Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 4294985 |

Sumera Dero, Liaquat Ali Lund, Zahir Shah, Ebenezer Bonyah, Wejdan Deebani, "Numerical Analysis of Hybrid Nanofluid of Streamwise and Cross Flow with Thermal Radiation Effect: Duality and Stability", Mathematical Problems in Engineering, vol. 2021, Article ID 4294985, 12 pages, 2021.

Numerical Analysis of Hybrid Nanofluid of Streamwise and Cross Flow with Thermal Radiation Effect: Duality and Stability

Academic Editor: Ahmed Zeeshan
Received12 Sep 2021
Accepted12 Nov 2021
Published02 Dec 2021


The motion of water conveying copper and aluminum nanoparticles on a heated moving sheet when thermal radiation and stretching/shrinking surface is significant and is investigated in this study to announce the increasing effects of volume fractions, thermal radiation, and moving parameters on this transport phenomenon. Furthermore, the flow of a /water hybrid nanofluid across a heated moving sheet has been studied in both cross and streamwise directions. Thermal radiation effect is also considered, as this effect along with cross flow has not yet been investigated for the hybrid nanofluid in the published literature. Two distinct types of nanoparticles, namely, (alumina) and (copper), have been used to prepare hybrid nanofluid where water is considered as a base fluid. The system of nonlinear partial differential equations (PDEs) has been transferred to ordinary differential equations (ODEs) by compatible transformations before solving them by employing the III-stage Lobatto-IIIa method in bvp4c solver in MATLAB 2017 software. Temporal stability analysis has been carried out in order to verify stable branch between two branches by obtaining the smallest eigenvalue values. The branches obtained are addressed in depth against every applied parameter using figures and tables. The results show that there are three ranges of branches, no solution exists when , dual branches exist when , and a single solution exists when . Moreover, thermal layer thickness declines initially and then enhances in the upper and lower solutions for the higher values of the thermal radiation parameter.

1. Introduction

Although numerous physical geometries cope with complicated phenomena depending on the industrial requirement, the near-surface flow is still an important aspect of aerodynamics and engineering. In aerodynamics, when the fluid moves on the sheet, the boundary layer which appears from the deformation of streamlines varies from the free flow based on the slope of pressure which is normal to the free flow direction. Pressure found inside the layer of boundary is not influenced by distance from sheet, although certain other characteristics play a part, such as a boundary surface, lower velocity, and smoother curve relative to free-flow velocity. Cross flow is also known as a secondary flow which happens when a component of velocity of the layer is perpendicular to the path of free stream. In such flows, transverse movement is expected to be completely formed. These kinds of flows are used in other technological contexts, such as aircraft, hydraulic, and wind movement phenomena. It seems that Sowerby [1] is the first scholar who studied that boundary layer on the cross flow. After that, the reverse flow of the boundary layer over the moving surface was examined by Klemp and Acrivos [2, 3]. They have concluded that flow consists of three overlapping domains such as the “external uniform flow, a conventional boundary layer with the reverse flow, and an inviscid collision region.” Hussaini et al. [4] considered the upstream moving wall of fluid and solved the modeled equations by Crocco formulation. Moreover, the nonuniqueness of the solutions was noticed. Recently, Altaç et al. [5] studied the mixed convection flow subjected to upward cross flow. Many other significant cross-flow studies can be obtained in these reported research papers [610].

In the present decade, the market for lightweight electronic devices and components has risen rapidly, needing more efficient thermal conductivity relative to the simple fluid. Thermal conductivity has a major role in heat transferring between the surface and medium. Technologists and acientists also developed a different form of fluid identified as nanofluid. The word nanofluid has been suggested as micromillimeter-sized particles that are dispersed in the conventional fluid such as water and engine oil [11]. As a consequence, nanofluid thermal conductivity is anticipated to improve heat transfer relative to normal heat transfer fluids [12]. Choi and Eastman [13] suggested the word nanofluid which describes the suspension of solid nanoparticles in the regular fluid. Furthermore, in a study of the flow of 29 nm nanofluid over a horizontal surface over the paraboloid of revolution subject to Lorentz force and thermoelectric, it was discovered that maximum temperature distribution with volume fraction is guaranteed at larger values of Lorentz force [14]. Mohamed et al. [15] investigated the effect of viscous dissipation on the water-based hybrid nanofluid, discovering that, as the volume fraction increases, so does skin friction. In addition, several experiments have been performed to demonstrate the versatility of nanofluid, taking into consideration different impacts, physical properties, surfaces, and shapes. Collections of nanofluid papers can be found in [1619].

