Heat and Mass Transfer Analysis of a Fluid Flow across the Conical Gap of a Cone-Disk Apparatus under the Thermophoretic Particles Motion

Abstract

This particular study focuses on investigating the heat and mass transport characteristics of a liquid flow across the conical gap (CG) of a cone-disk apparatus (CDA). The cone and disk may be taken as stationary or rotating at varying angular velocities. Consideration is given to heat transport affected by solar radiation. The Rosseland approximation is used for heat radiation calculations in the current work. To observe the mass deposition variation on the surface, the effect of thermophoresis is taken into account. Appropriate similarity transformations are used to convert the three-dimensional boundary-layer governing partial differential equations (PDEs) into a nonlinear ordinary differential equations (ODEs) system. Particularly for the flow, thermal and concentration profiles, plots are provided and examined. Results reveal that the flow field upsurges significantly with upward values of Reynolds numbers for both cone and disk rotations. The increase in values of the radiation parameter improves heat transport. Moreover, it is detected that the stationary cone and rotating disk model shows improved heat transport for an increase in the values of the radiation parameter.

1. Introduction

The cone-disk apparatus (CDA) has several applications, such as the stability scrutiny of the creeping flow of Oldroyd-B liquid [1], fluid viscosity estimation [2], and applications of bioengineering for propagating endothelium cells placed as a monolayer on the plate [3]. Cone-disk devices, where liquid flow occurs in a CG at angles of 1 to 5 degrees. The current research also focuses on the flow, mass, and heat passage in the CG between a CDA, co-counter rotating or both stationary at varied angular speeds. Shevchuk [4] was initially provided the self-similar versions of energy and Navier–Stokes equations (NSEs) for a heat transport and laminar flow in a CDA. The flow of the nanofluid (NF) was studied by Basavarajappa and Bhatta [5] in a CDA by pondering the suitable rotational Reynolds numbers and Buongiorno model. Wang et al. [6] scrutinized the impact of radiation on the NF flow among the CG of a CDA. Here, their research suggests that for higher values of the radiation parameter, the dynamics of flow with nanoparticle aggregation instance exhibits better-quality heat transmission. From the perspective of the stabilization of thermal energy, Alrabaiah et al. [7] addressed the microbes-based flow of hybrid NF using the CDA. Here, they discovered that, while the temperature at the outside edge is static, a circulating disk and a stationary cone may effectively cool the CDA. The flow of the hybrid NF in the conical space of CDA was examined by Gul et al. [8]. They stated that, for a revolving disk with a stationary cone, the required cooling of CDA may be achieved while the surface temperature stays constant.

A phenomenon of small-particle migration in the direction of a reducing thermal gradient is known as thermophoresis. The force felt by the suspended particle is referred to as thermophoretic force, and the velocity the particle acquires is referred to as thermophoretic velocity [9,10,11]. Small particles accumulate on the cold surfaces as a result of thermophoresis. Gowda et al. [12] explored the thermophoretic particle deposition (TPD) in unsteady hybrid NF flow over a revolving disk. According to their results, an increase in thermophoretic coefficient and parameter values causes a decrease in the mass transfer rate. Khan et al. [13] examined the interaction between TPD and the radiation effect along a stretched sheet in a two-dimensional thin-film stream of second-grade liquid. The TPD impact on the Glauert wall jet slip flow of NF was induced by Alhadhrami et al. [14]. Their results indicate that an increase in the heat transfer rate may be achieved by increasing the values of the radiation parameter and volume fraction simultaneously. Increasing the values of the thermophoretic parameter results in an increase in the rate of mass transfer. Abbas et al. [15] explored the combination of TPD and radiation effects on the convective flow of optically dense gray fluid. Using the TPD, Shah et al. [16] analyzed the effect of convective heating on the second-grade fluid flow past an upright surface.

