A sensitivity study on carbon nanotubes significance in Darcy–Forchheimer flow towards a rotating disk by response surface methodology

The current research explores incremental effect of thermal radiation on heat transfer improvement corresponds to Darcy–Forchheimer (DF) flow of carbon nanotubes along a stretched rotating surface using RSM. Casson carbon nanotubes’ constructed model in boundary layer flow is being investigated with implications of both single-walled CNTs and multi-walled CNTs. Water and Ethylene glycol are considered a basic fluid. The heat transfer rate is scrutinized via convective condition. Outcomes are observed and evaluated for both SWCNTs and MWCNTs. The Runge–Kutta Fehlberg technique of shooting is utilized to numerically solve transformed nonlinear ordinary differential system. The output parameters of interest are presumed to depend on governing input variables. In addition, sensitivity study is incorporated. It is noted that sensitivity of SFC via SWCNT-Water becomes higher by increasing values of permeability number. Additionaly, sensitivity of SFC via SWCNT-water towards the permeability number is higher than the solid volume fraction for medium and higher permeability levels. It is also noted that sensitivity of SFC (SWCNT-Ethylene-glycol) towards volume fraction is higher for increasing permeability as well as inertia coefficient. Additionally, the sensitivity of LNN towards the Solid volume fraction is higher than the radiation and Biot number for all levels of Biot number. The findings will provide initial direction for future device manufacturing.


List of symbols θ
Dimensionless temperature f Dimensionless velocity ρ n f Nanofluid's density µ n f Nanofluid's dynamic viscosity ρ p Density of nanomaterials β nf Is volume expansion coefficient of fluiď T Temperature of liquid τ Heat capacity ratio of nanomaterials by nanofluid σ Electrical conductivity of nanoliquiď T ∞ Ambient temperature φ Solid volume fraction ǔ,v,w Velocity components in (r, ϕ, z) directions respectively k CNT CNTs thermal conductivity, k f Base fluid's thermal conductivity (ρc p ) CNT  www.nature.com/scientificreports/ from unity leads to non-linear flowing. In certain circumstances, the consequences of inertia and limits can not be overlooked. The impacts of inertia and boundary can't be ignored under these circumstances. Forchheimer 22 incorporated a square velocity expression to Darcian velocity term to estimate inertia and boundary effects. Muskat 23 referred to this term as "Forchheimer term" that always holds for high Reynolds number. In fact, higher velocities of filtration in the momentum expression create quadratic drag for porous material. Seddeek 24 investigated the effects of viscous dissipation and thermophoresis in DF mixed convective flow saturated porous medium. Pal and Mondal 25 implemented DF law to studied hydromagnetic flow of variate viscosity fluid in a porous medium. Recently Shafiq et al. 26 analyzed the influence of convective conditions and thermal slip in 3D rotating DF nanoliquids. Latest accomplishments for further assessment relating to Darcy-Forchheimer are in Refs. [27][28][29][30][31][32][33][34][35] .
In comparison to traditional materials, carbon nanotubes are well-suited for practically any activity involving high strength, electrical conductivity, durability, thermal conductivity, and lightweight attributes. CNTs are currently primarily utilized as synthetic additives. CNTs are widely available as a powder, which means they are heavily tangled and agglomerated. CNTs must be untangled and uniformly distributed in the substrate in order for their unique properties to unfold. By keeping this in mind the intention of this study is to look into the significance of DF flow of Casson carbon nanotubes along a rotating disk utilizing convective boundary condition. Both types of carbon nanotubes such as SWCNT and MWCNT are taken into account. Water and Ethylene glycol are considered as basic fluid. The porous space representing the Darcy Forchheimer expression is filled by an incompressible Casson fluid. Results are observed and evaluated for both SWCNTs and MWCNTs. The method of shooting (RK-4) was then utilized to solve numerically transformed nonlinear ordinary differential system. This study is concerned with essential use of carbon nanofluids in design for industrial usages such as air conditioning and refrigeration, transportation, microelectronics and solar thermal. Furthermore, an experimental scheme (RSM) [36][37][38][39][40][41][42][43][44][45] intimately associated to a sensitivity study to examine dependence of interest bearing output parameters on input governing parameters. Remarkably, the authors conducted a sensitivity analysis based on the SFC and LNN for both types of carbon nanotubes (SWCNT and MWCNT). This study is linked with feasible rule in future gadget development. To date, such analysis is fresh and unfulfilled for the best systematic review uncovered.

