Lie symmetry analysis of bio-nano-slip flow in a conical gap between a rotating disk and cone with Stefan blowing

1 Introduction

This study is innovative as it incorporates distinctive elements, namely Stefan blowing and slip effects, into the analysis of bio-nano-convective flow within a cone–disk system (CDS). Unlike prior studies that have explored different facets of a nanofluid flow, this work uniquely applies the Buongiorno model to investigate the interaction between slip conditions and mass transfer driven by Stefan blowing. Furthermore, the use of Lie symmetries to derive invariant solutions introduces a novel methodological angle, increasing the theoretical foundation for understanding the complex fluid dynamics in three-dimensional geometries. Suspensions of nanoparticles with a diameter less than 10⁻⁹ nm in the base fluids are known as nanofluids. According to published studies, they may increase the transfer rates of heat, nanoparticle volume fraction (NPVF), and microorganisms in many industrial applications. Examples include heat exchangers, transportation, electronics, biomedicine, biomedical, drug delivery, and the food sector. Numerous researchers have investigated various aspects of nanofluids due to their many potential applications [1,2,3]. Assessment of heat transfer in the presence of hybrid nanofluid was carried out by Batool et al. [4]. Alnahdi and Gul [5] explored the hybrid nanofluid flow along a surface with slip effects.

The impulsive movement of minuscule liquids, such as dipped plumes, is called bioconvection. Bioconvection has become a sizable field of fluid dynamics, with exciting applications in fuel cell and green energy technologies. Owing to its many applications, bioconvection inside a nanofluid is an attractive research area. Experimental studies have demonstrated that adding microorganisms to nanofluids routinely increases thermal performances and environmentally friendly, sustainable properties. These advancements will create a solid foundation for the subsequent generation of biofuels. Bioconvection nanofluid flow has been the topic of researchers due to its many applications. Sheikhpour et al. [6] analyzed the applications of nanofluids in drug delivery and biomedical sectors. Mandal et al. [7] explained the surface undulation effects on combined bioconvective flow in a porous medium with a magnetic field. The influence of the magnetohydrodynamic field on bio-nano-convective flow has been studied by Jawad et al. [8]. Awais et al. [9] numerically studied the rheology of the MHD bio-nano-convective flow. Ferdows et al. [10] investigated nanofluid flow across a stretchy sheet because of its importance in various engineering and commercial applications. Wang et al. [11] reported bioconvective applications for nanofluid flow past a horizontal sheet. Unsteady bioconvective flow past a porous artery was investigated by Priyadharsini and Gururaj [12]. Their area of interest was the mathematical study of blood’s non-Newtonian behavior when bacteria and drug nanoparticles with magnetic and thermal radiation impacts were present.

The flow produced due to disk rotation received the attention of scholars due to its uses in geophysics, industries, rotating machinery, medical equipment, turbine systems, etc. The disk–cone technology, primarily employed in viscosimeters to determine fluid viscosity, has both practical and technological relevance. In recent times, the influence of nanoparticles on the radiative flow between the gap of a disk and cone was studied by Wang et al. [13]. The working of bio-convection power law slip flow along a needle placed in a porous medium with blowing effects was studied by Uddin et al. [14]. Convective heat/mass transfer analysis of a fluid flow across the conical gap of a cone–disk apparatus in the presence of the thermophoretic particle motion was studied by Srilatha et al. [15]. Farooq et al. [16] investigated the influence of a silver and cobalt ferrite-based hybrid nanofluid flow between a rotating disk and cone. CNT-nanofluid flow in a rotating system between the gap of a disk and a cone was studied by Gul et al. [17]. The fluid flow between a cone and a disk, both stationary or rotating with energy transfer, was studied by Turkyilmazoglu [18]. In another paper, Turkyilmazoglu [19] examined the fluid flow and heat transfer along a moving disk. The study of ferrofluid flow and heat transfer between the cone and disk was studied by Bhandari [20].

According to the monogram of Shevchuk [21], the flow of a fluid in the devices within a gap and a small angle γ = 1. 5° between a rotating cone and a static surface is employed in viscosimetry. In medical science, viscosimetry is utilized to develop endothelium cells, disposed of as a monolayer on the surface. At the same time, the gradually rotating cone prepares for the possible circulation of feeding culture medium in the gap. Many researchers have analyzed various aspects of this type of flow problem due to applications in medical devices, conical diffusers, and viscometers. Shevchuk [22] studied numerous phases of heat transport and fluid flow in CDSs using a self-similar solution technique. Many authors studied many aspects of the problems following his pioneering work. Turkyilmazoglu [19] analyzed the fluid flow and heat transfer past a rotating disk. In another paper, Turkyilmazoglu [18] studied the fluid flow and heat transfer between a cone and a stationery and rotating disk. Hybrid nanofluid flow in a rotating system between the gap of a disk and a cone was studied by Gul et al. [23]. The investigation of ferrofluid flow and heat transfer between cone and disk was experimented with by Bhandari [20]. Upadhya et al. [24] reported the investigation of the flow and heat transfer in a CDS, considering various physical features. Recently, Turkyilmazoglu [25] studied the flow and heat in the conical region of a rotating cone and an extendable disk. Basavarajappa and Bhatta [26] used scaling group transformation to radiative nanofluid flow in CDSs with hall current. Srilatha et al. [15] studied heat and mass transfer analysis of a fluid flow in the gap of CDSs.

