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Official websites use. Share sensitive information only on official, secure websites. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. The current study investigates the MHD flow of nanofluid across an elongating surface while taking into account non-uniform heat flux. For this, we have considered the flow of a boundary layer over a stretched sheet containing water-based Al 2 O 3 nanoparticles. The convective boundary conditions for temperature have been invoked. The flow created by a surface that is exponentially expanding in the presence of a magnetic field and a non-uniform heat flux has been mathematically formulated by using laws of conservation. The effects of emerging parameters on the fluid velocity and temperature profiles have been further described by plotting graphs. An experimental design and a sensitivity analysis based on Response Surface Methodology RSM are used to examine the effects of various physical factors and the dependence of the response factors of interest on the change of the input parameter. To establish the model dependencies of the output response variables, which include the skin friction coefficient and the local Nusselt number, on the independent input parameters, which include the magnetic field parameter, the nanoparticle volume fraction, and the heat transfer Biot number, RSM is used. Further, it is concluded that the sensitivity of the SFC, as well as the LNN through heat transfer Biot number, is greater than that of nanoparticle volume fraction and magnetic field parameter. The SFC is sensitive to all combinations of the input parameters. At high levels of heat transfer Biot number, the LNN displays negative sensitivity via magnetic field parameters. Research in the field of fluid mechanics is of more practical importance than research in several other fields because the most commonly encountered substances in human life are fluids like air, water, plasma, mud, colloidal suspensions, blood, etc. The industrial use of these fluids is not possible without complete knowledge of their transport properties. Flows over a stretching surface are encountered in several industrial processes like extrusion of polymer sheets, coating, and colouring of continuously moving metal sheets, drawing of copper wires, thin film coating on photographic films, polyvinyl chloride and plastic sheet extrusion, etc. The study of flows caused by stretching surfaces is a hot topic in current research because of its daily life applications. Sakiadis 1 was the first to explore the boundary layer flow on a continuously moving surface with a viscous fluid. For 2D flow constrained by a linearly stretching sheet, Crane 2 obtains a congested type solution. This ground-breaking research for stretching the surface of two-dimensional flows has been examined in a variety of ways. Andersson 3 has studied the slip effects on a stretching surface. Ariel 4 extends the work of Crane 2 to the 3D case. The heat transfer study for electrically conducting fluid of second grade across a stretched surface was given by Liu 5. Ishak 6 considers the radiation effects while presenting the MHD flow of the boundary layer owing to an exponentially extending surface. Bachok et al. Mukhopadhyay 8 employed suction and injection to explore the effects of slip and thermal radiation on boundary layer flow across an exponentially stretched surface using suction and injection. Awais et al. Ali et al. Waini et al. Stefan blowing effects for second-grade fluid flow across a curved stretching surface are studied computationally by Gowda et al. Gowda et al. Prasannakumara 14 discusses the heat transfer phenomenon of the flow of Maxwell nanofluid across a stretching surface and presents the numerical simulations by considering the magnetic dipole effects. The nanofluids are a novel type of heat transfer fluid that contains both nanoparticles and fluid, i. Some recent inventions describe how microparticles can be used in heat transfer applications. The dissipation of nano-size fusible particles in traditional heat transfer fluids is known as nanofluids. The disadvantage of utilizing micro-sized particles up to m is that waste forms along the flow channel, causing path attrition. This form of fluid has piqued the curiosity of researchers investigating fluids from all around the world. It's found in a wide range of modern technological applications that help people live better lives. Nanofluids are also used in medical applications, such as the treatment of cancerous tumours with gold nanoparticles and the creation of miniature explosives to destroy tumors. Later, Buongiorno 16 investigated convective heat transmission in nanofluids and discovered that, in the absence of turbulent processes, Brownian diffusion and thermophoresis play a significant role. Nadeem and Lee 17 investigated the boundary layer flow on an exponentially elongating surface while accounting for the nanofluid. Mustafa et al. Bhattacharyya and Layek 19 demonstrated the nanofluid phenomenon across a porous stretched surface. Heat and mass fluxes for nanofluids across a stretched surface are presented by Ghosh and Mukhopadhyay Sulaiman et al. The heat transfer