The heat and mass transport over the surface of a rotating disk with the fluid flow has attracted a considerable amount of consideration from researchers. Owing to the fact that it has several applications in technical and aerospace sciences, including thermal energy-producing systems, rotational equipment, steam turbine deltoids, the geothermal sector, and medical equipment. Karman [17], in his paper on the solution of NSEs by making use of the proper transformation, presented the concept of liquid flow across a rotating system for the very first time. Turkyilmazoglu [18] examined flow and heat transport using CDA. Shevchuk [19] studied the laminar heat and mass transference in a CDA. The flow of an NF caused by a revolving disk was studied by Tassaddiq et al. [20]. Turkyilmazoglu and Senel [21] conferred the heat and mass transference of the flow that occurred across a spinning, porous disk.

Thermal radiation is significant in satellites, aircraft missiles, gas turbines, spacecraft, and nuclear power plants. Shevchuk [22] investigated the effect of radiation on a self-similar solution across spinning CDA. The effects of heat radiation on NF flow were inspected by Kumar et al. [23]. Their results indicate that an increase in the values of the radiation parameter produces an improvement in the thermal profile as a result of the generation of internal heat. Kumar et al. [24] analyzed magnetic liquid flow when heat radiation occurred from an stretched sheet. In this study, the researchers primarily concentrate on a magnetic fluid that is electrically non-conducting, incompressible, which has a modest saturation magnetization, and has a low Curie temperature. Alzahrani et al. [25] investigated the effect of radiation on the plane wall jet flow of Casson NF. In this study, the researchers conclude that an increase in the Casson parameter’s value would result in a decrease in the heat transfer rate assumed for nanoliquids, while an increase in the radiation parameter’s value would show the opposite tendency. Jayaprakash et al. [26] conferred the impact of heat radiation on the NF flow past a sheet. They concluded that an improvement in the heat transfer process may be observed for increased levels of radiation and ferromagnetic interaction parameters.

The abovementioned studies show that no effort has been made to date to scrutinize the impact of TPD on the three-dimensional flow of fluid by considering the radiation effect concerning the disk and cone as moving or stationary, under the influence of thermal radiation. The main priority is to extend the idea of Refs. [5,18], which also consist of the most relevant studies related to the present model, and to develop a mathematical model for the rotating disk and cone, which are considered as moving or stationary, in both cases counter\rotating or co-rotating. Furthermore, a visual representation of the numerical outcomes are presented and discussed. The purpose of this study is to analyze the non-dimensional parameters that affect flow, heat, and mass transfer processes.

2. Mathematical Formulation

The conical gap between a cone and disk was considered, which was filled with an incompressible viscous fluid. The velocity components $\left(u,v,w\right)$ were considered along $\left(r,\phi ,z\right)$ an increasing direction. The axisymmetric flow assumption eliminated tangential component derivatives. It was assumed that both tools were either fixed or rotating at varying angular velocities, as indicated in Figure 1. There was a total of four distinct physical configurations that were considered, (i) including a rotating cone with static disk, (ii) static cone with rotating disk, (iii) the co-rotation of disk and cone, and (iv) counter-rotation of disk and cone (see Refs. [8,18,19]).

The generalized mathematical model, which is presented as follows:

$$\nabla . \overrightarrow{u}=0,$$

(1)

Equation (1) is the continuity equation that shows that the in-flow equals the out-flow.

The conservation of momentum is the following:

$${\rho }_f\left(\left(\overrightarrow{u}. \nabla \right)\overrightarrow{u}\right)=-\nabla p+{\mu }_f{\nabla }^2\overrightarrow{u},$$

(2)

Equation (2) is the momentum equation that represents the conservation law of momentum. The symbols on the left-hand side of Equation (2) result from inertial forces, the first term on the right-hand side is due to the pressure force, and the second term is due to viscous forces.

