Modeling Transient Free Convection and Nanofluid Effects in Follicular Dynamics: A Delay Differential Equation Approach to Fertility and Endocrine Regulation

Follicular growth and ovulation involve complex biochemical and biophysical interactions, including intra-follicular transport and inter-follicular communication. The thermal and convective properties of follicular fluid influence nutrient and hormone distribution, impacting follicle maturation. Recent advancements in nanofluid applications in biochemical processes necessitate advanced computational models. This study integrates transient free convection theory and delay differential equations (DDEs) to analyze follicular wave behavior, linking mathematical modeling with reproductive medicine.

Recent studies have enhanced understanding of follicular dynamics, particularly in free convection, nanofluid applications, and DDEs in fertility and endocrine regulation. For example, Tanveer et al. (2023) studied flow and heat transfer in the fallopian tube, emphasizing ciliary motion and mixed convection in ovum transport, highlighting the role of metachronal wave patterns in reproductive efficiency. Ibrahim and Gamachu (2019) analyzed nonlinear convection of Williamson nanofluids over a radially stretching surface, relevant to follicular fluid dynamics, while Benygger et al. (2023) studied natural convection in a porous medium using Buongiorno’s model, providing insights applicable to intra-follicular processes.

Furthering nanofluid research, Ramakrishna (2024) explored variable thermophysical properties in non-Newtonian fluids, focusing on the antiviral and antibacterial properties of silver nanoparticles, which may influence follicular fluid dynamics. Similarly, Falodun (2024), Akinremi (2024), and Omole (2024) investigated the effects of silver nanoparticles on follicular thermal regulation. Yasmi (2024) studied hybrid nanofluid flows containing Fe_3O_4 and Au nanoparticles, critical for modeling nanoparticle-enhanced heat transfer in ovarian follicles, while Adnan (2024) explored radiative ternary nanofluid flow under an induced magnetic field, offering insights into how magnetic fields regulate follicular fluid behavior.

From a computational perspective, Bani-Fwaz (2024) examined thermal modulation in radiated nanofluid systems, focusing on nanoparticle diameters (Al_2O_3 and H_2O) to optimize heat and mass transport in biological fluids, relevant to follicular metabolic efficiency. Jifeng Cui (2024) analyzed Carreau-Yasuda mixed convective flow in a porous medium, incorporating the Soret and Dufour effects to refine follicular thermoregulation models.

Biological studies further support these findings. Tanveer (2023) explored metachronal wave motion in cilia, highlighting its influence on ovum transport, while Padma (2023) investigated time-dependent magnetohydrodynamic (MHD) free convective heat circulation of hybrid nanofluids over porous plates, relevant to ovarian microenvironment regulation. Lavanya (2024) analyzed the effects of thermal radiation and shape factors on hybrid nanofluid flow, incorporating cross-diffusion effects and entropy generation to refine ovarian thermoregulation models.

Foundational studies on nanofluids and heat transfer, such as those by Glassl et al. (2010), who examined how nanofluid particle concentration affects thermal conductivity, and Demirkir and Erturk (2020), who investigated graphene-water nanofluid heat transfer, provide key insights. Bhargava and Chandra (2017) developed numerical methodologies for MHD boundary layer flow, while Ahmed and Podder (2024) analyzed mixed convection in Al_2O_3-water nanofluids, offering parametric insights applicable to ovarian follicles.

Understanding intra-follicular and inter-follicular communication is crucial for ovarian follicle development. Liu et al. (2019) explored intra-ovarian regulatory factors, emphasizing the balance between systemic hormones and local ovarian regulators in follicle selection. Qiao et al. (2023) investigated intra-pituitary follicle-stimulating hormone (FSH) signaling and its link to hepatic metabolism. A 2024 study in the Journal of Ovarian Research analyzed follicular fluid composition and its role in granulosa cell and oocyte communication, identifying biomarkers for fertility treatments. Integrating computational and mathematical models with biological research enhances understanding of fertility and endocrine regulation. Heat and mass transfer, nanofluid behavior, and convection dynamics provide novel insights into follicular development. This interdisciplinary approach informs fertility treatments and endocrine therapies, establishing a foundation for future reproductive medicine advancements.