The diffusion of the species generates a bulk motion of fluids, called the Stefan-blowing effect. Species transfer varies on the flow field, and Stefan blowing influences the bulk motion of fluids. Therefore, a pairing between momentum and NPVF is essential to model the flow [27]. Evaporation in process industries and soil science is a natural phenomenon that uses the idea of Stefan blowing. As per the literature, Stefan blowing arises in paper drying when mass flux exists from the surface to the surroundings. Note that Stefan blowing is different from conventional mass injection/blowing due to transpiration. Injection/blowing due to transpiration occurs from a permeable surface, while Stefan blowing occurs from an impermeable surface. Uddin et al. [14] recently studied bio-nano-convective micropolar fluid flow in a Darcy porous medium over a cone with Stefan blowing effects. Reddy et al. [28] examined Stefan’s blowing impacts on the flow of hybrid nanofluids past a moving slender needle.

In this study, the governing equations of the bio-nano-convective flow system, including continuity, momentum, energy, NPVF, and microorganism density equations, are solved using numerical techniques. The Runge–Kutta–Fehlberg (RKF-45) method is employed due to its adaptive step-size control and balance between accuracy and computational cost. This method is widely recognized for its efficiency in solving the systems of nonlinear differential equations, making it particularly suitable for complex fluid flow problems like those in the CDS. This method integrates the equations using a fixed initial step size of 0.001 and automatically adjusts the step size based on error estimation. This helps maintain the desired accuracy, with a convergence criterion set at 10⁻⁶, ensuring that the error remains below the threshold throughout the computation. This method is highly efficient for boundary value problems and is implemented here to ensure stability in solving the nonlinear partial differential equations derived from the Lie symmetry analysis. This research builds on prior studies and real-world applications, intending to investigate the 3-D flow characteristics of the Buongiorno bio-nanofluid within a cone–disc apparatus. Unlike the existing literature, this study incorporates novel elements, such as Stefan blowing and slip impacts, which have not been explored previously. The Buongiorno bio-nanofluid, combining nanoparticles and bio-based components, holds promise for increased heat transfer and potential applications in biotechnology and medicine. The chosen cone–disc geometry provides a specific framework for understanding fluid dynamics. Stefan blowing introduces a mass transfer aspect, and slip impacts consider the relative motion at fluid–solid interfaces. This comprehensive approach aims to contribute valuable insights into the intricate behavior of the bio-nanofluid within a previously unexplored context, emphasizing the novelty and potential advancements this study could bring to the scientific understanding of complex fluid systems.

2 Mathematical model

The coordinates ( r , φ, z ) and the cone–disk device are shown in Figure 1. Here, r is measured along the radial direction, φ is the azimuthal angle, and z is along vertically upward. The origin is located at the apex of the cone (Figure 1). The bio-nanofluid fills the gap between the cone and the disk. The gap height is h = z tan γ , where γ is the gap angle. The wall fluid temperature, NPVF, and the density of motile microorganisms (DMM) are assumed to follow power law variation in the radial direction. We presume the flow is axisymmetric, and the two-component in-homogeneous Buongiorno nanofluid is adopted. The thermophysical properties are assumed to be constants. Fluid velocity components along the coordinates ( r , φ, z ) are (u, v, w), respectively. The temperature, NPVF, and DMM are denoted by t, c, and n, respectively. It was postulated that the disk and cone could be either static or in rotation at different angular velocities (Figure 1).

Figure 1

Flow configuration and the coordinate system [26].

The following modeling assumptions are made:

• The flow is assumed to be axisymmetric for bio-nanofluids.

• The governing equations include continuity, momentum, energy, NPVF, and microorganism density.

• Buongiorno’s nanofluid model is employed to account for slip mechanisms and mass transfer at fluid–solid interfaces.

• Stefan blowing effects are included to model the interaction between the flow and thermal transport.

• The RKF-45 method with adaptive step sizing is used for numerical solutions to ensure stability and accuracy.

• Thermophysical properties of the bio-nanofluid are considered constant throughout the analysis.

• Rotational Reynolds numbers are varied to analyze different flow configurations.

• Gap angles for the CDS are calibrated against established benchmark data.

