Thermal Transport in Nonlinear Unsteady Colloidal Model by Considering the Carbon Nanomaterials Length and Radius

: Thermal transport analysis in colloidal suspension is signiﬁcant from industrial, engineering, and technological points of view. It has numerous applications comprised in medical sciences, chemical and mechanical engineering, electronics, home appliances, biotechnology, computer chips, detection of cancer cells, microbiology, and chemistry. The carbon nanomaterials have signiﬁcant thermophysical characteristics that are important for thermal transport. Therefore, the thermal transport in H 2 O composed by single and multiwalled carbon nanotubes is examined. The length and radius of the nanomaterials is in range of 3 µ m ≤ L* ≤ 70 µ m and 10 nm ≤ d ≤ 40 nm, respectively. The problem is modelled over a curved stretching geometry by inducing the velocity slip and thermal jump conditions. The coupling of Runge-Kutta (RK) and shooting technique is adopted for the solution. From the analysis it is perceived that the heat transfer at the surface drops for stretching. The heat transfer rate prevailed for Single walled carbon nanotubes SWCNTs-H 2 O colloidal suspension. The suction and stretching of the surface resist the shear stresses and more shear stress trends are investigated for larger curvature.


Introduction
The heat transfer analysis in the nanofluids attained much interest of researchers, engineers, and industrialists. Nanofluids are newly engineered fluids with potential heat transfer characteristics. A fluid composed by the tiny particles of the metals and their oxides and host liquids is called a nanofluid. Thermal conductivity is significant in the nanofluids for heat transfer due to high thermal conductance of the tiny particles of various metals and oxides. Therefore, Maxwell [1] thought that thermal transport characteristics could be enhanced and proposed a theoretical thermal conductance correlation. for micropolar fluid flow over an arched stretching/shrinking surface with permeable effects. 3D flow for stretching plate, flow over stretching cylinder, and boundary layer study over solid surface were described in [26][27][28], respectively.
It is clear that the analysis of thermal transport by incorporating the length and radius effects of carbon nanomaterials in the host liquid has not been attempted so far. Therefore, this analysis was carried out to fill this significant gap in the research field. The effects of velocity are incorporated in the flow conditions and the model treated numerically. Further, the results for the nanofluid velocity behavior, local thermal transport, and shear stresses for various flow parameters are decorated and discussed comprehensively in the presented analysis.

Statement and Geometry of the Model
The colloidal composition of CNTs-H 2 O nanofluid is considered over an arched surface with stretching property in curvilinear frame. It is assumed that the nanofluid obeys the characteristics of incompressibility. The radius of the surface is denoted by R and it is placed in a (r, s) coordinate system. The surface has stretching or shrinking properties with velocity U w which varies along the surface. Moreover, the velocity V w depends on s and t represents the injecting fluid for V w > 0 and suction of the fluid for V w < 0. Figure 1 elaborates the flow situation for the consideration nanofluid.
Energies 2020, 13, x FOR PEER REVIEW 3 of 15 regimes for micropolar fluid flow over an arched stretching/shrinking surface with permeable effects. 3D flow for stretching plate, flow over stretching cylinder, and boundary layer study over solid surface were described in [26][27][28], respectively. It is clear that the analysis of thermal transport by incorporating the length and radius effects of carbon nanomaterials in the host liquid has not been attempted so far. Therefore, this analysis was carried out to fill this significant gap in the research field. The effects of velocity are incorporated in the flow conditions and the model treated numerically. Further, the results for the nanofluid velocity behavior, local thermal transport, and shear stresses for various flow parameters are decorated and discussed comprehensively in the presented analysis.

Statement and Geometry of the Model
The colloidal composition of CNTs-H2O nanofluid is considered over an arched surface with stretching property in curvilinear frame. It is assumed that the nanofluid obeys the characteristics of incompressibility. The radius of the surface is denoted by R and it is placed in a (r, s) coordinate system. The surface has stretching or shrinking properties with velocity Uw which varies along the surface. Moreover, the velocity Vw depends on s and t represents the injecting fluid for Vw > 0 and suction of the fluid for Vw < 0. Figure 1 elaborates the flow situation for the consideration nanofluid.