The study by Ali, Khan, and Abbas (2023) presents a numerical framework for modeling the dynamics of microorganism movement on a Carreau-Yasuda layer, focusing on non-Newtonian fluids. This work is extended in the current study by incorporating nanofluid effects and DDEs to model follicular dynamics, capturing delayed hormonal responses. This represents a significant advancement over traditional fluid dynamics models, as it integrates biological processes with fluid mechanics, providing a more comprehensive understanding of follicular dynamics. Khan, Wang, and Zhang (2022) examined the impact of surface roughness on sperm motility, which is crucial for understanding reproductive biology. The current study shifts the focus to follicular fluid dynamics and nanofluid-enhanced heat and mass transfer, offering a broader perspective on reproductive health by integrating fluid mechanics with endocrine regulation. Zhang, Li, and Chen (2021) analyzed cilia-driven flows in shear-thinning fluids, relevant to ovum transport. The current study builds on this by incorporating nanofluid effects and thermal convection to model follicular wave interactions, offering a more comprehensive understanding of how fluid dynamics influence follicular development.

Ahmed, Khan, and Ali (2023) focused on the simulation of complex fluids using the Implicit Finite Difference Method (IFDM). The current study integrates biological processes such as follicular growth and hormonal regulation with fluid dynamics, providing a more holistic model for reproductive health. Wang, Zhang, and Li (2022) explored the electro-fluid dynamics of organisms in complex fluids. In contrast, the current study focuses on nanofluid-enhanced follicular dynamics and hormonal feedback loops, offering a novel approach to understanding fertility and endocrine regulation. Li, Chen, and Zhang (2023) examined low Reynolds number flows in complex geometries. The current study extends this by incorporating nanofluid effects and delay differential equations to model follicular wave behavior, providing a more nuanced understanding of follicular dynamics. Chen, Zhang, and Li (2022) investigated bacterial motion in viscoelastic fluids. In contrast, the current study focuses on follicular fluid dynamics and hormonal regulation, offering a unique perspective on reproductive health by integrating fluid mechanics with biological processes.

The study does not explicitly mention a sensitivity analysis. However, the numerical methodology involves grid independence tests and stability assessments using Courant-Friedrichs-Lewy (CFL) criteria to ensure convergence and stability. Future work could include a sensitivity analysis to assess how variations in parameters such as Grashof number, Prandtl number, and Schmidt number affect the model’s predictions. This would help evaluate the robustness of the model under different physiological conditions and ensure its reliability in real-world applications. The study suggests that future work should focus on experimental validation to confirm the model’s predictions against real-world data. This could involve comparing the model’s results with experimental data on follicular thermodynamics and hormone diffusion in reproductive fluids. Additionally, the study highlights the potential for real-world applications in assisted reproductive technologies (ART) and endocrine therapies. The model could be used to optimize fertility treatments by simulating different hormonal and thermal conditions to improve follicular development and ovulation timing. Future research could also explore the integration of machine learning techniques to enhance computational efficiency and predictive accuracy.

The study employs a combination of numerical methods, including the finite difference method (FDM) for discretization, the shooting method for boundary value problems, and the Runge-Kutta method for solving delay differential equations. While these methods are effective, the study acknowledges the need for improved computational efficiency, particularly for large-scale simulations. Potential improvements could include the use of parallel computing techniques, adaptive mesh refinement, and more advanced numerical solvers to reduce computational time and resource requirements. Additionally, the study suggests that future work could explore the use of machine learning algorithms to optimize the numerical solution process and improve the model’s scalability for complex biological systems.

Mathematical Formulation

The governing equations for fluid flow, heat transfer, and mass transport are derived based on the principles of free convection and nanofluid interactions. The system is modeled using the Navier-Stokes equations for momentum conservation (White, 2021), energy equation incorporating Newtonian heating (Incropera & DeWitt, 2020), and the concentration equation for mass diffusion and chemical reactions (Bird, Stewart, & Lightfoot, 2019). These equations are coupled with follicular growth dynamics using delay differential equations (Bellman & Cooke, 2022), enabling the study of delayed hormonal responses and feedback mechanisms in follicular waves.