Flow problem
A steady DF flow of Casson CNTs along a rotating disk is considered (see Fig. 1). Heat transport phenomenon is studied with subject to thermal radiation and viscous dissipation. The porous space representing the DF expression is filled by an incompressible Casson fluid. In this study, Ethylene glycol and pure water fluid are considered as base fluid and SWCNT/MWCNT is considered as nanomaterials. At z = 0 , disk spins with (constant angular velocity). The consequent governing equations are [9][10][11][12]14 ): (1) ∂ǔ ∂r + ∂w ∂z = −ǔ r , An important condition is incorporated at boundary, namely convective condition. The heat transport through surface improves temperature and hence thermal conductivity of nanofluids because of convective condition. The application of convective boundary condition is therefore best adapted as a standard compared to isothermal conditions. In these situations, lower surface is heated via hot liquid that have Ť f temperature with h f coefficient of heat transfer. In these situations, k n f is the nanofluid's thermal conductivity inside the boundary layer, u = ra is stretched velocity, v = r is rotational speed. Suction is considered in the current boundary, adding/removing reactants, reducing the drag, cooling the surface, fluid scaling or preventing corrosion. Consequently, suction can be used with stretching/shrinking surfaces to effectively control the growth/decay of the momentum boundary layer. Suction is adapted to established boundary that adds/removes reactants, reduces drag, cools surface, prevents fluid corrosion or scaling. Consequently, suction can be used with stretching/shrinking sheets to effectively control the growth/decay of the momentum boundary layer.
The effective characteristics of carbon nanotubes are given below 13 : where CNTs solid volume fraction is φ, CNTs thermal conductivity is k CNT , Base fluid's thermal conductivity is k f , nanofluid's dynamic viscosity is μ n f , nanofluid's density ρ n f , CNTs heat capacity (ρc p ) CNT . Thermophysical properties of different base liquids and CNTs are listed in Table 1. We take transformations into consideration www.nature.com/scientificreports/ where non-dimensional distance along axis of rotation is defined as η and f, g and θ are functions of η . Replacing the above mentioned transformations into Eqs. (1)-(6), we attain the following set of differential equations: Here , k 1 , F r , Pr, R * , S 1 , Ec, δ 1 and γ 2 are defined mixed convective number, permeability number, Inertia coefficient, Prandtl parameter, radiation parameter, suction parameter, Eckert number, stretching-strength parameter and Biot number respectively and describe as follows

Numerical computational simulation
A numerical computational simulation which interacts with quantity interpretation is basically known as mathematical experiment. It's a process containing of a series of data tests, using a computer program to mimic the behaviors of the real world scenario. A computational analysis is carried out to find out output result of a change in code, because of several input variables. Conclusion on importance and pertinent variables may also be concluded in the end study. The model dependence is defined using RSM (see 41,42 ) in terms of relationship among input factors and output response.
In the entire investigation, there are four interest parameters and total of "12 ′′ independent input parameters. However, we mainly highlighted sensitivity assessment for interest parameter named LSFC and LNN. Additionally, only selective inputs variables which are assumed to have significant variability on SFC and LNN are considered.
The full quadratic model is given by involving intercept, quadratic, linear and two-factor bilinear terms. Thus Ř defines local response of SFC and NN. It consists of three independent input parameters coded via (A, B, C) symbols (solid volume fraction, inertia coefficient and permeability parameter respectively) for skin friction and (A 1 , B 1 , C 1 ) (solid volume fraction, radiation and Biot parameters) for LNN (For simplicity for LNN we also use same symbols A, B, and C). According to RSM, twenty runs along with 19, DOF are suitable for chosen 3 stages of parameters. These quantities are small, medium and large as (−1, 0, 1). Table 4 shows input parameters according to its respective levels and symbols. In addition, CCD (Central Composite Design) for conduct of a numerical experiment is commonly used in R−programming. The series of twenty runs of experiments is planned to refer the term of 2 F + 2F + P, where P = 6 is center points number and F = 3 is number of factors. The sequence of experimental programs is given for SWCNT-Water, MWCNT-Water, SWCNT-Ethylene glycol, MWCNT-Ethylene glycol in Tables 4 and 5 for both SFC and LNN respectively. ANOVA is a statistical strategic significance for utility of uncertainty in dependency of defined variables on RSM model. ANOVA studies the RSM model's optimization criterion for degree of model accuracy by which numerical estimators are DOF, SS, MMS, F−value and p− value. Tables 6, 7, 8 and 9 demonstrate ANOVA analysis to point out corelations among SFC and LNN numbers to three independent input parameters for SWCNT and MWCNT for both type of base fluids.