Four unique physical formations: (i) a rotating cone with a stationary disk, (ii) a stationary cone with a rotating disk, (iii) the simultaneous rotation of the disk and cone, and (iv) the counter-rotation of the disk and cone [26] are used. The relevant equations are as follows:

Continuity

U-momentum

V-momentum

W-momentum

Energy

Microorganism

Nanoparticle volume fraction

The relevant boundary conditions are as follows [14]:

We propose the following non-dimensional variables:

The dimension-free forms of the above equations are as follows:

Continuity

u -momentum

v -momentum

w -momentum

Energy

Nanoparticle volume fraction

Microorganism

The boundary conditions become

The emerging dimensionless parameters arising in equations (11)–(17) are defined as Le = α D B (Lewis number), Pe = b W C α (Peclet number), Pr = ν α (Prandtl number), N b = τ D B Δ C α (Brownian motion parameter), N t = τ D T Δ T α T ∞ (thermophoresis), Lb = D m α (bioconvection Lewis number), a = ν N 1 L R (slip), s = − Δ c ( 1 − c w ) (blowing parameter), Re ω = ω R L 2 ν (Reynolds number for disk), and Re Ω = Ω R L 2 ν (Reynolds number for cone).

3 Symmetry analysis

To generate an invariant, we apply the general Lie-group technique. Lie group analysis was also significantly studied by Meena [29]. The readers are referred to the previous studies [30,31,32] for details of Lie group theory and its applications in various disciplines. Consider the flowing one-parameter ( ε ) Lie group infinitesimal transformations in ( R , Z , U , V , W , P , T , C , N ):

where ε is a real number known as the parameter of the group and the variables ( R , Z , U , V , W , P , T , C , N ) are transformed into ( R ⁎ , Z ⁎ , U ⁎ , V ⁎ , W ⁎ , T ⁎ , P ⁎ , C ⁎ , N ⁎ ) , which keeps the partial differential system (11)–(19) invariant. The system of partial differential equations is said to admit a symmetry generated by the vector field

if it is left by the transformation ( R , Z , U , V , W , P , T , C , N ) to ( R ⁎ , Z ⁎ , U ⁎ , V ⁎ , W ⁎ , T ⁎ , P ⁎ , C ⁎ , N ⁎ ) . By working on a long, boring algebra using Maple 18, the form of the infinitesimals is found as follows:

where c i ( i = 1 , 2 , 3 , … , 8 ) are constants. It is observed from (22) that the parameters c 2 , c 5 , c 7 correspond to the translation to the variable Z, T , and N , respectively. It is further observed that the parameter c 1 corresponds to the scaling in the variables R , Z , U , V , W , P , respectively, while c 3 , c 4 c 1 correspond to the scaling in the variables N , T , respectively.

From (22),

Solving the above equations, the following are the similarity transformations related to the infinitesimals in (22):

where m = c 4 c 1 . As the stream function is settled to an arbitrary constant, we may set c 3 = 0 . (F, G, H) represent dimensionless velocity while (T, C, N) represent temperature, NPVF, and DMM functions, respectively.

4 Transformed equations

By substituting (23) into equations (11)–(19), we obtain

The boundary conditions are transformed to

where η 1 = tan γ represents the position at the cone surface, and based on angular velocities ω and Ω, the rotational Reynold numbers are defined as Re ω = ω L 2 R 2 ν , Re Ω = Ω L 2 R 2 ν . In line with Shevchuk’s findings [21], specific rotational Reynolds numbers Re = Re w η 1 2 12 = 1 Re = Re Ω η 1 2 12 = 1 were selected to determine the cone position under varying rotational Reynolds numbers.

Shevchuk used Re from 12 to 2,463, relating to gap angles of 45° and 40°, respectively, in their investigation, while Basavarajappa and Bhatta [26] utilized values of 36 and 167.14 for rotational Reynolds numbers, aligning with gap angles of 30° and 15°, respectively. In our research, we extended the range of rotational Reynolds numbers from 2 to 12, and the corresponding gap angles are presented in Table 1.

5 Physical quantities

The physical quantities are shear stresses (tangential C f r d and radial C f θ d ), the heat transfer gradient (on the disk Nu d and cone Nu c ), the NPVF gradient (on the disk Sh d and cone Sh c ), and the DMM transfer gradients (on the disk Nn d and cone Nn c ). The shear stresses are defined as

In terms of dimensionless variables, shear stresses become

The heat, NPVF, and DMM transfer gradients are defined as

In terms of dimensionless variables, they are

This study involves solving similarity equations (24)–(30) and boundary conditions (31) and (32). The RKF-45 numerical method, with a fixed step size of 0.001, is chosen to solve these equations, balancing accuracy and computational cost. The adaptive step-size feature of RKF-45 adjusts dynamically based on accuracy requirements, with a convergence criterion set at 10⁻⁶. This criterion ensures that the solution is converged when the error falls below the defined threshold, and the method adapts the step size to maintain the desired accuracy. The details of RKF-45 can be found in previous studies [33,34].

6 Results and discussion

This research explores the behavior of a bio-nano fluid’s flow within a conical gap formed by a rotating disk and a cone, examining four distinct physical models. Within this conical gap, the nanofluid experiences fluctuations in the velocity, temperature, concentration, and DMM induced by the movement of both rotating and stationary surfaces, thereby significantly influencing the flow, heat, NPVF, and DMM transfer processes. The numerical results are reported in Tables 2–5 and Figures 2–23 and are discussed below.

Figure 2

Effects of velocity slip and Reynolds number on the radial velocity of a bio-nanofluid when the (a) disk and (b) cone are stationary.