The biological dynamics are incorporated through equations governing follicular growth, hormonal interactions, and ovulatory response. These equations account for the role of follicle-stimulating hormone (FSH) in granulosa cell proliferation and estradiol production, ensuring a physiologically accurate representation of the follicular cycle (Zhang & Wang, 2021).

Figure 1: Schematic representation of patterns of serum FSH and E2 concentrations, and gonadotropin dependent follicle growth,

The first four governing equations model computationally efficient and suitable for analyzing follicular fluid dynamics under different physiological conditions. The last four equations—Follicular Growth, Hormonal Regulation (FSH), Estradiol Production, and Ovulatory Surge (DDE for LH Response)—describe biological processes.

Continuity Equation:

\( \frac{\partial u}{\partial r} + \frac{\partial v}{\partial z} = 0 \)

Momentum Equation:

\( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial r} = -\frac{1}{\rho} \frac{\partial P}{\partial r} + \nu \frac{\partial^2 u}{\partial r^2} + g \beta (T – T_\infty) + S_c (C – C_\infty) \)

Describes the movement of follicular fluid under temperature gradients (buoyancy), concentration gradients, and external forces.

\( \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial r} = \alpha \frac{\partial^2 T}{\partial r^2} + Q(T – T_\infty) \)

Concentration Equation:

\( \frac{\partial C}{\partial t} + u \frac{\partial C}{\partial r} = D \frac{\partial^2 C}{\partial r^2} – kC \)

Follicular Growth Equation:

\( \frac{dG}{dt} = \lambda F G – \mu G^2 \)

where: G is granulosa cell mass, λ is the growth rate stimulated by FSH, and μ represents inhibition due to resource competition.

Hormonal Regulation Equation (FSH):

\( \frac{dF}{dt} = k_1 – k_2 F + k_3 \sum_i \frac{G_i}{1 + G_i} \)

where: F is the FSH concentration, k_1 represents basal FSH production, k_2 is the FSH clearance rate, G_i represents individual follicle growth rates.

Estradiol Production Equation:

\( \frac{dE_2}{dt} = \alpha G – \beta E_2 \)

where: α is estradiol synthesis per granulosa cell, β represents estradiol clearance.

Ovulatory Surge (DDE for LH Response):

\( F(t) = F_0 + \gamma E_2 (t – \tau) \)

where: τ represents the biological delay before LH surge.

Now, we will systematically apply the perturbation method to simplify the eight governing equations from partial differential equations (PDEs) to coupled ordinary differential equations (ODEs). we first introduce dimensionless variables to reduce the number of parameters and simplify the system. We define the following:

Dimensionless radial and axial coordinates:

\( \eta = \frac{r}{L}, \quad \xi = \frac{z}{L} \)

Stream function transformation:

\( u = \frac{\partial \psi}{\partial r}, \quad v = -\frac{\partial \psi}{\partial z} \)

Dimensionless temperature, concentration, and velocity variables:

\( \theta = \frac{T – T_\infty}{T_w – T_\infty}, \quad \phi = \frac{C – C_\infty}{C_w – C_\infty}, \quad f(\eta) = \frac{\psi}{\sqrt{\nu L}} \)

Perturbation expansion:

Assume small perturbation effects due to nanofluid interaction or weak convective forces, represented by a perturbation parameter ϵ: u, v, T and T in the first four equations and G and F in the last four equations, we use

\( u = u_0 + \epsilon u_1 + \epsilon^2 u_2 + \cdots \)

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\( v = v_0 + \epsilon v_1 + \epsilon^2 v_2 + \cdots \)

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\( T = T_0 + \epsilon T_1 + \epsilon^2 T_2 + \cdots \)

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\( C = C_0 + \epsilon C_1 + \epsilon^2 C_2 + \cdots \)

Now, we expand each of the eight (8) governing PDEs using the perturbation method,

Expanding G and F in perturbation form:

\( G = G_0 + \epsilon G_1 + \epsilon^2 G_2 + \cdots \)

