The physical phenomena of convective flow of Cross fluid containing carboxymethyl cellulose water over a stretching sheet with convective heating were studied. Cross nanofluid containing , nanoparticles, and based fluid of CMC water is used. Entropy generation minimization is examined in the current analysis. The system of PDEs is altered into a set of ODEs through suitable conversion. Further, these equations are computed numerically through the MATLAB BVP4c technique. The behavior of governing parameters on the velocity, temperature, entropy generation, and Bejan number is plotted and reported via graphs. It is found that the larger value of unsteady variable reduced the velocity, thermal layer, and entropy production. Surface drag frication of the and and + is enhanced with the more presence of unsteady parameter. Comparison of current results in a limiting case is obtained with earlier analysis and found an optimum agreement.

1. Introduction

Carboxylmethyl cellulose (CMC) is a water-soluble cellulose derivative [1], and it has many flow properties due to its greater stability and high viscosity. The stability of nanoparticles in CMC escalates the fluid behavior. It is engaged to increase lubricating effects such as polymeric structures [2, 3]. These multifunction aspects of various cellulose derivatives have many industrial and technical applications. To recognize the fluid flow with CMC study, research have been studied [46]. Saqib et al. [7] described the natural convective flow of CMC with carbon nanotube using a fractional derivative approach. The effect of slip velocity and non-Newtonian nanofluid contained with wt CMC water was discussed by Rahmati et al. [8]. Akinpelu et al. [9] explored the thermophysical metal properties in CMC. MHD flow of Casson nanofluid under heat transfer in CMC over a solid sphere was developed by Alwawi et al. [10].

Nanotechnology has been progressively more fascinated by the researchers because of their efficiency in several industrial processes such as microelectronic, oil emulsion, and molecular emulsion. Nanotechnology has the ability in suspending nanoscale particles in ordinary fluids, like ethylene glycol, oil, and water. The origin of nanotechnology was initiated by Choi and Eastman [11] in 1995. After, Buongiorno [12] developed a mathematical model of heat transfer with the addition of Brownian motion and thermophoresis effects. Tiwari and Das [13] investigated to examine the solid volume fraction in nanofluids. Devi and Devi [14] reported the numerical simulation of hybrid nanofluid over a porous surface with suction. Afridi et al. [15] carried out the heat transfer analysis in hybrid nanofluid under fraction heating. The effect of second law analysis with hybrid nanofluid and viscous dissipation due to rotating disk was scrutinized by Farooq et al. [16]. Devi et al. [17, 18] revealed the heat transfer of hybrid nanofluid flow with two different base fluids. Gorla et al. [19] addressed the impact of heat sink/source in the hybrid nanofluid past the permeable surface. Chamkha et al. [20] analyzed the time-dependent flow of mixed convective hybrid nanofluid over half cavity. More recently, Zainal et al. [21] disclosed the unsteady 3-D MHD stagnation point flow of hybrid nanofluid using stability analysis. Few more cutting edge research reports are seen in Refs. [2228].

The study of entropy optimization has broad features in the thermal engineering process such as heat pump, heat engine, solar power, and refrigerator. The improvement of the thermal system is enhanced due to the entropy production. The Bejan number [29] is a dimensionless quantity that represents overall entropy generation ratio of heat transmission and total entropy generation. Khana et al. [30] discussed the computational analysis of hybrid nanofluid with entropy generation due to rotating disk between parallel plates. Dawar et al. [31] surveyed the heat transfer analysis through SWCNTs/MWCNTs in entropy generation and activation energy over a moving wedge. The numerical study of second law analysis of nanofluid due to an inclined surface was discussed by Butt et al. [32]. Heat transfer in MHD third-grade nanofluid with convective condition and entropy generation over a stretching surface was encountered by Rashidi et al. [33]. The investigation of entropy production for the magnetic field, thermal radiation, and porous medium was reported by Makinde and Eegunjobi [34]. The impact of entropy generation on two permeable stretched surfaces was inspected by Khan et al. [35]. Afridi et al. [36] described the hybrid nanofluid flow over a thin needle with entropy generation. Reddy et al. [37] studied the entropy generation on Williamson nanofluid with thermal radiation and internal heat source over the lubricated surface.