Figure 3

Effects of velocity slip and Reynolds number on the radial velocity of a bio-nanofluid for (a) co-rotating and (b) counter-rotating disk and cone.

Figure 4

Effects of velocity slip and Reynolds number on the azimuthal velocity of a bio-nanofluid when the (a) disk and (b) cone are stationary.

Figure 5

Effects of velocity slip and Reynolds number on the azimuthal velocity of a bio-nanofluid for (a) co-rotating and (b) counter-rotating disk and cone.

Figure 6

Effects of velocity slip and Reynolds number on the axial velocity of a bio-nanofluid when the (a) disk and (b) cone are stationary.

Figure 7

Effects of velocity slip and Reynolds number on the axial velocity of a bio-nanofluid for (a) co-rotating and (b) counter-rotating disk and cone.

Figure 8

Effects of velocity slip and Reynolds number on the dimensionless temperature of a bio-nanofluid when the (a) disk and (b) cone are stationary.

Figure 9

Effects of velocity slip and Reynolds number on the dimensionless temperature of a bio-nanofluid for (a) co-rotating and (b) counter-rotating disk and cone.

Figure 10

Effects of velocity slip and Reynolds number on NPVF (concentration) of a bio-nanofluid when the (a) disk and (b) cone are stationary.

Figure 11

Effects of velocity slip and Reynolds number on the dimensionless concentration of a bio-nanofluid for (a) co-rotating and (b) counter-rotating disk and cone.

Figure 12

Effects of velocity slip and Reynolds number on rescaled DMM in a bio-nanofluid when the (a) disk and (b) cone are stationary.

Figure 13

Effects of velocity slip and Reynolds number on rescaled DMM in a bio-nanofluid for (a) co-rotating and (b) counter-rotating disk and cone.

Figure 14

Variation of radial skin friction at the disk surface with slip and blowing parameters when the (a) disk and (b) cone are stationary.

Figure 15

Variation of the radial skin friction at the cone surface with slip and blowing parameters when the (a) disk and (b) cone are stationary.

Figure 16

Variation of the tangential skin friction at the disk surface with slip and blowing parameters when the (a) disk and (b) cone are stationary.

Figure 17

Variation of the tangential skin friction at the cone surface with slip and blowing parameters when the (a) disk and (b) cone are stationary.

Figure 18

Variation of Nusselt number at the disk surface with slip and blowing parameters when the (a) disk and (b) cone are stationary.

Figure 19

Variation of Nusselt number at the cone surface with slip and blowing parameters when the (a) disk and (b) cone are stationary.

Figure 20

Variation of Sherwood number at the disk surface with slip and blowing parameters when the (a) disk and (b) cone are stationary.

Figure 21

Variation of Sherwood number at the cone surface with slip and blowing parameters when the (a) disk and (b) cone are stationary.

Figure 22

Variation of the motile density number at the disk surface with slip and blowing parameters when the (a) disk and (b) cone are stationary.

Figure 23

Variation of the motile density number at the cone surface with slip and blowing parameters when the (a) disk and (b) cone are stationary.

The impact of blowing and slip parameters on the radial and tangential skin friction at the surfaces of the disk and cone is detailed in Table 1 for models I and II and Table 2 for models III and IV, while other variables are held constant.

7 Velocity profiles

This study utilizes four distinct models to illustrate the effects of slip and Reynolds numbers on radial, azimuthal (tangential), and axial velocities. The radial velocity, representing the movement either outward or inward from the rotation axis, is shown in Figures 2 and 3 across various Reynolds numbers and slip parameters. Examining the impact of these factors, Figure 2(a) and (b) displays radial velocity profiles for bio-nanofluids in model I (rotating cone and stationary disk) and model II (rotating disk and stationary cone). In model I, the radial velocity decreases at the disk, subsequently increasing the gap between the disk and cone with increased cone rotational velocity. This pattern aligns with expectations, as higher Reynolds numbers, associated with increased disk rotational speed, lead to augmented fluid motion and elevated velocities near the surface. Within model II, the flow over the cone exhibits a radially inward direction, while the flow over the disk manifests as radially outward. The observed increase in the radial velocity at the disk with higher disk rotational velocity and slip parameter can be attributed to the rotational dynamics of the system. As the disk’s rotational speed increases, the fluid experiences increased motion due to the higher angular velocity imparted by the rotating disk. The slip parameter, which accounts for the relative motion between the fluid and the solid surface, also contributes to the overall velocity increase. Notably, when the cone rotates faster than the disk (i.e., at higher Reynolds numbers), the fluid exhibits a distinct radial flow pattern. In this scenario, the fluid flows radially outward past the cone and radially inward over the disk. This behavior results from the relative rotational speeds of the cone and disk, causing the fluid to move outward along the cone’s surface and inward along the disk’s surface.