Governing Model and Similarity Transformations
The following unsteady nondimensional flow model represents the nanofluid flow of CNTs-H2O over a curved surface [29].

Governing Model and Similarity Transformations
The following unsteady nondimensional flow model represents the nanofluid flow of CNTs-H2O over a curved surface [29].
The conditions at the surface and at the free surface are: for any r and s and t < 0 In Equation (6), the velocities U w = as (1 − αt) −1 , V w = −aν f (1 − αt) −1 S are s and t dependent, surface stretching and shrinking is represented by λ < 0 and λ > 0, respectively. The accelerated and decelerated flow situations depend on the value of α. The positive α shows the accelerating and negative α corresponds to decelerating flow situation. Further, suction or injection of the nanofluid denoted by S and L 1 * and L 2 * are the temperature jump and velocity slip parameters, respectively.
For nondimensionalization of the model, the following are defined similarities [29]:

Effective Nanofluid Models
To improve the thermophysical characteristics, the following models are utilized [20]: k nf =k where,ρ nf is the density, ρ C p nf heat capacitance,μ nf , dynamic viscosity, andk nf is the effective thermal conductivity of the nanofluid. Further,ρ f ,μ f ,k f are the density, dynamic viscosity, and thermal conductivity of the fluid phase andk s is thermal conductivity of the carbon nanotubes. The volume fraction is φ and the length and radius of the Carbon nanotubes (CNTs) are L and R, respectively. The thermal and physical characteristics of the fluid phase and carbon nanotubes are described in Table 1. With the help of Equation (11), Equations (1)-(4) are transformed into the following model in the presence of length and radius of the carbon nanotubes incorporating in the energy equation: The boundary conditions described in Equations (5) and (6) after the implementation of the similarity transformation reduced into the following version: Energies 2020, 13, 2448 6 of 14

Significant Quantities from Engineering Aspects
The shear stresses and local Nusselt number can be defined by the formulas: Here, the wall shear stress and the wall heat flux are denoted by τ w and q w , respectively. These are as follows: Finally, the self-similar form for shear stress and local heat transfer are obtained in the following way:

Mathematical Analysis
Consideration of SWCNTs-H 2 O and MWCNTs-H 2 O is a tedious, highly nonlinear, and coupled system of ODEs. In such situations, the analytical solutions are not reliable or do not even exist. Therefore, numerical techniques are better to tackle the model. Thus, the coupling of RK and shooting techniques ( [14,15,30]) is adopted to capture the behavior of nanofluids flow regimes for multiple physical quantities. In ordered to initiate the algorithm, the following transformations are required depending upon the order of the self-similar momentum and energy equations described in Equations (12) and (13) along with boundary domains embedded in Equations (14) and (15). These transformations reduced the model into first order initial value problem which is then solved: From Equation (19), From Equations (19) and (20), we obtained; In Equation (21), L and N are obtained by rearranging Equations (12) and (13) as below; Energies 2020, 13, 2448 Or, Now, the resultant initial value problem is attained by including the values from Equations (19) and (21) in Equations (22) and (24). Then, this was solved successfully by means of Mathematica software 10.0.