Sensitivity is extensively described in terms of model variables as derivative of response function. Sensitivity research explores the eccentric prerequisites provided by model output assigned by input variables, that compared to estimation of model vigor.

Discussion
The governing transformed differential system (9-11) with boundary conditions (12) are solved via Runge-Kutta Fehlberg technique. The boundary layer thickness η ∞ is putting 10. Tables 2 and 3 show numerical values of SFC and LNN correspond to SWCNT and MWCNT by considering water and Ethylene glycol as base fluid, for various values of φ, γ 1 , k 1 , F r , M 1 , 1 , R * , Ec and γ 2 . Tables 7, 8, 9 and 10 are related to ANOVA study, to set up correlations among SFC and LNN to independent input factors. In study of ANOVA, F-value is estimation of data variance over average value, whereas p-value is probability validation of model accuracy from statistical context. High F-value labels a significant outcome while small p-value shows sufficient support to significance of outcome. Therefore, F-value is often utilized to offer sufficient evidence on the importance of outcome alongside the p− value. Accordingly, effect of linear, www.nature.com/scientificreports/ two-factor bilinear and square terms are known to be statistically meaningful for response parameters (SFC and LNN), with good evidence of high F−value and low p-value. Particularly, residual error is unspecified data point via regression line, whereas lack of fit depicts if model neglects to display functional connectedness between input and output response. Figs. 2 and 3 show normal Q−Q residual plot for SFC and LNN correspond to SWCNT and MWCNT (Base fluid: Water and Ethylene glycol). The plots that appear with a straight line indicating the errors are normally distributed. Hence, regression model is properly fitted.
Regression coefficients for responses (SFC and LNN) via its corresponding p-value for non-linear polynomial model in (17) are given in Tables 11, 12, 13 and 14 for SFC and LNN corresponds to SWCNT and MWCNT (Base fluid: Water and Ethylene glycol). It is noteworthy that large p−value is considered to be statistically insignificant, indicating no relative change in output can be noted due to change in input. Further, a term with low p−value (≤ 0.05) that is statistically important elsewhere can be overlooked. As a consequence, A, A 2 , AB, AC B, and C corresponds to SWCNT-Water while A, C, A 2 , AB and AC corresponds to MWCNT-Water are significant factors for SFC. On the other hand, A, C, A 2 and AC corresponds to SWCNT-Ethylene glycol while A, C, A 2 and AC corresponds to MWCNT-Ethylene glycol are significant factors for SFC. For LNN, A, C, A 2 and AC corresponds to both SWCNT and MWCNT (Water as base fluid) are important terms. On the other side, same terms are important for both SWCNT and MWCNT when Ethylene glycol is considered as base fluid.