<br>

\( F = F_0 + \epsilon F_1 + \epsilon^2 F_2 + \cdots \)

 on expanding equation (1) using perturbation terms as in (12a), it becomes

\( \frac{\partial (u_0 + \epsilon u_1 + \cdots)}{\partial r} + \frac{\partial (v_0 + \epsilon v_1 + \cdots)}{\partial z} = 0 \)

At leading order (O(1)),

\( \frac{\partial u_0}{\partial r} + \frac{\partial v_0}{\partial z} = 0 \)

At first-order perturbation (O(ϵ)),

\( \frac{\partial u_1}{\partial r} + \frac{\partial v_1}{\partial z} = 0 \)

These equations (14a and 14b) simplify mass conservation and decouple velocity components.

Similarly, equation (2) on Substituting the perturbation expansions, it becomes

\( \frac{\partial (u_0 + \epsilon u_1 + \cdots)}{\partial t} + (u_0 + \epsilon u_1 + \cdots) \frac{\partial (u_0 + \epsilon u_1 + \cdots)}{\partial r} = -\frac{1}{\rho} \frac{\partial (P_0 + \epsilon P_1 + \cdots)}{\partial r} + \nu \frac{\partial^2 (u_0 + \epsilon u_1 + \cdots)}{\partial r^2} + g \beta (T_0 – T_\infty) + S_c (C_0 – C_\infty) \)

At O(1):

\( \frac{\partial u_0}{\partial t} + u_0 \frac{\partial u_0}{\partial r} = -\frac{1}{\rho} \frac{\partial P_0}{\partial r} + \nu \frac{\partial^2 u_0}{\partial r^2} + g \beta (T_0 – T_\infty) + S_c (C_0 – C_\infty) \)

At O(ϵ):

\( \frac{\partial u_1}{\partial t} + u_0 \frac{\partial u_1}{\partial r} + u_1 \frac{\partial u_0}{\partial r} = -\frac{1}{\rho} \frac{\partial P_1}{\partial r} + \nu \frac{\partial^2 u_1}{\partial r^2} + g \beta (T_1 – T_\infty) + S_c (C_1 – C_\infty) \)

This yields a coupled system where the first equation (16a) describes primary flow, and the second equation (16b) introduces nonlinear effects. Also, using the perturbation expansion, energy equation (3) becomes

\( \frac{\partial T_0}{\partial t} + u_0 \frac{\partial T_0}{\partial r} = \alpha \frac{\partial^2 T_0}{\partial r^2} + Q(T_0 – T_\infty) \)

<br>

\( \frac{\partial T_1}{\partial t} + u_0 \frac{\partial T_1}{\partial r} + u_1 \frac{\partial T_0}{\partial r} = \alpha \frac{\partial^2 T_1}{\partial r^2} + Q(T_1 – T_\infty) \)

And, using the perturbation expansion, concentration equation (4) becomes

At O(1):

\( \frac{\partial C_0}{\partial t} + u_0 \frac{\partial C_0}{\partial r} = D \frac{\partial^2 C_0}{\partial r^2} – k C_0 \)

At O(ϵ):

\( \frac{\partial C_1}{\partial t} + u_0 \frac{\partial C_1}{\partial r} + u_1 \frac{\partial C_0}{\partial r} = D \frac{\partial^2 C_1}{\partial r^2} – k C_1 \)

on expanding equation (5) using perturbation terms as in (12b), it becomes

At O(1):

\( \frac{dG_0}{dt} = \lambda F_0 G_0 – \mu G_0^2 \)

At O(ϵ):

\( \frac{dG_1}{dt} = \lambda (F_0 G_1 + F_1 G_0) – 2 \mu G_0 G_1 \)

This results in a coupled system where G_0 represents primary follicular growth, while G_1 captures nonlinear effects such as nutrient competition.

on expanding equation (6) using perturbation terms as in (12b), it becomes

At O(1):

\( \frac{dF_0}{dt} = k_1 – k_2 F_0 + k_3 \sum_i \frac{G_{i0}}{1 + G_{i0}} \)

At O(ϵ):

\( \frac{dF_1}{dt} = -k_2 F_1 + k_3 \sum_i \frac{G_{i1}}{(1 + G_{i0})^2} \)

This system now describes FSH dynamics, where F_0 in 20a governs baseline hormone fluctuations, while F_1 in 20b describes nonlinear follicle-hormone interactions.