The conservation of energy is:

$${\left(\rho C_p\right)}_f\left(\left(\overrightarrow{u}. \nabla \right)T\right)=k_f{\nabla }^{2}T,$$

(3)

the conservation of the nanoparticles volume fraction is:

$$\left(\left(\overrightarrow{u}. \nabla \right)C\right)=D_f{\nabla }^{2}C.$$

(4)

Consideration is given to heat transport affected by solar radiation. The Rosseland approximation was used for heat radiation calculations in the current study. Equation (3) takes the following form:

$$\left(u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}\right)=\frac{k_f}{{\left(\rho C_p\right)}_f}\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial z^2}\right)+\frac{16{\sigma }^{*}{T}_{\infty }^3}{{\left(\rho C_p\right)}_{f}3k^{*}}\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial z^2}\right),$$

(5)

Equation (5) represents the conservation law of energy. Here, the last term on the right-hand side shows the additional effect of thermal radiation. To observe the variation in mass deposition on the surface, the effect of TPD was considered. The particle mass flux was negligible to prevent thermophysical processes from having an influence on the main stream speed and temperature field. Because of its low concentration, the particle’s surface concentration was zero. The particles were absorbed by the surface after impact and did not bounce back. Here, Equation (4) takes the following form:

$$\left(u\frac{\partial C}{\partial r}+w\frac{\partial C}{\partial z}\right)=D_f\left(\frac{{\partial }^{2}C}{\partial r^2}+\frac{1}{r}\frac{\partial C}{\partial r}+\frac{{\partial }^{2}C}{\partial z^2}\right)-\frac{\partial }{\partial z}\left(C{W}_T\right)$$

(6)

The boundary conditions are:

$$\begin{array}{l}\left(u,v,w,T\right)=\left(0,\omega r,0,T_w\right),C=C_w=0 \left(\mathrm{fully} \mathrm{absorbing} \mathrm{and} \mathrm{clean} \mathrm{surface}\right) at z=0,\\ \left(u,v,w,T,C\right)=\left(0,\Omega r,0,T_{\infty },C_{\infty }\right) at z=r\tan \gamma .\end{array}\}$$

(7)

The impacts of thermophoresis are accounted to observe mass deposition alternation on surface. The main stream speed and temperature field are unaffected by thermophysical processes because particles mass flux is reasonably small. Only the thermophoretic velocity component that is normal to the surface is significant because the thermal gradient in the z-direction is much greater than that in the r-direction due to the behavior of the boundary layer. The thermophoretic deposition velocity in the z-direction is presented by (see Refs. [27,28,29,30])

$${\mathrm{W}}_T=-\frac{{\nu }_f{k}^{*}}{T}\left(\frac{\partial T}{\partial z}\right)$$

(8)

We remark that only the velocity component presented by (8) is to be considered within the boundary-layer assumptions.

Where the thermophoretic co-efficient $k^{*}$ can take on any value between 0.2 and 1.2.

The similarity variables can be illustrated in the following manner (see Refs. [8,18,19]):

$$u=\frac{{\nu }_{f}F}{r},C=\chi C_{\infty },T=T_{\infty }+\theta (T_w-T_{\infty }),w=\frac{{\nu }_{f}H}{r},v=\frac{{\nu }_{f}G}{r},\eta =\frac{z}{r},p=\frac{{\rho }_f{\nu }_f^2}{r^2}P.$$

which reduces Equations (1)–(7) as follows:

$$H^{\prime }-\eta F^{\prime }=0$$

(9)

$$(1+{\eta }^2)F^{″}+3\eta F^{\prime }+F\left(\eta F^{\prime }+F\right)-H{F}^{\prime }+G^2+2P+\eta P^{\prime }=0,$$

(10)

$$(1+{\eta }^2)G^{″}+3\eta G^{\prime }+\eta F{G}^{\prime }-H{G}^{\prime }=0,$$

(11)

$$(1+{\eta }^2)H^{″}+3\eta H^{\prime }+H-P^{\prime }+FH+\eta F{H}^{\prime }-H{H}^{\prime }=0,$$