Researchers became interested in nanotechnology and nanoparticles’ research in the last decade of the twentieth century due to their wide range of applications in almost all fields of science. Many researchers conducted several experimental and computational studies in this regard in order to obtain better results in the form of the heat transfer rate. To improve the heat transfer performance of conventional base fluids, solid nanoparticles and base fluids were combined in the preparation of nanofluids as discussed above. Typically, researchers focused on aluminum oxide and copper particles for computational studies due to their good thermophysical properties. Aluminum oxide is used to improve the rheological and filtration properties of drilling fluids because of its high thermal conductivity, which allows it to effectively dissipate heat from the fluid via Brownian motion [20]. Moreover, it is used as a nanoparticle in polymer products to strengthen ceramics and increase density, smoothness, crack hardness, creep resistance, thermal fatigue resistance, and wear resistance. Its most common application is in integrated circuit baseboards. It has extremely high melting and boiling points of 29770°C and 20400°C, respectively, which increase the thermal inertia of the nanofluid [21]. Copper nanoparticles are a fascinating class of material with multifunctional properties that have prospective uses in catalysts, batteries, magnetic storage media, solar energy, and superconductors [22]. Because of their high surface area, these nanoparticles have significant benefits over traditional materials. Recently, researchers introduced a new kind of nanofluid, namely, “hybrid nanofluid” due to the demand for the high thermal conductivity and heat transfer rate by the growing technology of the world. Elasticity was predicted to be higher in thermal conductivity relative to standard fluid and nanofluid. The hybrid nanofluid concept is meant to describe a mix of superior properties at an acceptable expense with two or more specific forms of scattered nanoparticles in the base fluid. In this regard, many experimental and computational modified models have been purposed from these basic models [2326]. Devi and Devi [27, 28] proposed a computational hybrid nanofluid model and concluded that it is the most appropriate model for the hybrid nanofluid by comparing the results with the practical investigational work of Suresh et al. [29]. Yan et al. [30] used the same properties of thermophysical and found double solutions over the exponential surface. This model has been used by many researchers to deal with the hybrid nanofluid such as [3137].

In this study, the characteristics of flow and heat transfer of the hybrid nanofluid in the directions of streamwise and cross flow are investigated. To the best of the authors' knowledge, no such investigation on numerous solutions has previously been conducted. The current study has four main objectives: first, an analysis of the flow of the streamwise and cross-flow hybrid nanofluid model with thermal radiation impact, second, to adopt the Devi and Devi model for hybrid nanofluid, third, to obtain multiple branches, and fourth, to conduct a stability analysis.

2. Problem Formulation

Three-dimensional uniform steady transverse and streamwise flow of /water hybrid nanofluid flow on the heated flat plate flowing with continual velocity out and in of origin centered at has been considered, where is reference determined on a flat wall, is the fixed velocity of the outward flow, and indicates the moving parameter. The constant temperature of the plate is , whereas the uniform temperature of the free stream is (refer Figure 1). It is, therefore, supposed that the cross-flow is infinite in the direction of spanwise and hence completely formed. The fields of temperature and velocity are thus independent of coordinates, such that the boundary layer equation of energy with the term of thermal radiation may be written as follows [6, 34, 38]:subject to the boundary conditions:where , and are the dynamic viscosity, heat capacity, density, and thermal conductivity of hybrid nanofluid, respectively. Furthermore, subscription indicates the thermophoresis properties of hybrid nanofluid. Moreover, and indicate the corresponding spanwise and normal coordinate in the direction of plane and transverse flow, corresponding velocity along , and axes is indicated by , and , the transverse velocity at origin is , and the temperature of fluid is denoted by . It should be noted that when (), then plate passes out (into) of the origin. For this analysis, the unique type of properties of Yan et al. [30] is used to analyze the model of cross flow. Table 1 lists the thermophysical properties of the hybrid nanofluids that will be used in this study. Table 2 shows the thermophysical properties of water and nanoparticles.

Hybrid nanofluid

Dynamic viscosity
Thermal conductivity, where
Heat capacity
Electrical conductivity, where

FluidsCopper (Cu)Alumina ()Water ()

(J/kg K)3857654179
k (W/m K)400400.613

Looking for similarity transformations to convert PDEs to ODEs, we have

Substituting equation (6) in equations (1)–(4), equation (1) is immediately satisfied, and equations (2)–(4) assume the following nondimensional form:subject to the boundary conditions:where prime denotes the differentiation with respect to ; denotes Prandtl number.

The physical quantities of interest are the coefficient of skin friction and Nusselt number which are described as

By putting equation (6) in equation (12), one obtainswhere is the local Reynold number.

3. Stability Analysis

By taking the instructions of Khan et al. [39] to conduct a temporal study of reliability of branches, we ought to add a new nondimensional time-dependent similarity transformation by assuming such that we have to obey new similarity transformation variables:

The unsteady model of the governing equations is written as

By replacing equation (14) in equations (15)–(17), we obtainsubject to the boundary conditions:

To test the stability of steady flow branches, where , and satisfying BVP (7)–(9), it can be written aswhere the corresponding small relatives of , and are , and , and is an undefined eigenvalue quantity that needs to be calculated. Through replacing equation (22) in equations (18)–(20) with , we are presented with the following linearized problem of eigenvalue:subject to boundary conditions:

As per Khashi’ie et al. [33], one boundary condition must be modified to achieve the lowest . It should be remembered that there is no impact on the effects of this relaxation in boundary conditions [37]. For this specific problem, the boundary condition of is relaxed to .