Additionaly, the values of R 2 and R 2 − adj =R 2 , are also dispensed in Tables Tables 15 and 16) correspond to SWCNT and MWCNT when water and Ethylene glycol are considered as base fluid. It is noticed that a +ve sensitivity esteem showing increase of regressor induces an  Figure 4 shows the sensitivity outcomes for SFC via SWCNT for the case where water is taken as base fluid. The overall trend shows that the sensitivity of the SFC rises with increment in governing variables under all values     www.nature.com/scientificreports/ of permeability parameter. However, the sensitivity of the SFC via SWCNT becomes higher by increasing values of permeability number from 0.2 to 0.6 i.e. C = −1 to 1. Additionally, sensitivity of SFC via SWCNT-water towards the permeability number is higher than the solid volume fraction for C = 0 and C = 1 . For the case of lower permeability number (C = −1) , the SFC (SWCNT-water) seems to have a high sensitivity towards the solid volume fraction instead of permeability and inertia coefficient (see Fig. 4a,b). On the other side, it is noted from Fig. 4c that the SFC (SWCNT-water) has a higher sensitivity for permeability as compare to solid volume fraction and inertia coefficient for C = −1. Sensitivity of SFC via MWCNT for the case where water is taken as base fluid at various values of permeability parameter is shown in Figs. 5a-c. It is noted that the sensitivity of SFC via MWCNT for the case where water is taken as base fluid falls with the increment in parameters under all values of solid volume fraction. Although, the SFC (MWCNT-water) have a low sensitivity corresponds to inertia coefficient for the increasing values of permeability as well as inertia coefficient. Additionally, it is observed that the sensitivity of SFC via MWCNT-water have a higher sensitivity towards permeability parameter as compare to inertia and solid volume fraction for C = 0 and C = 1 (see Fig. 5a-c). Furthermore, for the lower permeability parameter i.e. C = −1 , the sensitivity of SFC towards inertia is higher instead of permeability and solid volume fraction (see Fig. 5b,c). Whereas, sensitivity of SFC is lower towards inertia instead of permeability and higher in case of solid volume fraction (see Fig. 5a).    Fig. 6, the sensitivity of SFC (SWCNT-Ethylene-glycol) towards volume fraction is higher for increasing permeability as well as inertia coefficient. It is also noted that the sensitivity of SFC (SWCNT-Ethylene-glycol) is approximately equal for permeability under all levels of permeability number (see Fig. 6a-c). Meanwhile, same pattern is noted for inertia under all levels of permeability. Similar behaviour is observed for sensitivity of SFC (MWCNT) when Ethylene-glycol is taken as base fluid (Fig. 7).
Sensitivity of LNN via SWCNT and MWCNT by considering two type of base fluids towards various parameter at different levels of Biot number are plotted in Figs. 8, 9 and 11. In general, the sensitivity of LNN via SWCNT-water towards solid volume fraction increases by increasing Biot number. But the sensitivity of LNN (SWCNT) towards Biot is approximately equal despite the increment in Biot and radiation parameter. Figure 8a shows the similar positive sensitivity at low radiation parameter under all levels of Biot. However, very small sensitivity of LNN towards radiation is noted at higher Biot number ( C = 1 ) see Fig. 8b-c. Figure 9 is drawn to see the sensitivity results for the LNN for MWCNT and water is taken as base fluid. Overall, trend noted from figures 9a-c shows that sensitivity of the LNN that rises with increment in parameters under all values of Biot number. Yet, sensitivity of LNN remains approximately constant with increasing Biot number from 0.2 to 0.6 ( C = −1 to 1). Additionally, the sensitivity of LNN towards the Solid volume fraction is higher than the radiation and Biot number for C = −1, 0, 1 (see Figs. 9a-c. Similar behavior is noted for Figs. 10 and 11 for both types of carbon nanotubes, when base fluid is Ethylene glycol. Only for the case of lowest Biot number the LNN seems to have a higher sensitivity towards Biot number instead of radiation and solid volume fraction. The predicted non-dimensional SFC as a function of the solid volume fraction (A) , inertia coefficient (B) and permeability parameter (C) are shown in Fig. 12 for SWCNT-Water. The effects of inertia coefficient and permeability parameter on non-dimensional SFC for A = 0 (φ = 0.1) are shown in Fig. 12a. It is noted that maximum non-dimensional SFC occurs near higher level for inertia coefficient (B) and permeability parameter (C) and vice versa. On the other side, the maximum average of SFC occurs near the high and low levels for solid volume fraction (A) and inertia coefficient (B) (see Fig. 12b). But the moderate value of SFC occurs at the middle levels for solid volume fraction (A) . Moreover same behavior is observed in Fig. 12c.