The behavior of non-Newtonian fluid models, like second grade, power law, and Williamson, was investigated by many researchers in the past few years due to their vital role in engineering and industrial applications. However, these models cannot be recognized to analyze the behavior of fluid at higher and lower shear rates. To illustrate the behavior of fluid at a very low and high shear rate, the Cross fluid model has been introduced by Cross [38]. The Cross fluid model has optimum potential to trounce the challenges that are overlooked while the shear rate is highly accelerated or depreciated. Few recent developments under this direction are collected in [39, 40]. The effect of heat source/sink on Cross fluid with thermal radiation was studied by Nazeer et al. [41]. Sabir et al. [42] scrutinized the heat transfer phenomena through radiation and activation energy over an inclined sheet. Yao et al. [43] investigated the magnetic dipole effect for Cross fluid through spectroscopy. Khan et al. [44] interpreted the effect of thermal radiative and activation energy on Cross fluid near the stagnation point. Reddy and Ali [45] constructed the MHD Cross nanofluid under Cattaneo-Christov double diffusion theory over a vertical stretching sheet.

The abovementioned studies reveal the focus on the heat transport analysis of a CMC-nanofluid, but no authors examined the CMC-hybrid nanofluid in the presence of unsteady Cross fluid with the effect of mixed convection. So, the authors attempted to investigate the heat transfer analysis of CMC-based Cross hybrid nanofluid with convective heating. The system of PDEs is transformed into ODEs through the suitable transformation, and these ODEs are tackled through BVP4c for numerical solution. The entropy analysis is implemented in the present study. This combination is more useful in thermal and aerospace engineering.

2. Mathematical Formulation

Let us consider the unsteady incompressible mixed convective flow of Cross hybrid nanofluid over a stretching sheet with surface heating. Moreover, Cartesian coordinates have been taken in the -axis along the sheet, and -axis is perpendicular to surface as seen in Figure 1. The radiation can only travel a distance within thick nanofluid; so, the Rosseland approximation is considered into account for radiative heat transfer.

Under the above assumptions, the flow model can be extract as follows:

The boundary constraints are applied as follows:

Here, , ,,, , and are horizontal velocity and vertical velocity, viscosity, density, thermal conductivity, specific heat capacity, and thermal radiative for hybridnanoflud, respectively. is the heat transfer coefficient.

2.1. Suitable Transformation for Unsteady Flow

It is relevant to use the following appropriate transformation:

Using the suitable transformation described in Eq. (5), to Eq. (2), Eq. (3) altered into the following ordinary differential equations with respect to parameter :

The transformation boundary conditions are stated as follows:

Nondimensionless governing variables are Weissenberg number , unsteady parameter , mixed convection , Prandtl number , radiation parameter , the skin friction coefficient , and the local Nusselt number which are presented by

Wall shear stress and heat flux are as follows:

In view of Eqs. (5) and (10), we get

3. Entropy Generation

The appearance of entropy production for Cross hybrid nanoliquid over a stretching sheet is defined as [46, 47]:

The characteristics entropy generation is described below:

The dimensionless form of entropy generation is .

is the entropy generation due to heat transfer, and = is the entropy generation due to fluid friction.

The Bejan number is defined by

4. Numerical Investigation

The set of an altered system of highly nonlinear ODE’s equations (6)–(7) with subject to the boundary condition (8) has been numerically computed with aid of the BVP4c method. For this purpose, first, we converted the higher order derivative into first order.

Converted boundary conditions are as follows:

The iterative process has been used, and the accuracy of the solution is obtained to .

5. Result and Discussion

In this segment, we examine the variations of , , , and for different flow variables, such as Weissenberg number , Biot number Prandtl number , thermal radiation , nanoparticle volume fraction (), and mixed convection parameter (). For performing graphical study, single variable varies, whereas all the physical variables were kept in constant values such as . Table 1 demonstrates the thermophysical properties of , , and +. Tables 2 and 3 show the comparison outcome of against against with the limiting case . From these tables, it is found that our computations are optimum one. Table 4 shows the impact of , , , and on skin friction coefficient for , , and +.