The Velocity Distribution
This section describes the results for velocity, temperature, shear stresses, and local heat transfer rate by altering the parameters concealed in the velocity and energy equation. How the volume fraction affects the thermal and physical characteristics of the host fluid and carbon nanotubes is also explored. Furthermore, the impacts of the length and radius of the carbon nanotubes are investigated for the enhancement of thermal conductivity.
The behavior of velocity against suction S > 0 and stretching and shrinking of the surface (λ > 0 and λ < 0) are captured in Figure 2. It is perceived that the nanofluid velocity declines for a larger stretching parameter. Physically, larger stretching surface parameters produce more free space and the nanofluid particles are scattered at the surface therefore the velocity L'(η) drops. The decreasing behavior of MWCNTs-H 2 O is quite abrupt in comparison with SWCNTs-H 2 O. The reason behind these trends in the velocity are their densities. The nanofluid velocity asymptotically vanishes apart from the surface. On the other side, the velocity L'(η) enhances when the surface shrinks. Physically, when the surface shrinks then the curvature decreases and, consequently, the velocity increases. This behavior of the velocity L'(η) is shown in Figure 2a.
The influences of surface curvature on the nanofluid velocity are captured in Figure 2b. It is perceived that the nanofluid velocity L'(η) enhances for more stretching and larger curvature of the surface. Physically, the fluid particles adjacent to the surface drag when the surface stretches, therefore the velocity of the nanofluid increases. For smaller curvature, slow increment in the velocity behavior is captured. The velocity of SWCNTs-H 2 O and MWCNTs-H 2 O nanofluids asymptotically vanishes beyond η ≥ 4.0. the nanofluid particles are scattered at the surface therefore the velocity L'(η) drops. The decreasing behavior of MWCNTs-H2O is quite abrupt in comparison with SWCNTs-H2O. The reason behind these trends in the velocity are their densities. The nanofluid velocity asymptotically vanishes apart from the surface. On the other side, the velocity L'(η) enhances when the surface shrinks. Physically, when the surface shrinks then the curvature decreases and, consequently, the velocity increases. This behavior of the velocity L'(η) is shown in Figure 2a. The influences of surface curvature on the nanofluid velocity are captured in Figure 2b. It is perceived that the nanofluid velocity L'(η) enhances for more stretching and larger curvature of the surface. Physically, the fluid particles adjacent to the surface drag when the surface stretches, therefore the velocity of the nanofluid increases. For smaller curvature, slow increment in the velocity behavior is captured. The velocity of SWCNTs-H2O and MWCNTs-H2O nanofluids asymptotically vanishes beyond η ≥ 4.0.
The velocity of SWCNTs-H2O and MWCNTs-H2O against λ1 for stretching and shrinking is painted in Figure 3. The reversal behavior of the velocity is examined for higher values of λ1. Further, it is noted that the nanofluid flows slowly for stretching while abrupt increasing trends are seen for the shrinking case. Asymptotic behavior of the velocity L'(η) is detected beyond η ≥ 5.0 for both SWCNTs-H2O and MWCNTs-H2O nanofluids.

Streamlines and Thermophysical Characteristics
The influence of parameters concealed in the streamlines pattern and the effective thermal conductivity for CNTs, effective density, and heat capacitance are incorporated in this subsection. The parameter α > 0 highlights the accelerated flow and decelerated flow corresponds to negative α. Figure 4 portrays the flow pattern for accelerated and decelerated cases by keeping the surface curvature k = 4. In Figure 4a, the streamlines pattern is parabola opening downward over a curved surface. The flow is accelerated due to positive α. The streamlines expand at the free surface. It is investigated that there is no intersection of the streamlines which elaborates the phenomena of laminar flow. Figure 4b describes the flow pattern for the decelerated case. In the decelerated situation, the streamlines stretch towards a curved surface. The streamlines in the vicinity of the surface are exactly a parabolic shaped and far from the surface these become steeper. The streamlines behavior for larger surface curvature is depicted in Figure 5 for both accelerated and decelerated situations. The surface has the property of stretching which increases the surface curvature. Due to large curvature, the nanofluids flow over the surface along large curve which shrinks the streamlines pattern. An almost similar flow pattern is investigated for k = 8 in comparison with k = 4 for the decelerated case.