When both the disk and cone are rotating in the same direction, Figure 3(a), the flow is radially outward over the cone and radially inward over the disk. This pattern arises due to the relative rotational speeds of the two surfaces. As the disk rotates faster than the disk, the fluid near the cone’s surface experiences a higher radial velocity, causing it to move radially outward. On the other hand, the fluid near the disk’s surface, which rotates at a slower speed, exhibits a lower tangential velocity, resulting in a radially inward flow. This behavior is a consequence of the rotational dynamics of the system. This is observed when the relative angular velocities of the rotating components dictate the direction of radial fluid movement. When the disk and cone are rotating in opposite directions, Figure 3(b), the flow pattern is influenced by the relative rotational speeds of the two components. In such a configuration, when the cone rotates faster than the disk, the flow tends to be radially inward over the cone and radially outward over the disk. This is because the higher rotational speed of the cone imparts more incredible tangential velocity to the fluid near its surface, causing it to move radially inward. Conversely, the fluid near the disk’s surface, rotating slower, exhibits a lower tangential velocity, resulting in a radially outward flow. The specific direction of the flow is determined by the relative magnitudes of the angular velocities of the rotating cone and disk, with the faster-rotating component influencing the radial direction of the fluid movement.

The azimuthal or tangential velocity, denoting the speed in the circumferential direction around the axis of rotation, holds significance in a rotating disk and cone system. This velocity dynamically varies with the radial distance from the axis. The role of slip and rotational velocity in influencing tangential velocity is crucial. Figure 4(a) and (b) for models I and II presents the effects of slip parameters and rotational cone velocity. Figure 4(a) shows that the tangential velocity increases from the stationary disk to the cone rotational velocity. This is because the rotational motion of the cone imparts higher tangential velocity to the fluid near its surface. In the absence of slip, tangential velocity is higher and decreases with increasing slip, reflecting the impact of slip on the relative motion between the fluid and solid surfaces.

Conversely, when the cone is stationary, the velocity at the disk increases with the rotational velocity of the disk and decreases to zero at the stationary cone. This demonstrates the influence of disk rotation on tangential velocity. As expected, slip reduces tangential velocity at the disk surface with increasing slip, underscoring the complex interplay between slip, rotational velocities, and tangential fluid dynamics in the analyzed system.

The significance of slip and rotational velocity in tangential velocity becomes apparent when the disk and cone rotate in the same or opposite direction in a co-rotating system. Figure 5(a) and (b) for models III and IV illustrate the effects of the slip parameter and rotational cone velocity. In Figure 5(a), it is observed that the tangential velocity increases from the rotating disk to the cone rotating at a higher angular velocity in the same direction. This is expected as the increased rotational motion of the cone imparts more incredible tangential velocity to the fluid near its surface. In the absence of slip, the tangential velocity is higher and decreases with increasing slip, highlighting the influence of slip on the relative motion between the fluid and solid surfaces. Conversely, when the cone is rotating at a higher velocity in the opposite direction, the velocity at the cone decreases to the cone rotational velocity. This is a consequence of the counteracting rotational speeds, which reduces tangential velocity at the cone surface (Figure 5(b)). As anticipated, slip reduces the tangential velocity in the gap between the disk and cone with increasing slip, emphasizing the complex interplay between the slip, rotational velocities, and tangential fluid dynamics in the examined co-rotating system.

The significance of slip and rotational velocity in the axial direction becomes evident when the disk is stationary and the cone is rotating with varying angular velocities, as depicted in Figure 6(a) and (b) for models I and II, respectively. Figure 6(a) shows that the axial velocity increases from the stationary disk to a maximum and then decreases as the cone rotates at different angular velocities. This behavior is expected, as the rotational motion of the cone influences the axial velocity. The maximum velocity is higher at higher rotational cone velocity and decreases with decreasing cone rotational velocity, reflecting the direct correlation between the angular speed and axial fluid motion. In the absence of slip, the axial velocity is lower and increases with increasing slip, underscoring the impact of slip on the relative motion between the fluid and solid surfaces.

Conversely, when the cone is stationary and the disk rotates with varying angular velocities, the velocity in the axial direction remains zero. This is because the rotational motion of the components influences the axial velocity. In the gap between the disk and cone, the axial velocity is initially adverse, attains its minimum, and then increases toward the stationary cone. The relative angular velocities of the rotating components influence this complex pattern. As expected, slip reduces the axial velocity in the gap between the disk and cone with increasing slip, emphasizing the complex interdependence of the slip, rotational velocities, and axial fluid dynamics in the analyzed system.

In the scenario where both the disk and cone are rotating in either the same direction (model III) or opposite directions (model IV), the interplay between the slip, rotational cone velocity, and axial fluid dynamics is shown in Figure 7(a) and (b). Notably, in Figure 7(a), the axial velocity increases from the stationary disk to a maximum and then decreases to the stationary cone. This behavior is influenced by the rotational motion of the cone, with the maximum velocity occurring at lower rotational velocities and decreasing with both decreasing cone rotational velocity and the gap angle between the cone and disk. The complex relationship between these factors suggests that lower rotational velocities and smaller gap angles contribute to higher axial fluid velocities. In the absence of slip, the axial velocity is lower and increases with increasing slip, emphasizing the impact of slip on the relative motion between the fluid and solid surfaces. Similarly, when the cone is stationary, and the disk rotates at varying angular velocities, Figure 7(b), the axial velocity remains zero, showcasing the influence of rotational motion on axial fluid dynamics. This pattern is consistent with the effects of slip and rotational angular velocity observed earlier, highlighting the robustness of the observed trends across different configurations in the analyzed system.