(12)

$$\frac{1}{\mathrm{Pr}}\left(1+R\right)\left(\eta {\theta }^{\prime }+\left(1+{\eta }^2\right){\theta }^{″}\right)+\eta {\theta }^{\prime }F-H{\theta }^{\prime }=0,$$

(13)

$$\frac{1}{Sc}\left(\eta {\chi }^{\prime }+\left(1+{\eta }^2\right){\chi }^{″}\right)+\eta {\chi }^{\prime }F-H{\chi }^{\prime }+k^{*}{N}_t{(1-N_t\theta )}^{-1}\left({\theta }^{″}\chi -N_t{(1-N_t\theta )}^{-1}{{\theta }^{\prime }}^2\chi +{\theta }^{\prime }{\chi }^{\prime }\right)=0.\}$$

(14)

The modified boundary constraints are as follows:

$$\begin{array}{l}F\left(0\right)=0,G\left(0\right)={\mathrm{Re}}_{\omega },H\left(0\right)=0,\theta \left(0\right)=1,\chi (0)=0,\\ F\left({\eta }_0\right)=0,H\left({\eta }_0\right)=0,G\left({\eta }_0\right)={\mathrm{Re}}_{\Omega },\theta \left({\eta }_0\right)=0,\chi ({\eta }_0)=1.\end{array}\}$$

(15)

The following is a list of dimensionless parameters for the proposed study:

$${\mathrm{Re}}_{\omega }=\frac{r^2\omega }{{\nu }_f},Sc=\frac{{\nu }_f}{D_f},N_t=\frac{T_{\infty }-T_w}{T_{\infty }},R=\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}{k}_f},\mathrm{Pr}=\frac{{\mu }_f{C}_p}{k_f},{\mathrm{Re}}_{\Omega }=\frac{r^2\Omega }{{\nu }_f}.$$

(16)

The skin-friction coefficients at the disk surface are as follows (see Ref. [5]):

$$\tan \mathrm{gential} \mathrm{skin}-\mathrm{friction}=G^{\prime }(0),$$

(17)

$$\mathrm{radial} \mathrm{skin}-\mathrm{friction}=F^{\prime }(0).$$

(18)

The heat transfer occurring at the surface of the cone and disk are as follows (see Refs. [5,6]):

$$N{u}_d=-\left(1+R\right){\theta }^{\prime }(0),N{u}_c=-\left(1+R\right){\theta }^{\prime }({\eta }_0).$$

(19)

$$S{h}_d=-{\chi }^{\prime }(0),S{h}_c=-{\chi }^{\prime }({\eta }_0).$$

(20)

3. Numerical Procedure

The shooting approach was used to numerically solve the system of nonlinear ODEs (Equations (9)–(14)) and the boundary conditions (Equation (15)). Initially, first-order differential equations were used to solve the two-point boundary value issue (initial value problem). Additionally, a shooting technique was used to predict the initial conditions that were missed. Later, the resulting one was integrated using the RKF-45 approach (see Refs. [23,24,25,26]). Here, a fifth-order Runge–Kutta technique was used together with a fourth-order Runge–Kutta procedure. Subtracting these two numbers reveals the algorithm’s error, which can then be used to determine the appropriate step size. The RKF-45 process algorithm is as follows:

$$\begin{array}{l}{k}_0=F\left(x_k, y_k\right),\\ k_1=F\left(x_k+\frac{1}{4}h, y_k+\frac{h}{4}{k}_0\right),\\ k_2=F\left(x_k+\frac{3}{8}h, y_k+h\left[\frac{3}{32}{k}_0+\frac{9}{32}{k}_1\right]\right),\\ k_3=F\left(x_k+\frac{12h}{13}, y_k+h\left(\frac{1932k_0}{2197}-\frac{7200k_1}{2197}+\frac{7296k_2}{2197}\right)\right),\\ k_4=F\left(x_k+h, y_k+h\left[\frac{439k_0}{216}-8k_1+\frac{3680k_2}{513}-\frac{845}{4104}{k}_3\right]\right),\\ k_5=F\left(x_k+\frac{h}{2}, y_k-\frac{8h}{27}{k}_0+2h{k}_1-\frac{3544}{2565}h{k}_2+\frac{1859}{4104}h{k}_3-\frac{11}{40}h{k}_4\right).\end{array}$$

$$\begin{array}{l}{y}_{k+1}=y_k+\frac{25}{216}h{k}_0+\frac{1408}{2565}h{k}_2+\frac{2197}{4109}h{k}_3-\frac{1}{5}h{k}_4,\\ y_{k+1}=y_k+\frac{16}{135}h{k}_0+\frac{6656}{12825}h{k}_2+\frac{28561}{56430}h{k}_3-\frac{9}{50}h{k}_4+\frac{2}{55}h{k}_5.\end{array}$$

The problem domain was considered as $\left[0,{\eta }_{max}\right]$ in the calculations, where ${\eta }_{max}$. The step size was carefully selected as $\mathsf{\Delta }\eta$ $=0.001$. An assessment of the RKF procedure using the shooting method and bvp4c is graphically presented to further explain the verification of the utilized model. With the aid of MATLAB software, the numerical results were obtained, and the numerical results were validated against previously published work. Additionally, the numerical results of the present study were compared to earlier studies, and the outcomes agreed with one another (see

Table 1. Comparison of the current study’s coefficient of heat transfer rate when R = 0.

Models	$\mathit{R}\mathit{e}$	Shevchuk [19]	Turkyilmazoglu [18]	Basavarajappa and Bhatta [5]	Present Results
I. Rotating cone with a stationary disk.	2463	13.401	13.4006970	13.40069715	13.40069716
	12	0.954	0.95405487	0.95405477	0.954054771
II. Stationary cone with a rotating disk.	2463	15.353	15.3528734	15.35287341	15.35287342
	12	1.041	1.04080471	1.04080467	1.040804672
III. Co-rotating cone and disk.	2463	14.346	14.3466439	14.34664704	14.34664705
	12	1.001	1.00087052	1.00087491	1.000874913
IV. Counter-rotating cone and disk.	2463	14.440	14.4395241	14.43952407	14.43952409
	12	0.989	0.98884832	0.98884832	0.988848323

and

Table 2. Comparison of the current study with previous research for a few limited cases.

Models		Turkyilmazoglu [18]	Basavarajappa and Bhatta [5]	Present Results
I	$N{u}_c$	0.83028093	0.83028103	0.83028104
	$N{u}_d$	1.09328442	1.09328437	1.09328437
II	$N{u}_c$	0.78069847	0.78069848	0.78069849
	$N{u}_d$	1.17198527	1.17198529	1.17198530
III	$N{u}_c$	0.80178312	0.80177021	0.80177022
	$N{u}_d$	1.13538224	1.13540222	1.13540222
IV	$N{u}_c$	0.81339331	0.81339331	0.81339332
	$N{u}_d$	1.12618149	1.12618149	1.12618150

). We repeated the method with several parameter values to fully verify the results of the suggested model.

4. Results and Discussion

This section presents a detailed observation of the impact of dimensionless parameters on velocity, temperature, and concentration fields. Here, the computations to Reynolds numbers were restricted to 12 corresponding to the gap angle $\pi /45$ representing a flow configuration with a small gap. In the event of co-gyrating, the ratio of Reynolds numbers was set to 12, whereas in the case of counter-gyrating, it was set to −12. Throughout the calculations, some of the parameters’ values were fixed as $Pr=7, k^{*}0.5, R=0.5, N_t=0.1,$$Sc=2.5, {\mathrm{Re}}_{\omega }=1, R{e}_{\Omega }=2$. The values of these parameters did not change, unless they are shown in the graphs and tables.