Figures 2(a)2(d) display the fluctuation of Weissenberg () on velocity distribution , temperature field , entropy production , and Bejan number for nanofluids () and hybrid nanofluid (). The fluid velocity and entropy generation reduce when augments. However, fluid temperature and Bejan number enhance when enhancing the quantity of . Physically, the Weissenberg number means shear rate time which helps to rise the fluid thickness, and this causes to depreciate fluid velocity. The variations of on , , , and are illustrated in Figures 3(a)3(d) for nanofluids and hybrid nanofluid. It is seen from these figures that the fluid velocity, fluid temperature, and entropy production decline when increasing the magnitude of , and Bejan number raises when rising the values of Figures 4(a) and 4(b)present the consequences of and on and for nanofluids and hybrid nanofluid for nanofluids and hybrid nanofluid. It is seen that the fluid velocity and fluid temperature upsurge when mounting the quantity of and . The impact of radiation on , , and was portrayed in Figures 5(a)5(c) for nanofluids and hybrid nanofluid. It is concluded that the fluid temperature, entropy production, and Bejan number are increasing function of radiation parameter. Physically radiation parameter enhances the rate energy transport to the fluid and thereby enriching the fluid temperature and thicken the thermal boundary layer. Figures 6(a) and 6(b) provide the changes of on and on for nanofluids and hybrid nanofluid. It is detected that the momentum boundary layer thickness escalates when enriching the values, see Figure 6(a). The fluid temperature raises when raising the Biot number, see Figure 6(b). Physically, Biot number leads to enrich the heat transfer coefficient, this leads to enhance the fluid thermal state, and this causes to improve the fluid temperature and thicker the thermal boundary layer thickness. Figure 7 displays the effect of on for nanofluids and hybrid nanofluid. It is found that the fluid temperature and its associated boundary layer thickness downturn when strengthening the Prandtl number.

Figure 8 shows the influence of and on skin friction coefficient for nanofluids and hybrid nanofluid. It is proved from this figure that the skin friction coefficient enriches when strengthening the values, and it is almost fixed when changing the values. Further, the skin friction coefficient is low in hybrid nanofluid than the nanofluids case. The local Nusselt number for various values of and for nanofluids and hybrid nanofluid is plotted in Figure 9. It is seen that the heat transfer gradient depresses when enriching the and for all cases. In addition, the less local Nusselt number is attained in hybrid nanofluid than the nanofluids case. Finally, Figures 10 and 11 present the streamline pattern for numerous values of nanoparticle volume fraction, Stedy and unstedy flows.

6. Final Remarks

The two-dimensional mixed convection flow of Cross fluid is based on CMC-water with nanoparticles and with thermal radiation over a stretching sheet. The second law analysis has been made. The physical model is computed via the MATLAB BVP4c function. The numerical and graphical results for flow and energy transfer are produced for diverse values of dimensionless variables. Moreover, skin friction and Nusselt number have been computed. The main findings of this work are as follows: (i)Momentum boundary layer thickness of , , and + reduces as the Weissenberg number is enhanced.(ii)The temperature profile , , and + is reduced for both and (iii)Entropy generation and Bejan number of , , and + are quite similar trends for and .(iv)Temperature distribution, entropy generation, and Bejan number , , and + are enhanced as increases the value of thermal radiation .(v)Both Biot number and mixed convection are enhanced for temperature and velocity distribution of , , and +.(vi)The drag friction and Nusselt number have an increasing effect for nanofluid and hybrid nanofluid.


a:Stretching rate
:Mixed convection parameter
:Effective thermal conductivity
:Reference density of fluid
:Reference density of solid
:Prandtl number
:Biot number
:Bejan number
:Total entropy generation
:Viscosity of fluid
:Weissenberg number
:Unsteady parameter
:Power-law index
:Thermal conductivity of base fluid
:Velocity along the -axis
:Velocity along the -axis
:Density of fluid
:Effective viscosity of nanofluid
:Nusselt number
:Thermal conductivity of fluid
:Thermal conductivity of solid
:Local Reynolds number
:Fluid temperature
:Temperature of the hot fluid


PDE:Partial differential equations
ODE:Ordinary differential equations
CMC:Carboxylmethyl cellulose.

Data Availability

The raw data supporting the conclusions of this article will be made available by the corresponding author without undue reservation.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.