8 Temperature profiles

Temperature profiles in a rotating disk–cone system refer to the distribution of temperatures within the fluid or material in the system as a function of spatial coordinates. In a rotating disk–cone system, these profiles describe how temperature varies across different locations, such as radial and axial positions. The temperature profile is affected by factors such as rotational speeds, slip and solid surfaces, and the overall geometry of the system. In Figure 8(a) and (b) for models I and II, the influence of slip and rotational velocity on the dimensionless temperature is shown when the disk is stationary and the cone rotates at different angular velocities. Figure 8(a) shows that the dimensionless temperature is maximum at the stationary disk and decreases to zero at the cone rotating at various angular velocities. This pattern reveals the impact of rotational motion on temperature distribution, with the maximum temperature occurring at the stationary disk surface. In the case of no slip, the temperature is minimal and increases with both increasing slip and decreasing rotational cone velocity. This signifies that slip and reduced cone rotational speed contribute to higher temperatures within the system.

Similarly, when the cone is stationary and the disk rotates at different angular velocities, the dimensionless temperature decreases in the gap between the disk and cone (Figure 8(b)). This is consistent with the notion that increased rotational disk velocity and velocity slip lead to lower temperatures in the gap region. The complex relationships depicted in these figures underscore the importance of slip and rotational velocity in shaping the temperature profiles within the rotating disk–cone system, providing valuable insights into the thermal dynamics of the configuration.

In Figure 9(a), where both the disk and cone are rotating in the same direction, the dimensionless temperature distribution is analyzed for model III. At the disk surface, the temperature is at its maximum and decreases within the gap, reaching the cone surface. This variation is attributed to the rotational angular velocity of the cone; as the cone rotates, the temperature decreases in the gap. The absence of slip results in a low dimensionless temperature, and as the slip increases, the temperature increases. At a low rotational cone velocity, the dimensionless temperature is higher and diminishes with an increasing cone Reynolds number within the gap. Conversely, in Figure 9(b), where the cone rotates in the opposite direction, the behavior of the dimensionless temperature is reversed compared to Figure 9(a). The specifics of this reversal are not explicitly mentioned, but the change in the rotational direction likely changes the heat transfer dynamics, leading to a different temperature distribution pattern between the disk and cone surfaces in the gap.

The dimensionless NPVF profiles in a rotating disk–cone system describe the concentration variation of nanoparticles within the fluid relative to a reference concentration. These profiles are essential for understanding how the concentration of a solute or component changes across different spatial coordinates in the system. In Figure 10(a) and (b) for models I and II, the impact of slip and rotational velocity on the dimensionless concentration is explored when the disk is stationary, and the cone is rotating at various cone angular velocities. The observations from Figure 10(a) reveal that the dimensionless concentration is maximum at the stationary disk and decreases to zero at the cone, dependent on its rotational velocity. This behavior indicates the influence of rotational motion on concentration distribution, with the maximum concentration occurring at the stationary disk surface and decreasing along the cone. The dimensionless concentration profiles further demonstrate interesting behaviors at different Reynolds numbers. At low Reynolds numbers, the dimensionless concentration decreases linearly up to the cone surface. In contrast, at higher Reynolds numbers, the concentration decreases in a more complicated manner, suggesting the influence of fluid dynamics on solute transport. In the absence of slip, the dimensionless concentration shows variations depending on the position of the disk or cone, highlighting the impact of slip-on concentration distribution. Additionally, the cone Reynolds number alters the direction of dimensionless concentration in the gap between the disk and cone, reflecting the complex interplay between rotational speeds and fluid behavior.

When the cone is stationary and the disk rotates at different angular velocities, the dimensionless concentration decreases from maximum to zero concentration at the stationary cone. This is consistent with the expected patterns influenced by the rotational disk velocity and slip. The observed behaviors underscore the complex relationship between the slip, rotational velocity, and dimensionless concentration, providing valuable insights into the mass transport dynamics within the rotating disk–cone system.

The impact of slip parameter and rotational velocities on the dimensionless concentration is shown in Figure 11(a) for model III and Figure 11(b) for model IV. As depicted earlier, the dimensionless concentration exhibits a decrease from the disk, rotating at a certain angular velocity to the rotating cone with varying cone velocities, as shown in Figure 11(a). This performance can be attributed to the dynamics of fluid flow and the effects of rotational motion on the concentration distribution. The positions at the cone surface under different cone Reynolds numbers are detailed in Table 1, providing insight into the spatial variations. In the absence of slip, the dimensionless concentration is lower, and it increases with an increase in slip, indicating the influence of slip on the concentration profile. Notably, when the cone rotates in the opposite direction, the dimensionless concentration demonstrates a similar behavior. This consistency suggests that the rotational direction of the cone plays a significant role in governing the concentration distribution, with slip further modulating the concentration levels within the system.