The predicted SFC as a function of solid volume fraction (A) , inertia coefficient (B) and permeability number (C) are plotted in Fig. 13 for MWCNT-Water. The effects of inertia coefficient and permeability number on non  www.nature.com/scientificreports/ dimensional SFC for A = 0 (φ = 0.1) are shown in Fig. 13a. It is noted that maximum non-dimensional SFC occurs near large level for inertia coefficient (B) and higher and lower levels for permeability parameter (C) . On the other side, the moderate level of SFC occurs near the extreme level of inertia coefficient (B) and the moderate level of permeability coefficient (C). In addition, the maximum average of SFC is observed at the extreme levels of A and C, on the contrary the oposite behaviour observed for lower levels of A and C and low levels for solid volume fraction (A) and inertia coefficient (B) (see Fig. 13b). Also the same pattern is observed in Fig. 13c. The predicted non-dimensional SFC as a function of A, B and C are shown in Fig. 14 for the case of SWCNT and Ethylene glycol is taken as base fluid. The impact of inertia coefficient and permeability parameter on nondimensional SFC for solid volume fraction (A = 0 (φ = 0.1)) are shown in Fig. 14a for SWCNT-Ethylene glycol. It is noted that maximum non-dimensional SFC is noted near all the levels for inertia coefficient (B) and extreme high and low levels for permeability parameter (C) . On the other hand, moderate behavior is observed at the moderate level of C and all the levels of A. In Fig. 14b, at the extremely higher level of A and C the maximum non-dimensional SFC is reflected. In addition the maximum average SFC is examined near the extreme level for solid volume fraction (A) and all levels for inertia coefficent (B) (see 14(c)).
The predicted non-dimensional SFC density as a function of solid volume fraction (A) , inertia coefficient (B) and permeability parameter (C) are analyzed in Fig. 15 for the case of MWCNT and Ethylene glycol is taken as base fluid. The strenght of inertia coefficient and permeability parameter on non-dimensional SFC (MWCNT-Ethylene glycol) for A = 0 are drawn in Fig. 15(a). It is indicated that average maximum non-dimensional SFC (MWCNT-Ethylene glycol) is examined at the extreme level of C and all levels of B. While the maximum level of non-dimensional SFC for MWCNT-Ethylene glycol is noted on moderate level of A and higher level of permeability parameter (C) (see 15(b)). In addition the maximum average SFC (MWCNT-Ethylene glycol) is analyzed near the higher level for solid volume fraction (A) and all levels for inertia coefficent (B) (see 15(c)).
In Figs. 16 and 17, residual histograms along with the density function are shown for both local SFC and NN via SWCNT and MWCNT using water and Ethylene glycol as a base fluids. It is noted from these figures that behavior of the residual histogram is less skewed distribution and shown the behaviors which are almost similar to a symmetrical distribution. The results in Table 17 were determined to validate the current results with previously reported results. In this case, we can see that the current numerical solution agrees with previous solution by 46 in a limited context.

Concluding remarks
A numerical investigation on heat transfer improvement corresponds to Darcy-Forchheimer flow of carbon nanotubes along radiative stretched rotating disk using response surface methodology (RSM). The traditional heat transfer liquids such as water, thermal liquids and ethylene glycol, are widely used in various industrial processes involving refrigeration and air conditioning, transportation, solar thermal and microelectronics. Here we us water and ethylene glycol are considered a basic fluid. Main findings are listed below, which offered preliminary guideline for lab-based experimenters in future device of solar-thermal, air-conditioning, refrigeration, transportation and microelectronics: • The normal Q − −Q residual plot presents the best fitted regression model for SFC and LNN for both SWCNT and MWCNT when water and ethylene glycol are taken as base fluids. • The factors A, C, A 2 and AC corresponds to SWCNT-Ethylene glycol and MWCNT-Ethylene glycol are significant for skin friction coefficient. • The factors A, C, A 2 and AC corresponds to both SWCNT and MWCNT (for both Water and Ethylene glycol as base fluid) are important for local nusselt number. • Sensitivity of SFC via SWCNT-water towards the permeability parameter is higher than solid volume fraction for C = 0 and C = 1. • For lower permeability number, the SFC for SWCNT-water seems to have a high sensitivity towards the solid volume fraction instead of permeability and inertia coefficient. • The sensitivity of LNN for SWCNT-water towards Biot number is approximately equal despite the increment in Biot and radiation number. • Sensitivity of LNN for MWCNT-water towards the solid volume fraction is higher than the radiation and Biot number for C = 1, 0, 1.   www.nature.com/scientificreports/ Table 5. Design of experments and response results.  Table 6. Design of experments and response results.    www.nature.com/scientificreports/       www.nature.com/scientificreports/ Table 17. Comparative values of f ′′ (0) and g ′ (0) for value of F r = 0.2 when γ 1 → ∞, = 0.2, S 1 = 0 = M 1 = φ = δ 1 .

Present results
Naqvi et al. 46