Streamlines and Thermophysical Characteristics
The influence of parameters concealed in the streamlines pattern and the effective thermal conductivity for CNTs, effective density, and heat capacitance are incorporated in this subsection. The parameter α > 0 highlights the accelerated flow and decelerated flow corresponds to negative α. Figure 4 portrays the flow pattern for accelerated and decelerated cases by keeping the surface curvature k = 4. In Figure 4a, the streamlines pattern is parabola opening downward over a curved surface. The flow is accelerated due to positive α. The streamlines expand at the free surface. It is investigated that there is no intersection of the streamlines which elaborates the phenomena of laminar flow. Figure 4b describes the flow pattern for the decelerated case. In the decelerated situation, the streamlines stretch towards a curved surface. The streamlines in the vicinity of the surface are exactly a parabolic shaped and far from the surface these become steeper. The streamlines behavior for larger surface curvature is depicted in Figure 5 for both accelerated and decelerated situations. The surface has the property of stretching which increases the surface curvature. Due to large curvature, the nanofluids flow over the surface along large curve which shrinks the streamlines pattern. An almost similar flow pattern is investigated for k = 8 in comparison with k = 4 for the decelerated case.
Energies 2020, 13, 2448 9 of 14 situation, the streamlines stretch towards a curved surface. The streamlines in the vicinity of the surface are exactly a parabolic shaped and far from the surface these become steeper. The streamlines behavior for larger surface curvature is depicted in Figure 5 for both accelerated and decelerated situations. The surface has the property of stretching which increases the surface curvature. Due to large curvature, the nanofluids flow over the surface along large curve which shrinks the streamlines pattern. An almost similar flow pattern is investigated for k = 8 in comparison with k = 4 for the decelerated case.          The impacts of length, radius, and volume fraction of CNTs on thermal conductivity of SWCNTs and MWCNTs are shown in Figure 8. High volume fraction of CNTs increases thermal conductivity of SWCNTs and MWCNTs. It is investigated that thermal conductivity for MWCNTs arises very rapidly in comparison with SWCNTs. The volume fraction φ is in direct proportion to thermal conductivity of CNTs. Like volume fraction, the length of the carbon nanotubes also favors the thermal conductivity for both types of carbon nanotubes. Less thermal conductivity is observed for a smaller length of carbon nanotubes. The radius of carbon nanotubes opposes the thermal conductivity. The carbon nanotubes having a larger diameter opposes the thermal conductivity. However, thermal conductivity of the carbon nanotubes can be enhanced for smaller diameters. The impacts of length, radius, and volume fraction of CNTs on thermal conductivity of SWCNTs and MWCNTs are shown in Figure 8. High volume fraction of CNTs increases thermal conductivity of SWCNTs and MWCNTs. It is investigated that thermal conductivity for MWCNTs arises very rapidly in comparison with SWCNTs. The volume fraction ϕ is in direct proportion to thermal conductivity of CNTs. Like volume fraction, the length of the carbon nanotubes also favors the thermal conductivity for both types of carbon nanotubes. Less thermal conductivity is observed for a smaller length of carbon nanotubes. The radius of carbon nanotubes opposes the thermal conductivity. The carbon nanotubes having a larger diameter opposes the thermal conductivity. However, thermal conductivity of the carbon nanotubes can be enhanced for smaller diameters. The volume fraction which has a central role in the study of nanofluids is very significant for heat capacity of the nanofluids. These influences are shown in Figure 9. For smaller values of ϕ, heat capacity rises very rapidly. The high-volume fraction opposes the effective heat capacity of the carbon nanotubes. The relation of ϕ and effective density of the carbon nanotubes is in direct proportion. The density of the carbon nanotubes increases for high volume fraction. These effects are elucidated in Figure 10. The volume fraction which has a central role in the study of nanofluids is very significant for heat capacity of the nanofluids. These influences are shown in Figure 9. For smaller values of φ, heat capacity rises very rapidly. The high-volume fraction opposes the effective heat capacity of the carbon nanotubes. The relation of φ and effective density of the carbon nanotubes is in direct proportion. The density of the carbon nanotubes increases for high volume fraction. These effects are elucidated in Figure 10.

Local Heat Transfer Rate and Skin Friction
Local Nusselt number or simply local heat transfer rate and skin friction are important quantities from an industrial point of view. The significance of these quantities cannot be denied due to industrial usage. Figure 11 depicts the heat transfer rate versus curvature, suction of the nanofluid, and stretching of the surface. The heat transfers at the surface drops for larger curvature and enhances for smaller curvature. For SWCNTs-H2O, nanofluid is better for more heat transfer. The larger impacts were inspected for suction of the nanofluid. For higher suction of the nanofluids, more fluid particles transfer at the surface which intensify the heat at the surface. The intensifications in the heat transfer are almost alike. The heat transfer and stretching (λ > 0) of the surface are in assisting condition. For a more stretched surface, there is a larger amount of the heat flow at the surface. The more escalations in the heat transfer were observed for the SWCNTs-H2O nanofluid.