The rescaled DMM in a bio-nanofluid holds significant implications for understanding the behavior of microorganisms in a fluid environment. In this study, the effects of the slip parameter and rotational cone velocity on this rescaled density are shown in Figure 12(a) for model I and Figure 12(b) for model II. Similar to dimensionless temperature and concentration, the rescaled DMM showcases a decrease from the stationary disk to the cone surface rotating at different velocities, as shown in Figure 12(a). This behavior indicates the fluid dynamics influenced by the rotation of the cone and highlights how microorganisms are distributed within the fluid. In the absence of slip, the rescaled DMM is lower, and it increases with an increase in slip, demonstrating the role of slip in modulating the distribution of microorganisms. Additionally, an increase in the cone Reynolds number correlates with a decrease in the rescaled DMM, suggesting a complex interplay between the fluid flow and microorganism behavior. Interestingly, when the cone is stationary, the rescaled DMM exhibits a similar trend, underlining the significance of rotational motion in influencing microorganism distribution (Figure 12(b)).

The impact of slip parameter and rotational cone velocity on the rescaled DMM is illustrated in Figure 13(a) for model III and Figure 13(b) for model IV. As previously observed, the rescaled DMM exhibits a decrease in the gap between the disk and cone, which rotate at varying cone velocities, as depicted in Figure 13(a). This behavior is likely influenced by the fluid dynamics resulting from the rotation of the components. Similar to dimensionless temperature and concentration, the rescaled DMM follows a consistent pattern. The commonality in this behavior suggests that the slip parameter and rotational velocity play pivotal roles in governing the distribution and movement of motile organisms within the bio-nanofluid. The observed trends underscore the interconnected nature of these factors, providing valuable insights into the intricate dynamics of microorganism transport in such systems.

The radial skin friction in the flow of a bio-nanofluid within a conical gap between a rotating disk and cone refers to the shear force exerted parallel to the radial direction on the fluid at the interface of the rotating components. The variation of the radial skin friction with the slip parameter is explored for different values of the blowing parameter in Figure 14(a) for model I (stationary disk) and Figure 14(b) for model II (stationary cone). In model I, where the disk is stationary, it is observed that radial skin friction increases with both the slip and blowing parameters. This trend can be attributed to the increased fluid motion caused by the slip, leading to increased friction, while increased blowing increases the interaction between the fluid and the rotating components, thereby increasing skin friction. Conversely, for model II with a stationary cone, the radial skin friction decreases with the slip parameter and increases with the blowing parameter, as shown in Figure 14(b). Here, slip reduces the frictional forces at the interface, while increased blowing intensifies the interaction, resulting in increased radial skin friction. The comparison of Figure 14(a) and (b) reveals higher friction at the disk when the cone is stationary and the disk is rotating.

The consistent trend observed in Figure 15(a) and (b) in both models for the radial skin friction indicates a shared behavior regardless of the specific configuration. Notably, the radial skin friction is maximum at the cone surface when the disk is stationary. This phenomenon can be attributed to the obscure interplay of fluid dynamics and the stationary disk’s influence on interacting with the rotating cone. When the disk is stationary, the motion and distribution of the bio-nanofluid are primarily influenced by the rotational velocity of the cone. The stationary disk creates conditions that increase the friction at the cone surface, maximizing the radial skin friction. This observation underscores the significance of the relative motion and arrangement of the rotating components in influencing the distribution of skin friction within the conical gap, with the maximum occurring at the cone surface in the presence of a stationary disk.

The tangential skin friction in the flow of a bio-nanofluid within a conical gap between a rotating disk and cone refers to the shear force exerted parallel to the circumferential direction on the fluid at the interface of the rotating components. The variation of the tangential skin friction with the slip parameter is shown in Figure 16 at the disk surface and in Figure 17 at the cone surface for models I and II, respectively. When the disk is stationary and the cone rotates at a lower velocity, as depicted in Figure 16(a), the tangential skin friction decreases with both the slip and blowing parameters. This decrease is attributed to the influence of slip, which reduces the frictional forces at the disk surface, and blowing, which diminishes the interaction between the fluid and the stationary disk. In Figure 16(b), tangential skin friction decreases with the slip parameter but increases with the blowing parameter, highlighting the intricate relationship between the slip, blowing, and tangential skin friction. The comparison of models I and II shows higher tangential skin friction at the disk surface in model I, indicative of the impact of the stationary disk on frictional forces. At the cone surface, the potential skin friction increases with the slip parameter in both cases, as shown in Figure 17(a) and (b). However, the tangential skin friction at the cone surface increases with the blowing parameter when the disk is stationary but decreases when the cone is stationary. This complex interplay reflects the dynamic nature of the bio-nanofluid flow system and underscores the importance of considering both the slip and blowing parameters in understanding tangential skin friction variations.

