The homogeneousheterogeneous reaction in the boundary layer flow of a waterbased nanofluid in the stagnationpoint region of a plane surface is investigated. The type of small particles explored here is the singlewalled carbon nanotubes. The homogeneous nanofluid model is employed for description of behaviours of nanofluids. Here, the homogeneous (bulk) reaction is isothermal cubic autocatalytic, while the heterogeneous (surface) reaction is single, isothermal, and first order. The steady state of this system is analysed in detail, with equal diffusion coefficients being considered for both reactants and autocatalysts. Multiple solutions of the reduced system are captured for some particular sets of physical parameters, which seem to be overlooked in all previous published works with regard to studies of homogeneousheterogeneous reactions modeled by homogeneous nanofluid models. Besides, we discover the significant limitation of previous conclusion about that the solutions by homogeneous nanofluid flow models can be recovered from those by regular fluids.
Chemically reacting systems in liquids or biochemical systems such as catalysis and burning admit both homogeneous and heterogeneous reactions. However, the correlation between homogeneous and heterogeneous reactions is rather complex to handle. The reaction heat that is generated or absorbed during the chemical reactions has strong influence on the flow and heat transfer in the surrounding fluid that in turn affects the local concentrations of the reactants and the products and eventually the performance of the chemical reactions.
Motivated by experimental observations of Williams et al. [
It has been known that heat transfer rate of base fluids could be enhanced significantly as highly conductive solid particles are added into them [
The suspended nanoparticles in nanofluids can be either metal, metal oxide, carbon nanotubes, or the combinations of them. Recently, carbon nanotubes (CNTs) have attracted much attention since they were discovered by Iijima [
As aforementioned, chemical reactions in liquids are complex, and different reaction rates can happen even under the same conditions. From the modelling point of view, this means that the solutions of the theoretical model for the chemical reaction of interest may not be unique. Though a few studies on homogeneousheterogeneous chemical reactions in the boundary layer flow and heat transfer based on the homogeneous nanofluid model [
Consider a steady boundary layer flow and heat transfer of a SWCNTnanofluid in the stagnationpoint region towards a plane surface in the presence of homogeneousheterogeneous chemical reactions. The homogeneousheterogeneous reaction model proposed by Chaudhary and Merkin [
In the aforementioned equations,
It is known that the homogeneous flow model [
Thermophysical characteristics of base fluid and SWCNTs.
Physical characteristics  Base fluid (water)  Nanoparticles (SWCNTS) 


997.1  2600 

4179  425 

0.613  6600 
Correspondingly, the governing equations describing the conservations of total mass, momentum, and energy, as well as chemical reaction diffusion, in the presence of singlewall carbon nanotubes (SWCNTs), in the framework of the boundarylayer approximations, are written as
In the studies of flow and heat transfer in the boundary layers, it is a common practice to introduce the boundary layer assumptions [
We utilize the same idea to handle the problem considered in this work. In doing so, we introduce the following similarity variables:
Substituting the similarity variables (
In many practical applications, the diffusion of species
Using equations (
It is worth mentioning to this end that Magyari [
Substituting equation (
To resume the governing equations to Newtonian fluid’s ones, it must hold
From equation (
However, from equation (
If only the flow field is considered, then the fixed relation between homogeneous modelling nanofluid and regular fluid’s solutions is readily discovered from correlation (
Solutions of governing equations (
It has been already known from Chaudhary and Merkin [
We start our discussion by considering the variation of
The bifurcation points for possible ranges of
Minimum and maximum values (say
Bifurcation point 





2.696  2.659  — 

14.922  6.505  — 
We then consider the effect of the nanoparticle volume fraction
Minimum and maximum values of
Bifurcation point 





2.723  2.659  2.568 

6.738  6.505  6.174 
The influence of
It has been reported by Chaudhary and Merkin [
(a) Concentration profile
From equation (
Using equation (
Here, the prime denotes the differentiation with respect to
Equation (
Substituting equation (
Taking boundary condition (
As briefly discussed in [
Using the method proposed by Billingham and Nadeem [
At
Since equation (
However, as concluded by Chaudhary and Merkin [
Note that this relation still holds when the effect of nanofluids is considered.
It is worth mentioning to this end that the nanofluid is considered in our case; therefore, the nanoparticle volume fraction plays a role on
Autoignition takes place at
As shown in Figure
Autoignition: (a) a graph of
Particularly, for
By solving this equation numerically, we obtain
For example, asymptotic expression (
As expected, the critical value


Chaudhary and Merkin [  




 
0.3  3.413  3.358  3.277  
0.5  3.1589  3.0989  3.0108  
1  2.783  2.719  2.627  2.785 
1.5  2.556  2.492  2.401  
2.5  2.273  2.211  2.123 
It is obvious that
To check the significance of
Equation (
From equation (
However, equation (
The hysteresis bifurcation point can then be calculated by solving equation (
As shown in Figure
Graphs of (a)
In this case, the reaction region is thicker than the region of the boundary layer flow. Since the thick outer region is of extent
Substituting equation (
In the inner boundary layer region, we expand
At the order of
On the contrary, based on equations (
Note that the inner region has to satisfy the boundary conditions (
The solution of this equation is
Thus, the appropriate inner condition for equation (
The solutions of equation (
Equation (
The solution at the leading order is simplified as
However, at the order of
To balance the outer region with the inner boundary condition, we obtain
We, therefore, are able to find the solution of equation (
It is now expected that equation (
The homogeneousheterogeneous reactions in a nanofluid with the suspended singlewalled carbon nanotubes at the stagnationpoint region of a plane surface have been investigated. The homogeneous (bulk) reaction is assumed to be isothermal cubic autocatalytic, while the heterogeneous (surface) reaction is single, isothermal, and first order. The homogeneous nanofluid model is employed with equal diffusion coefficients being considered for both reactants and autocatalysts. By doing similarity transformation, multiple solutions of the model have been captured for particular sets of physical parameters, which are overlooked in previous publications on modelling homogeneousheterogeneous reactions. Furthermore, we have discovered the limitation of Magyari’s conclusion [
In summary, the main novel aspects of this study are as follows:
Multiple solutions for modelling homogeneousheterogeneous reactions are found for some sets of physical parameters, and their origins are discussed
Homogeneous and heterogeneous reaction rates
Nanoparticle volume fraction
Magyari’s conclusion [
The data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.
This work was partially supported by the National Natural Science Foundation of China (Grant no. 11872241). This work was supported in part by the Australian Research Council through the Centre of Excellence grant CE140100003 to QS.