Local Heat Transfer Rate and Skin Friction
Local Nusselt number or simply local heat transfer rate and skin friction are important quantities from an industrial point of view. The significance of these quantities cannot be denied due to industrial usage. Figure 11 depicts the heat transfer rate versus curvature, suction of the nanofluid, and stretching of the surface. The heat transfers at the surface drops for larger curvature and enhances for smaller curvature. For SWCNTs-H2O, nanofluid is better for more heat transfer. The larger impacts were inspected for suction of the nanofluid. For higher suction of the nanofluids, more fluid particles transfer at the surface which intensify the heat at the surface. The intensifications in the heat transfer are almost alike. The heat transfer and stretching (λ > 0) of the surface are in assisting condition. For a more stretched surface, there is a larger amount of the heat flow at the surface. The more escalations in the heat transfer were observed for the SWCNTs-H2O nanofluid.

Local Heat Transfer Rate and Skin Friction
Local Nusselt number or simply local heat transfer rate and skin friction are important quantities from an industrial point of view. The significance of these quantities cannot be denied due to industrial usage. Figure 11 depicts the heat transfer rate versus curvature, suction of the nanofluid, and stretching of the surface. The heat transfers at the surface drops for larger curvature and enhances for smaller curvature. For SWCNTs-H 2 O, nanofluid is better for more heat transfer. The larger impacts were inspected for suction of the nanofluid. For higher suction of the nanofluids, more fluid particles transfer at the surface which intensify the heat at the surface. The intensifications in the heat transfer are almost alike. The heat transfer and stretching (λ > 0) of the surface are in assisting condition. For a more stretched surface, there is a larger amount of the heat flow at the surface. The more escalations in the heat transfer were observed for the SWCNTs-H 2 O nanofluid. The fluctuations in the shear stresses by altering curvature, suction of the nanofluids, and stretching parameters are elucidated in Figure 12. The shear stresses escalate for the less stretched surface. Higher stretching surfaces lead to drops in the shear stresses. Similarly, the skin friction drops for more suction. The variations in the skin friction for SWCNTs-H2O are rapid in comparison with the MWCNTs-H2O nanofluid. The curvature parameters and the friction are inversely proportional to each other. By decreasing the curvature of the surface, the skin friction escalated, and dominant behavior is observed for the SWCNTs-H2O nanofluid.

Conclusions
Thermal transport investigation in colloidal suspension of water suspended by carbon nanomaterials was examined over curved geometry. To improve the thermal transportation in the The fluctuations in the shear stresses by altering curvature, suction of the nanofluids, and stretching parameters are elucidated in Figure 12. The shear stresses escalate for the less stretched surface. Higher stretching surfaces lead to drops in the shear stresses. Similarly, the skin friction drops for more suction. The variations in the skin friction for SWCNTs-H2O are rapid in comparison with the MWCNTs-H2O nanofluid. The curvature parameters and the friction are inversely proportional to each other. By decreasing the curvature of the surface, the skin friction escalated, and dominant behavior is observed for the SWCNTs-H2O nanofluid.

Conclusions
Thermal transport investigation in colloidal suspension of water suspended by carbon nanomaterials was examined over curved geometry. To improve the thermal transportation in the

Conclusions
Thermal transport investigation in colloidal suspension of water suspended by carbon nanomaterials was examined over curved geometry. To improve the thermal transportation in the nanofluid, a thermal conductivity correlation comprising the effects of length and radius of carbon nanomaterials was plugged into the energy equation. Then, the colloidal model was treated numerically and captured the results for the velocity, local thermal transport, and shear stresses against multiple physical parameters. It was found that: • The stretching of the surface provides more flowing area, due to which the nanofluid particles scattered at the surface and, consequently, the velocity of the nanofluids declines over the surface.

•
The nanofluids flow abruptly over a shrinking surface and asymptotically vanish apart from the surface.

•
The greater heat transfer rate at the surface is perceived for the suction and stretching/shrinking parameters.

•
The decreasing trends in the shear stresses over the curved surface are examined for larger curvature, whereas they increase for the stretching/shrinking and suction parameters. Funding: This research received no external funding.

Conflicts of Interest:
There is no conflict of interest regarding this publication.