The Nusselt number represents the ratio of the convective to conductive heat transfer at the surface of the components. The variation in the Nusselt number with the slip for different blowing parameters at the disk and cone surfaces is shown in Figures 18 and 19 for models I and II, respectively. When the disk is stationary, the slip parameter tends to decrease the Nusselt number at the disk surface Nu d (Figure 18a) while increasing it when the cone is stationary (Figure 18b). This behavior is indicative of the influence of the slip-on heat transfer characteristics, with reduced slip leading to a decrease in the convective heat transfer. In both cases, the blowing parameter reduces Nu d , suggesting that increased blowing diminishes the convective heat transfer at the surface. The comparison between models I and II reveals higher Nu d when the cone is stationary, emphasizing the impact of the components’ relative motion on the heat transfer.

Figure 19(a) and (b) further explains the variation in the Nusselt number with slip and blowing parameters at the cone surface. In model I, Nu c increases with both slip and blowing parameters, signifying the increase of convective heat transfer. Conversely, for model II, the slip parameter tends to reduce Nu c , while the blowing parameter increases it. The comparison indicates higher Nu c when the disk is stationary, reflecting the intricate interplay between the slip, blowing, and heat transfer characteristics in the bio-nanofluid flow system.

The Sherwood number (Sh) characterizes the mass transfer rate in the context of a bio-nanofluid flowing in a gap between a disk and cone. It is a dimensionless quantity representing the ratio of convective to diffusive mass transfer at the surface of the components. The variation in the Sherwood number with the slip parameter for different values of the blowing parameter at the disk and cone surfaces is shown in Figures 20 and 21 for models I and II, respectively. When the disk is stationary, the slip and blowing parameters tend to decrease the Sherwood number at the disk surface Sh d (Figure 20a). In contrast, the slip parameter increases when the cone is stationary (Figure 20b). This behavior reflects the influence of slip and blowing on the mass transfer characteristics, with reduced slip and increased blowing leading to a decrease in the convective mass transfer. In both cases, the blowing parameter reduces the Sherwood number, indicating that increased blowing diminishes the convective mass transfer at the surface. The comparison between models I and II reveals higher Sh d when the cone is stationary, emphasizing the role of the components’ relative motion in mass transfer.

Figure 21(a) and (b) further explains the modification in the Sherwood number at the cone surface ( Sh c ) with slip and blowing parameters. In Model I, Sh c increases with both slip and blowing parameters when the disk is stationary, signifying the increase of convective mass transfer. Conversely, for model II, the slip parameter tends to reduce Sh c , while the blowing parameter increases it. The comparison indicates higher Sh c when the disk is stationary, showcasing the intricate interplay between the slip, blowing, and mass transfer characteristics in the bio-nanofluid flow system.

The motile density number of microorganisms or microorganism transfer rate ( Nn ) in the flow of the bio-nanofluid between a disk and cone is defined as a dimensionless quantity representing the ratio of the convective to diffusive microorganism transfer at the surfaces of the rotating components. The variation of the microorganism transfer rate due to slip and blowing parameters is depicted in Figure 22 for the microorganism transfer rate at the disk ( Nn d ) and in Figure 23 considering models I and II for the microorganism transfer rates at the cone. For model I, it is observed that the slip and blowing parameters tend to reduce Nn d (Figure 22a), while the slip helps in increasing Nn d and the blowing parameter reduces Nn d (Figure 22b). This behavior suggests that increased slip increases the convective microorganism transfer, whereas blowing diminishes it at the disk surface.

Similarly, at the cone surface, depicted in Figure 23(a) for model I and Figure 23(b) for model II, both velocity slip and blowing parameters tend to increase the microorganism transfer rate ( Nn c ) for model I, indicating that increased slip and blowing increase the convective microorganism transfer at the cone surface. Conversely, for model II, velocity slip reduces Nn c , while the blowing parameter increases it, emphasizing the complex interplay between the slip, blowing, and microorganism transfer characteristics in the bio-nanofluid flow system.

9 Conclusion

The study provides valuable insights into the combined effects of flow, heat, mass, and microorganism transport in a bio-nanofluid system between a rotating cone and disk (CDS) under various rotational configurations. The key findings across the four models can be condensed into the following common effects:

• Blowing parameter: Increasing the blowing parameter generally increases the radial skin friction at both the disk and cone surfaces, while its effect on the tangential skin friction varies – typically increasing at the cone surface but decreasing at the disk surface. In all models, an increase in the blowing parameter tends to reduce the transfer rates of heat, mass, and microorganisms from the rotating disk, with a slight increase in these rates at the stationary cone.

• Slip parameter: A higher slip parameter increases the radial skin friction at both the cone and disk surfaces, while its impact on the tangential skin friction is mixed, leading to a decrease at the disk surface and variable effects at the cone surface. The slip parameter also has a notable influence on the transfer rates of heat, mass, and microorganisms, typically reducing the transfer rates from the disk but increasing them from the cone.

These findings highlight how varying the slip and blowing parameters significantly influences the skin friction, as well as heat, mass, and microorganism transport across the cone and disk surfaces, providing a deeper understanding of the bio-nanofluid dynamics within the CDS.