Inclined magnetic field and nanoparticle aggregation effects on thermal Marangoni convection in nanoliquid: A sensitivity analysis
Introduction
The improved heat transfer achieved by nanofluids has led to greater efficiency in various industrial processes, heat exchangers, electronic devices, and cooling processes. The nanofluids (introduced by Choi and Eastman [1]) are prepared by dispersing 1–100 nm sized particles (such as metals or metal oxides) in base fluids such as water, ethylene glycol, or oil. Researchers widely use the consideration of nanofluid as a unique homogeneous phase that includes the thermophysical properties of nanofluids to model their characteristics [2]. Sheikholeslami and Vajravelu [3] analyzed the effect of an irregular magnetic source on the flow of nanofluids heated from below. Rostami et al. [4] explored the heat transfer in ethylene-glycol based alumina nanoliquid under magnetic forces in a complex enclosure. The heat transfer rate is reported to improve with an increment of nanoparticle volume fraction. Mebarek-Oudina [5] performed a comparative analysis of the heat transfer of titania NP dispersed in various basic liquids. Because of this, Hosseinzadeh et al. [6] studied the nanofluid flow over a vertical plate for a ferrofluid using different base fluids. The lamina-shaped nanoparticle was reported to have a higher temperature profile for the water-based nanofluid.
Various studies have reported that the nanoparticles (NP) form aggregates in nanofluids (see Fig. 1). He et al. [7] included the aspect of NP aggregation in the heat transfer analysis of titania nanofluid in a pipe. The increase of the viscosity with NP aggregation was reported. Ellahi et al. [8] considered the NP aggregates as chains when the heat transfer of alumina-water nanofluid past a wedge. The NP aggregation of ethylene glycol-based titania nanoliquid was considered by Mackolil and Mahanthesh [9] to explore the heat transfer in Marangoni convective flow. They reported the change in the thermophysical properties with the aggregation of NP. Recently, a comprehensive study on the NP aggregation impact on the radiative properties of nanofluids was conducted by Chen et al. [10].
The temperature variation can lead to surface tension gradients at the interface which in turn induce convection in fluids. This is called thermal Marangoni convection. This is predominantly observed in welding, solvent and oil extraction, crystal growth, and soap film stabilization. Studies on boundary layer flow due to Marangoni convection were initiated by Napolitano [11]. Lin et al. [12] examined the impact of thermal radiation on the Marangoni convective flow of a pseudo-plastic nanofluid. They concluded that the titania NP has a better improvement in heat transfer when compared to and alumina NP. The Marangoni convective flow of a carbon-water nanofluid was studied by including the effect of thermal radiation by Hayat et al. [13]. Some more recent explorations on Marangoni convective flows can be seen in [14], [15], [16] and the references therein.
The heat sources of various types (like THS and ESHS) occur frequently in engineering applications. They enhance the temperature profiles as they add additional energy to the system. Such heat sources play a significant role in the rate of heat transfer. The effect of a variable linear heat source on the nanofluid transport with external thermal radiation and magnetism was explored by Saleem et al. [17]. As expected, the increase in the heat source parameter led to an increment in the temperature profile and a decrement in the concentration profile. Zia et al. [18] studied the radiated 3D Casson liquid flow with an ESHS. The heat transfer rate was reported to decrease with the increment in the ESHS factor. Some recent works on the heat source effects on nanofluid flows can be seen in [19], [20], [21].
Motivated by previous studies, it is noted that the inclined magnetic field effect on the Marangoni convection of a nanoliquid with distinct heat sources and thermal radiation has not yet been studied. The nanoparticle aggregation kinematics aspect of this problem is also to be explored in detail. The sensitivity analysis of the heat transport used to study the interactive effects of pertinent parameters on the heat transfer rate leads to important findings. So, this work aims to:
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Model the thermal Marangoni convection with an inclined magnetic field and distinct heat sources.
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Conduct a comparative analysis of the nanoliquid flow with and without considering nanoparticle aggregation.
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Scrutinize the individual effects of the parameters on the profiles by using the percentage of the augmentation.
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Analyze the effect of various parameters on flow profiles and comparing the results with the limiting case from the literature.
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Explore the interactive impact of the heat sources and thermal radiation on the heat transfer rate through the sensitivity analysis of the Response Surface Methodology (RSM) model.
Section snippets
Mathematical formulation
The 2D steady laminar thermal Marangoni boundary layer flow of titania-ethylene glycol nanoliquid is considered. The fluid is assumed to be optically thick and electrically conducting. The temperature at the surface () is dependent on the space variable and is of the form , where is the ambient temperature, is a positive dimensional constant, and is the exponent of temperature variation. The coordinate system is chosen such that the -axis is along with the fluid and
Similarity transformation
The following equations represent the similarity variables (see [9]) used to convert the governing equations to ordinary differential equations. where,
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Here, denotes the stream function with and ; is the similarity variable, is a constant quantity, is the dimensionless stream function and is the dimensionless temperature. Using (7), the governing equations become:
Thermophysical properties
The thermophysical properties are chosen based on the aggregation property of the NP. In the absence of aggregation, the Brinkman and Maxwell models are employed for effective viscosity and thermal conductivity respectively. Table 1 summarizes the effective properties of nanoliquids.
As shown in Table 1, the Krieger-Dougherty model was modified to include the effect of nanoparticle aggregation. Chen et al. [10] concluded that this model was appropriate for the ethylene glycol-based titania
Numerical solution
Using the substitutions and , the nonlinear boundary value problem in (8) - (10) is converted into the following system of differential equations:
The above system of equations in (13) is solved using the conditions in (14) using the bvp5c routine in MATLAB. This code is
Parametric analysis
The impact of effective parameters on the velocity and temperature profiles are visualized in Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11. The Prandtl number () is fixed at 150.4583, calculated for the base fluid ethylene glycol at 300 K.
Inferences through Response Surface Methodology (RSM)
RSM is a statistical and mathematical method for analyzing the interactive influence of parameters on a response variable. Here, a second-order model is accounted which includes linear, interactive, and quadratic terms as follows:where, represent the regression coefficients.
This section aims to compare the interactive effects of the heat sources on the heat transfer rate. Hence, the response variable is and the parameters are and
Sensitivity analysis
The quadratic model of the coded coefficients is used to find the sensitivity of the heat transfer rate. The quadratic model with the coded coefficients is given below:
Now, the sensitivity functions are calculated by partially differentiating the quadratic model with respect to the coded variables as follows:
The nature of the
Concluding remarks
The following are the major conclusions drawn from this study:
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The models with aggregation yield a higher temperature profile than that using the homogeneous model.
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The heat source effects and the thermal radiation enhance the temperature profile.
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The angle of the inclination of the magnetic field has a strong diminishing effect on the fluid velocity.
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The heat transfer rate is positively sensitive towards thermal radiation and negatively sensitive towards the linear and exponential heat sources.
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Declaration of Competing Interest
The authors declare no conflict of interest in this manuscript.
Acknowledgement
The authors are indebted to the management of CHRIST (Deemed to be University). The authors also thank the Editor and the anonymous reviewers for their valuable comments to improve this article.
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