Figure 2a–d exhibits the influence of Reynolds numbers on $G\left(\eta \right)$ for all four models mentioned above. The scenarios for what time the disk was at rest and the revolving cone are shown in Figure 2a. Between the spaces in the disk and cone, the fluid flows. A positive change in cone velocity values improved the velocity profile $G\left(\eta \right)$, although the greatest flow intensity was located around the cone. The flow field presented in Figure 2b,d exhibits its supremacy over $R{e}_{\Omega }$ and $R{e}_{\omega }$, since both the surfaces spin in the same direction and encounter the least resistance. Figure 2c illustrates how, with the least amount of resistance, the co-gyration of the disk and cone efficiently enhances the fluid velocity.

Figure 3a–d shows the effect of $R$ on $\theta \left(\eta \right)$ four different models. Here, the escalation in values of $R$ improves $\theta \left(\eta \right)$ for all four cases. The radiation parameter $R=\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}{k}_f}$ in the energy Equation (20) is demarcated as the ratio of radiation heat contribution relative to thermal conduction. Strengthening the fluid’s thermal diffusivity and accelerating the thermal energy inside the related layer are the effects of increasing the thermal radiation contribution with an increase in R. As a result, as illustrated in the figures, θ(η) increases with an upsurge in R. From Figure 3a–d, it is detected that the stationary cone and rotating disk model shows improved heat transport for an increase in values of $R$. Figure 4a–d and Figure 5a–d show the impact of $N_t$ and $k^{*}$ on mass transport, respectively, for four different models. Here, the increase in values of both $N_t$ and $k^{*}$ declines the mass transport activity in all four cases. According to Figure 4a–d, the mass transport field for $N_t$ decreases as the temperature ratios increase because the thermophoretic force causes more particles to approach the apparatus at higher temperatures, which reduces the concentration further from the cool surface. The concentration profiles decrease as the $k^{*}$ value increases as the suction-like behavior acts on particles for a cold surface in thermophoresis. This supports the regulation of the heat gradient on the microfluidic scale, which is used in microdevices. Figure 6a–d portrays the influence of $Sc$ on $\chi \left(\eta \right)$ for the four different models. An increase in $Sc$ improves $\chi \left(\eta \right)$ in all four cases. The “Schmidt number” refers to a dimensionless number that establishes a bond between momentum and mass diffusivities in order to produce a fluid flow. On a more technical level, these two concepts are referred to as the hydrodynamic thickness and mass transportation layers. The value of $Sc$ corresponds to the highest possible concentration. When there is an increase in $Sc$, an increase in the mass transfer occurs as a result of mass diffusion. The effects of ${\mathrm{Re}}_{\Omega }$ and ${\mathrm{Re}}_{\omega }$ on $G^{\prime }(0)$ and $F^{\prime }(0)$ are displayed in

Table 3. Effects of ${\mathrm{Re}}_{\Omega }$ and ${\mathrm{Re}}_{\omega }$ on $G^{\prime }(0)$ and $F^{\prime }(0)$.

${\mathbf{Re}}_{\mathbf{\Omega }}$	${\mathbf{Re}}_{\mathit{\omega }}$	${\mathit{G}}^{\prime }(0)$	${\mathit{F}}^{\prime }(0)$
$12$	$11.8812$	$0.4581$	$58.104$
$-12$	$12$	$-65.8021$	$15.2682$
$0$	$12$	$-26.7494$	$14.3176$
$12$	$0$	$39.1299$	$35.1944$

. The effects of ${\mathrm{Re}}_{\Omega }$, ${\mathrm{Re}}_{\omega }$, and $R$ on $N{u}_d$ and $N{u}_c$ are displayed numerically in

Table 4. Effects of ${\mathrm{Re}}_{\Omega }$, ${\mathrm{Re}}_{\omega }$, and $R$ on $N{u}_d$ and $N{u}_c$.