analysis of nanofluid across an exponentially diminishing surface with heat and mass fluxes was described by Ghosh and Mukhopadhyay A nanofluid phenomenon over an exponentially stretching surface has been presented by Ali et al. Oldroyd-B fluid flows across an exponentially extending surface for 3D. Recently, the entropy generation analysis of peristaltic flow for nanofluid has been investigated by Ali et al. In order to study the effects of a porous medium on the nanofluid flow on a rotating disk, Kumar et al. A discussion on the entropy generation phenomena for the Marangoni flow by considering the nanoparticles of aluminium oxide and copper has been presented by Li et al. Yusuf et al. Mahanthesh 31 , 32 studied the aggregation kinematics of nanoparticles as well as the flow and heat transfer analysis of nanomaterials with quadratic radiative heat flux. They use a finite volume technique for simulations. Sheikholeslami et al. They consider a two-phase model of nanofluid with a mixture of nanoparticles of oil and CuO. They conclude that by increasing the Reynolds number from to 20,, friction factor is reduced by Sheikholeslami 35 also presented the air conditioning unit that involves porous media, which consists of five-lobed cylinders containing paraffin and nano-sized particles of ZnO. Sheikholeslami 37 — 39 also discusses the numerical analysis of a solar system equipped with a novel turbulator and analyze the melting process of paraffin using a honeycomb-shaped heat storage device. Additionally, for the purpose of melting, they statistically analyze the solar energy storage within a double pipe. The study of the movements of electrically conducting fluids is known as magnetohydrodynamic MHD. Solitons, liquid crystals, seawater, and electrodes are just a few examples of this type of fluid. The term magnetohydrodynamic is made up of three words: magneto which refers to a magnetic field ; hydro which refers to a liquid ; and dynamic which refers to motion. According to Faraday's law of electromagnetic induction, when an electric field and a magnetic field move relative to one another, a potential is formed in the conductor, causing current to flow between the endpoints. This law is used to generate MHD electricity. Magnetic fields may cause currents to flow through a flowing electrically conductive fluid, polarizing the fluid and changing the magnetic field in the process. Alfven 40 used the term magnetohydrodynamics MHD to describe such a fluid. Many studies 41 — 46 have been conducted to better understand the transport mechanisms and novel applications of MHD flows, which have proven to be advantageous in a variety of industrial processes such as polymer extrusion, hot rolling, and stretching of plastic films, wire drawing, metal extrusion, glass-fiber, and metal spinning. Benos et al. Researchers have recently concentrated on using statistical tools to analyze the effects of various physical characteristics. To address the issues raised by earlier reviewers, it was decided to do a statistical analysis of the findings of the accessible publications in the literature. The statistical analysis would allow for the representation of observations on complete charts histograms and scatter diagrams , allowing for a more solid and mathematically reliable extraction of findings. All of the observations accessible in the literature are given the same amount of weight in this literature evaluation. In industrial and applied research, experimental design is crucial. When a combination of input parameter values is applied to an experimental unit, one or more responses are measured over the experimental unit. Different studies using RSM have been done; a few of them can be seen in 48 — 52 and in their references. Researchers can utilize RSM to help them create a list of experimental designs which can be used to predict response. It might be beneficial to alter the theoretical constraints to examine a certain model term or interaction. Furthermore, it may recommend the best amount or value of input parameters to maximize the answer. In related research, RSM is commonly used to find the optimal parameter level. Sensitivity analysis is a procedure that involves changing one or more variables in a problem to assess how such changes may affect a result or quantity of interest. Such a procedure has likely been used in all branches of science for a very long time. The impact of a problem restriction on the optimality of a cost or benefit function via shadow pricing, for instance, or the role and function of a model parameter in producing a model output, are a few examples. LSA is easy to use, intuitive, and suitable for extremely certain situations. Although it has been criticized for simply offering a confined picture of the issue space, it has still been utilized frequently and much more widely, particularly when employed in the context of examining parameter relevance in mathematical modelling. Additionally, sensitivity analysis is frequently used in conjunction with an experimental approach like RSM to assess how much the response depends on the input parameters 53 — The sensitivity analysis of nonlinear convective heat transfer in hybrid nanomaterial inside two concentric