4. Analysis and Discussion of Results

In this section, two branches have been obtained. Figure 2 portrays presence of the dual branches for a system of dimensionless ODEs (7)–(9) along with boundary conditions (10). These equations are solved by applying the III-stage-Lobatto-IIIa formula in bvp4c solver in MATLAB 2017 software. This technique has been commonly applied by various scholars and academics to get the solutions of the boundary layer problems. One must make an initial supposition at the initial mesh point and adjust the phase size in order to achieve the required consistency of the solutions. This method is explained extensively in the book of Shampine et al. [40].

The nondimensional profiles of streamwise velocity , cross velocity , and temperature are drawn in Figure 2 for numerous values of moving parameter () combinedly. It is detected that stream and cross velocities as well as temperature increase (decrease) in the upper (lower) branch when parameter of moving is augmented. However, thicknesses of momentum and thermal boundary layers are thicker in the lower solution as associated to the upper solution. Figure 3 shows the effect of on and . It can be observed easily that the profile of stream function is an increasing function of in the upper branch and vice versa.

Figures 46 demonstrate the behavior of , and for different values of , respectively. When volume fraction of is augmented then velocity, cross-flow velocity and temperature of /water hybrid nanofluid decrease in the upper as well as in lower branches as expected because lower thickness of the boundary helps to raise the heat transfer rate. Furthermore, the behavior of the thickness of the lower branch is the same as it is noticed in Figure 2.

Figure 7 is plotted for the effect of on when the values of the other all parameters are fixed. It is seen that the temperature profiles of /water hybrid nanofluid decline monotonically with the increment of for the first solution. In the lower solution, thermal boundary layer thickness and temperature of /water hybrid nanofluid rises at first and then tends to decline (refer to Figure 8). Figure 9 reveals that thickness of the thermal layer declines primarily and then enhances in the upper and lower solutions for the higher values of the thermal radiation parameter. It must be noted that when , lower branch does not converge at zero asymptotically. Therefore, it may be stated that nonuniqueness solutions also depend on the thermal radiation parameter.

Figures 811 portray to check the behavior of , and against the volume fraction of the hybrid nanofluid (copper) for different values of . It has been noticed that two branches of solutions are obtained when the parameter , while the behavior of the shows the existence of one branch when . It must be distinguished that dual branches exist for and when , but the profile of the temperature does not converge asymptotically at ; therefore, solutions of and could not be considered since we have solved the system of coupled ODEs. It is significant to remember that upper solution of dual branches is stable, whereas lower solution is unpredictable and unstable. These two branches are found in range , and is a critical value where both branches exist, and it is the minimum value where solutions exist. The critical value of (i.e., ) is the same as stated in article of Weidman et al. [6] which gives us the confidence to use our numerical method in this study. Furthermore, the friction coefficient grows in upper branch for the higher values of (see Figure 8). Then, again, an opposite pattern is seen for and ; however, the rate of heat transfer and skin friction boost as the volume fraction of the copper particles is enhanced during the examination of the /water hybrid nanofluid (refer to Figures 10 and 11). In practice, increases in magnitude are caused by an increase in the heat capacity of the nanofluid, which can also be seen from the volume fraction relation shown in Tables 1 and 2.

At last, the linearized equations (23)–(25) along with boundary conditions (26) have been solved in order to perform the temporal stability of the upper and lower branches. Our goal of carrying out this analysis is to obtain the minimum eigenvalue . The values of are tabulated in Table 3. This is worth examining those values of which are equal to zero when . Henceforth, changes sign from zero to positive for the upper branch (stable) and negative for the lower branch (unstable) which means that is a turning point.

Upper branchLower branch


5. Conclusions

The heat transfer of the /water type hybrid nanofluid of cross flow in the presence of radiation impacts was investigated. Similarity variables have been applied to change the governing equations into ODEs. The modified ODEs were then solved using the 3-stage Lobatto 3a method in bvp4c solver in MATLAB. The pointwise conclusion of this study is as follows:(1)Dual branches depend on a certain range of (2)There occur triple ranges of solution, no branch solution exists when , two branches occur when , and a single branch exists when (3)The heat transfer improved due to the volume fraction of in a hybrid nanofluid(4)In the thermal radiation presence, the dual behavior is noted in the rate of heat transfer in both branches(5)The results of stability analysis suggest that upper branch is feasible and stable; the lower branch is unrealizable and unstable


:Aluminum oxide
:Ambient temperature
:Coefficient of mean absorption
:Density of hybrid nanofluid
:Dimensionless temperature
:Dimensionless velocity
:Dynamic viscosity of hybrid nanofluid
T:Fluid temperature
:Transverse velocity
:Local Nusselt number
:Local Reynolds number
ODEs:Ordinary differential equations
PDEs:Partial differential equations
Pr:Prandtl number
:Radiation flux
:Skin friction coefficient
:Unknown eigenvalue
:Smallest eigenvalue
:Heat capacity of hybrid nanofluid
:Stability similarity variable
:Stream function
:Moving parameter
:Constant outward flow velocity
:Thermal conductivity of hybrid nanofluid
:Thermal radiation
:Stefan–Boltzmann constant
:Similarity variable
:Variable temperature at the sheet
x, y, z:Cartesian coordinates
:Velocity components
:Volume fraction

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.


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