${\mathbf{Re}}_{\mathbf{\Omega }}$	${\mathbf{Re}}_{\mathit{\omega }}$	$\mathit{R}$	$\mathit{N}{\mathit{u}}_{\mathit{d}}$	$\mathit{N}{\mathit{u}}_{\mathit{c}}$
$2$	$1$	$0$	$1.4871$	$0.6545$
		$0.1$	$1.61348$	$0.73106$
		$0.2$	$1.73988$	$0.80772$
		$0.3$	$1.86641$	$0.88452$
		$0.4$	$1.99304$	$0.96124$
$12$	$11.8812$	$0.5$	$11.7474$	$0.00015$
$-12$	$12$	$0.5$	$6.459$	$0.01155$
$0$	$12$	$0.5$	$3.8691$	$0.5757$
$12$	$0$	$0.5$	$9.03975$	$0.00135$

. Here, the increase in $R$ improves the heat transfer rate on both surfaces. Moreover, the fluid flow over disk surface shows a better-quality heat transference rate than the fluid flow over the cone surface for an increase in $R$ values.

Table 5. Effects of ${\mathrm{Re}}_{\Omega }$, ${\mathrm{Re}}_{\omega }$, $N_t$, and $k^{*}$ on $S{h}_d$ and $S{h}_c$.

${\mathbf{Re}}_{\mathbf{\Omega }}$	${\mathbf{Re}}_{\mathit{\omega }}$	${\mathit{N}}_{\mathit{t}}$	${\mathit{k}}^{*}$	$\mathit{S}{\mathit{h}}_{\mathit{d}}$	$\mathit{S}{\mathit{h}}_{\mathit{c}}$
$2$	$1$	$0.1$		$-1.2815$	$-0.767$
		$0.2$		$-1.2412$	$-0.7925$
		$0.3$		$-1.1932$	$-0.8221$
		$0.4$		$-1.1356$	$-0.8564$
			$0.1$	$-1.2105$	$-0.7965$
			$0.2$	$-1.1135$
			$0.3$	$-1.0243$	$-0.8981$
			0.4	−0.9422	−0.9483

displays the effects of ${\mathrm{Re}}_{\Omega }$, ${\mathrm{Re}}_{\omega }$, $N_t$, and $k^{*}$ on $S{h}_d$ and $S{h}_c$. Here, it is clear from the table that the increase in values of $N_t$ and $k^{*}$ improves $S{h}_d$ but decreases $S{h}_c$. Moreover, the fluid flow over the cone surface shows a better-quality mass transference rate than the fluid flow over the disk surface for an increase in the values of $N_t$ and $k^{*}$.

5. Conclusions

The heat and mass transport characteristics of a liquid flow with TPD across the conical gap of a CDA was analyzed in this study. The disk and cone may be taken as rotating or stationary at varying angular velocities. The Rosseland approximation was used for heat radiation calculations in the current work. To further understand the model’s features, the nonlinear differential equations with associated boundary conditions were solved using the RKF-45 strategy and shooting technique. The main conclusions of the present study are that the fluid particles were aroused by the increasing velocity of both the disk and cone, which is what led to the improvement of the velocity profiles. The velocity profile dramatically increased with increasing values for both cone and disk rotations. The present CDA-type with a radiation effect functioned well in terms of heat transmission. The values of $N_t$ and $k^{*}$ have a similar qualitative impact on mass transportation for all four models, seeing that the mass transportation decreases as $N_t$ and $k^{*}$ increase. The fluid flow over the disk surface shows an improved heat transfer rate compared to the fluid flow over cone surface for an increase in $R$ values. The fluid flow over the cone surface shows a better-quality mass transference rate than the fluid flow over the disk surface for an increase in the values of $N_t$ and $k^{*}$.