cylinders with non-uniform heat sources is covered by Thriveni and Mahanthesh A sensitivity analysis is carried out by Mackolil and Mahanthesh 60 to optimize heat transmission in the flow of thermal Marangoni convective hybrid nanomaterial. In a hybrid nanofluid with a thermal radiation effect, melting heat transfer is demonstrated by Basir et al. Using the modified Buongiorno model, Mahanthesh et al. The sensitivity analysis of the flow of nanofluid in the presence of thermal Marangoni convection and inclined magnetic field effects was studied by Mackolil and Mahanthesh The main goal of this study is to examine the influence of different physical factors and the dependence of the response parameters of interest on the change of input parameters through experimental design and sensitivity analysis based on response surface methodology. Experimental design is important in both applied and industrial research. In a designed experiment, when a variety of input parameter levels are applied to the testing unit, one or more responses are over the experimental units. Motivated by all these applications and by utilizing the knowledge of the above mentioned literature, the objective of this study is to apply an experimental design and a sensitivity analysis to the problem of MHD flow of nanofluid across an exponentially stretching surface in the presence of non-uniform heat flux. A nanofluid is a novel type of heat transfer fluid that comprises nanoparticles in the base fluid. It has many industrial applications, such as transportation, power generation, pharmaceutical processes, micro-manufacturing, etc. It can be used for thermal therapy for cancer treatment. Because of its vast variety of industrial uses, the flow of nanofluid across a stretched surface has become highly popular among contemporary researchers. Plastic sheet extrusion, hot rolling, wire drawing, paper manufacture, plastic film drawing, fiber glass, and cooling of a metallic plate are examples of such mechanisms. Scientists are still looking for the best ways to use these nanofluids and are attempting to minimize any bad features that may arise when they are used in a project or system. This study also looked at the impact of non-uniform heat flux on the flow of nanofluid through an elongating surface. In the current analysis, mathematical formulation was performed using conservation laws and appropriate transformations to simplify our set of nonlinear coupled partial differential equations into a set of coupled nonlinear ordinary differential equations. The effects of emerging parameters have been further explained by plotting graphs and analyzing the results using physical descriptions. Further, a data set is developed through a numerical technique because we have used an experimental design-based approach to determine the significance of specific input parameters that may improve response. The experimental design-based analysis is the main achievement of this research for the flow of nanofluid over an exponentially stretched surface. To understand the properties of the thermal transfer mechanism along an exponentially extending surface, we have considered a steady, incompressible, 2D flow of viscous base nanofluid across an exponentially stretched surface. A uniform magnetic field has been applied normal to the surface to examine the effects of MHD. The effects of non-uniform heat flux and the convective boundary conditions are also considered. A cartesian coordinate system has been used in which the x —axis is taken parallel to the stretched surface and the y —axis is normal to it. Figure 1 depicts the geometry of the model under consideration. The following are the mass, momentum, and energy equations after using boundary layer approximations for nanofluid flow past a stretching surface with a magnetic field and non-uniform heat flux effects 23 :. Under the given assumptions, conservations laws of mass, momentum and energy Eqs. The following are the boundary conditions for the considered problem 23 :. Let us define the following non-dimensional variables and transformations to get the non-dimensional form of Eqs. The continuity Eq. Similarly, by using 9 in place of boundary conditions 8 , we get the following non-dimensional form of boundary conditions:. In the non-dimensional Eqs. For this problem, we used nanoparticles of aluminium oxide Al 2 O 3 and water H 2 O as the base fluid. Table 1 depicts the fundamental thermo-physical characteristics of nanofluids, where the subscripts n f , f , s signify nanofluid, base fluid, and nanoparticles, respectively. Table 2 shows the numerical values for the thermo-physical properties of nanoparticles and base fluid utilized to generate the computational results. This section shows how various quantities affect the flow profiles generated by a numerical technique. Multistep techniques try to enhance effectiveness by retaining and using previous phase information instead of deleting it, and hence make use of such a large number of earlier points and derivative values. For the validation of the results obtained from the Adam-Bashforth predictor—corrector technique, we have compared our results with those obtained from the explicit Runge—Kutta method. In