entire approach and where the method shows an excellent correlation with the previously published results.

Computational investigation of magnetized hybrid nanofluids heat transport and flow through elongational surface with thermal radiation and wall slip

The improved thermal performance of recently discovered hybridized nanofluids has become essential in large scale thermal processes. In fact, this is highly efficient technique to introduce the thermal efficiency of tranditional heat transferring fluids. The behavior of the nanofluid can be significantly impacted by the unsteady heating and magnetic field effects that may be present in many applications. Therefore, the current study investigat the unsteady magnetized flow of hybrid nanofluid with heat transport characteristics subject to thermal radiation and slip at the surface wall. The shrinking/stretching surface is chosen as a flow source, which is frequently occure in polymer technology, which deals with the deformability of elastic sheets, and in metallurgy, where continued strips are cooled. The novel form of shrinking surface flow is fundamentally a reverse flow and exhibits physical characteristics that differ significantly from the channel flow scenario. The distinctive features of this scruinity is the use of empirical relations to approximate the optimum thermophysical attributes of a Cu−Al2O3/ water hybrid nanofluid in order to model the 2-dimensional flow past a flat shrinking/stretching sheet under the action of radiation, Lorentz forces and realastic boundary condition responses. The governing system of modelled equation are assembled using the Tiwari-Das model in conjunction with a hybrid mass-based nanofluid model. The bvp4c algorithm is employed within the computer MATLAB programme. The hybrid nanofluid flow shows conclusive improvement in the frictional coefficient and heat transport performance. However, the effectiveness the unsteadiness parameter deteriorates the heat transmission. In the contiguity of a suction parameter, multiple outcomes appear to arise for both stretched and shrinking instances. The coefficient of energy transport improves as the magnetic factor is augmented, however the skin coefficient of friction exhibits dual behavior for the second solutions. A time-dependence investigation is undertaken to figure out the reliability of the twin solutions, and it is discovered that merely one of them remains stable and aesthetically credible.

Dual solution of melting heat transfer efficiency in radiative hybrid (Cu-[formula omitted]/water) nanofluid flow

Hybrid nanofluids have been proposed as a new class of nanofluids, whose thermal properties and potential utility have been identified to serve the purpose of enhancing heat transfer rates. The main objective of the present analysis is to study the emerging unsteady hybrid nanofluid flows through porous medium with the impacts of melting temperature and slip velocity on the surface. Additionally, the heat transportation is analyzed with the presence of, thermal radiation, Joule heating and heat source/sink over a stretched/shrunk surface. Here two distinct nanoparticles (Cu- Al2O3) considered to examine the flow field over the shrinking/stretching surface. The Navier-Stokes equations are used in the process of developing the mathematical model under the mentioned assumptions. By utilizing the suitable transformations, the modeled equations are converted into non-linear ordinary differential equations. The formulated mathematical differential system is solved numerically by bvp4c technique in MATLAB. Dual nature study is focused in which solid lines show the first solution and dash lines for the second solutions. The behavior of hybrid nanofluids in terms of velocity and temperature distribution have been visualized graphically. For better understanding of the flow features and heat transfer rate, variation in skin friction coefficients and the Nusselt number of hybrid nanofluids are closely examined. We find from the present study that the rate of heat transfer is enhanced in the hybrid nanofluids by the impact of melting temperature. Moreover, in the first solution, skin friction coefficient rises with the enlargement of magnetic field parameter while heat transfer rate reduces in this situation. Additionally, the velocity profile reduces by the influence of unsteadiness parameter in the first solution while rises in the second solution. The heat transfer process at the surface is improved when a stronger Melting temperature is taken into account.

Unsteady thin film radiative nanofluids flow over time-dependent inclined stretching surfaces with Lorentz force and heat generation

This study aims to explore the unsteady magnetohydrodynamic thin film nanofluids flow over an inclined stretching surface in the presence of thermal radiation. The stretching velocity, Lorentz force and heat generation impacts together with the distributions of the temperature and nanoparticles at the surface are considered time-dependent. During the problem formulation, the haphazard motion and thermal migration are not neglected. The solution methodology is based on the Runge–Kutta method and it is divided into two steps, namely, solution of the momentum equations to obtain the nanofluid film thickness [Formula: see text] corresponding to the unsteadiness parameter S and the second step is using the values of [Formula: see text] to solve the entire system of equations. The major findings revealed that the nanofluid film thickness [Formula: see text] is reduced as the unsteadiness and magnetic field parameters are growing. Also, the rate of the heat transfer is diminishing as either the radiation parameter or the inclination angle increases. Furthermore, the increase in the buoyancy parameter from 0 to 1.5 gives an enhancement in the velocity up to 0.63%.