Fig. It is observed that the absolute error between the results obtained from the Adam-Bashforth method and the explicit Runge—Kutta method is around 10 - 6 - 10 - 10 , which is in a negligible vicinity, and hence validates our results. Velocity profile and absolute error for varying values of M. Physical changes in velocity and temperature fields are depicted in Figs. Figure 3 depicts the dependence of velocity on the magnetic field parameter. The velocity field is seen to decrease as M increases. The Lorentz force operates as a decelerator, reducing the speed of the fluid and the momentum boundary layer thickness. As a result, larger M values intensify the resistive force, which resists the magnetic forces with dominant retarding effects and reduces fluid velocity. Figures 4 , 5 , 6 , and 7 show the impact of different values of space-dependent and time-dependent parameters A and B on fluid temperature for both cases of heat generation and internal heat absorption. The existence of A physically reduces the quantity of heat available to the system, resulting in a degraded transportation mechanism. The temperature of the fluid tends to rise due to the existence of a heat source in the boundary layer that generates energy. Figures 6 and 7 depict the effects of time-dependent heat source and sink parameters on fluid temperature. Heat generation a non-uniform heat source parameter greater than zero boosts fluid temperature by adding extra heat into the system and thickens the thermal boundary layer. The existence of a heat source in the interior causes the flow field to transmit extra heat, which results in the thermal boundary layer thickness. Improved heat conductivity is one of the most essential features. Thermal conductivity enhancements can result in slight efficiency gains due to more efficient fluid heat transfer. Increasing the quantity of nanoparticles added to the base fluid improves the heat transmission properties of the material, resulting in a higher temperature profile. The impact of the heat transfer Biot number on the dimensionless temperature field is seen in Fig. The flow in the boundary layer moves as a function of the velocity of the stretched sheet. As a result, increasing the Biot number reduces the impact of sheet motion on the boundary layer, slowing the flow. According to research, the temperature rises as we increase the quantity of nanoparticles in the base fluid. This is because increasing the number of nanoparticles improves the thermal properties of fluids. Furthermore, adding nanoparticles to the base fluid boosts the heat transmission capabilities of the material, resulting in a higher temperature profile. In mathematical modelling, an experimental scheme is important because numerical computational simulations have a better chance of being interpreted. A data set is developed through computer programming and applied to real-world scenarios. Response Surface Methodology RSM is an experimental design-based approach to determine which input parameter is significant and which particular level of an input parameter may improve response, since numerous input parameters might affect response. The sensitivity analysis is also performed on one of the numerous model coefficients and parameters, but only for the relevant response variables and parameters indicated above. The typical nonlinear polynomial model is used to study and assess the correlation between the specified response variable model parameters, which is presented as:. There is an intercept r 0 , three linear terms r 1 , r 2 , r 3 , three square terms r 11 , r 22 , r 33 , and three interaction terms r 12 , r 23 , r 13 in this response surface Eq. For two responses, two response surface equations are investigated. Table 3 shows the input values as well as the nomenclature for them. A response surface element has been introduced to determine the experimental outcomes. It is determined that 20 runs and 19 degrees of freedom are appropriate for three levels of specified parameters. Table 4 displays the outcomes of these runs. Table 5 shows their estimates, as well as t -values and p -values, to study the relevant input parameters further. A level of flexibility, SS sum of squares , MS mean of sum of squares , F -statistics, and p -value are among the outputs. In an ANOVA investigation, the estimation of data variance across average value and the probability validation of model correctness are represented in statistical contexts by F -value and p -value, respectively. A high F -value and a modest p -value provide sufficient evidence for the importance of the outcome. As a result, both F -value and p -value are frequently used to assess the significance of response variables. In the same way, the model parameter has a quadratic influence on all the outcomes. Several variables are evaluated in order to examine the quality of fit. Second, adjusted R 2 is used to indicate how well the models describe the deviation in response. As a consequence, all models account for a significant fraction of the whole range in their respective replies. Thirdly, the quality of fit is evaluated using a traditional residual Q - Q plot. In an appropriate model that suitably reflects the functional connection between input parameters and response, theoretical and observed quantiles have a one-to-one correspondence. Figure 10 depicts the standard residual Q - Q plot for the two models. This means that the theoretical and observed quintiles are roughly one-to-one for both models. The straight line in the graph shows that the errors are normally distributed, which indicates that our models are properly fitted. Figure 11 depicts the residuals of the fitted models, which indicate that they all have a normal distribution. According to these assessments, both models are appropriate. The fitted models are:. A derivation of the response function is a common definition of sensitivity in terms of model variables. When compared to model vigour estimates, sensitivity analysis investigates the distinct demands offered by model output spread by input variables. Figures 12 and 13 exhibit the bar chart findings of the sensitivity analysis for better understanding. Positive sensitivity levels are depicted by upward bars in this bar plot, whereas, negative sensitivity values are represented by downward bars. We observed that for all levels low, medium, and high of magnetic field parameter B and Biot number C , there is positive sensitivity for C f Re x. When the Biot number has no impact, the SFC is maximized at the maximum level of the magnetic field parameter and nanoparticle volume fraction. The SFC is maximized at the highest level of the magnetic field parameter and Biot number when the nanoparticle volume fraction is held constant. When the magnetic field parameter remained zero, the SFC was maximized at any Biot number and maximum nanoparticle volume fraction. When the effects of the Biot number are maintained to a minimum, the LNN is maximize at the lowest nanoparticle volume fraction and highest magnetic field parameter values. The LNN is maximized for all Biot number and magnetic field parameter values when the effects of nanoparticle volume fraction are maintained constant. Similarly, by keeping the magnetic field parameter constant at zero, the LNN is maximized at all Biot number levels and at the maximum level of the nanoparticle volume fraction. In this work, we investigated the MHD flow of nanofluid across an exponentially extending surface with non-uniform heat flux effects. The governing equations after employing the boundary layer approximations are transformed to a system of non-dimensional equations which have been analyzed numerically, and the impacts of emerging parameters were illustrated using graphs. An experimental design and a sensitivity analysis based on Response Surface Methodology RSM were used to investigate the impacts of several physical factors on the skin friction coefficient and local Nusselt number, for instance, magnetic field parameter, nanoparticle volume fraction, and heat transfer Biot number. The magnetic field parameter and the volume fraction of nanoparticles reduce fluid velocity. As the temperature of the fluid rises, so does the nanoparticle volume fraction and the heat transfer Biot number. Based on well-defined statistical measures Q - Q residual plots, hypothesis testing via p -value, and adjusted R 2 , our models for skin friction coefficient and local Nusselt number are shown to be the best fitted models. The skin friction coefficient is significantly influenced by nanoparticle volume fraction, Biot number, and square of the Biot number at less than a 0. Among the significant parameters, Biot number has the highest impact on skin friction coefficient and local Nusselt number, with a coefficient of 8. The sensitivity of the skin friction coefficient, as well as the local Nusselt number through Biot number, is greater than that of nanoparticle volume fraction and magnetic field parameter. For example, the results through partial derivatives for magnetic field parameter, nanoparticle volume fraction, and Biot number are 0. Similarly, it can be shown at other levels of magnetic field parameter as well as for local Nusselt number. The skin friction coefficient is positively sensitive to all combinations of all input parameters. At high levels of Biot number, the local Nusselt number displays negatively sensitive magnetic field parameters. Hussain and K. Rasheed and A. Ali done Investigation, Methodology S. Hussain and A. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This section collects any data citations, data availability statements, or supplementary materials included in this article. As a library, NLM provides access to scientific literature. Sci Rep. Find articles by Shahid Hussain. Find articles by Kianat Rasheed. Find articles by Aamir Ali. Find articles by Narcisa Vrinceanu. Find articles by Ahmed Alshehri. Find articles by Zahir Shah. Received Jun 20; Accepted Oct 21; Collection date Open in a new tab. Numerical values of the thermo-physical properties of nanoparticles and base fluid. Evaluations of fitted model terms, along with t -value and p -value. Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Similar articles. Add to Collections. Create a new collection. Add to an existing collection. Choose a collection Unable to load your collection due to an error Please try again. Add Cancel. Cf Re x.