Similarly, on expanding the Estradiol Production Equation (7) using perturbation terms as in (12b), it becomes

Expanding E_2 and G in perturbation form:

\( E_2 = E_{20} + \epsilon E_{21} + \cdots \)

<br>

\( G = G_0 + \epsilon G_1 + \cdots \)

At O(1):

\( \frac{dE_{20}}{dt} = \alpha G_0 – \beta E_{20} \)

At O(ϵ):

\( \frac{dE_{21}}{dt} = \alpha G_1 – \beta E_{21} \)

This system captures estradiol synthesis dynamics, where E_20 governs primary estradiol levels, and E_21 models’ small variations due to follicular interactions. And finally, on expanding the Ovulatory Surge Equation (8) using perturbation terms as in (12b), it becomes

\( F = F_0 + \epsilon F_1 + \cdots \)

<br>

\( E_2(t – \tau) = E_{20}(t – \tau) + \epsilon E_{21}(t – \tau) + \cdots \)

At O(1):

\( F_0(t) = F_0 + \gamma E_{20}(t – \tau) \)

At O(ϵ):

\( F_1(t) = \gamma E_{21}(t – \tau) \)

This transforms the delay equation into an iterative system, where each term describes progressive hormone responses to past estradiol levels. The equations now reduce to a set of nonlinear ODEs for velocity u, temperature T, concentration C, and pressure P, forming a coupled system that can be solved numerically:

\( \frac{du}{d\eta} = f_1(u, T, C) \)

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\( \frac{dT}{d\eta} = f_2(u, T) \)

<br>

\( \frac{dC}{d\eta} = f_3(u, C) \)

and, the biological equations are reduced to a set of ODEs that describe hormonal, follicular, and fluid dynamics:

\( \frac{dG}{dt} = f_1(G, F) \)

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\( \frac{dF}{dt} = f_2(G, F, E_2) \)

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\( \frac{dE_2}{dt} = f_3(G, E_2) \)

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\( F(t) = f_4(E_2(t – \tau)) \)

This system provides a fully coupled model integrating fluid dynamics and hormonal regulation in follicular wave processes. This simplification makes the model computationally efficient and suitable for analyzing follicular fluid dynamics under different physiological conditions. The last four equations—Follicular Growth, Hormonal Regulation (FSH), Estradiol Production, and Ovulatory Surge (DDE for LH Response) describe biological processes rather than fluid dynamics. These equations can also be simplified using perturbation techniques to yield a set of coupled ordinary differential equations (ODEs).

To solve the governing equations, a combination of numerical methods is employed. The finite difference method (FDM) is used for the discretization of the partial differential equations governing fluid flow, temperature, and concentration distribution (Smith, 2023). A uniform computational grid is employed, and central differencing is used for spatial derivatives, while explicit and implicit schemes are implemented for temporal discretization.

For the boundary value problems encountered in the transformed equations, the shooting method is utilized (Keller, 2023). This approach allows for efficient resolution of nonlinearities present in the governing equations by converting them into initial value problems. Additionally, the Runge-Kutta method (Butcher, 2021) is applied to solve the coupled delay differential equations, ensuring stability and accuracy in the representation of hormonal delays.

The computational domain is carefully selected to ensure convergence and stability. Grid independence tests are performed by refining the spatial resolution until numerical variations are minimized. The stability of the numerical schemes is assessed using Courant-Friedrichs-Lewy (CFL) criteria, ensuring reliable time-step selection. Boundary Conditions The boundary conditions are defined as: u(0)=0, T(0)=T_w, C(0)=C_w u(∞)=0, T(∞)=T_∞, C(∞)=C_∞ where T_w and C_w represent follicular wall temperature and concentration, respectively and Stream Function Formulation A stream function ψ is introduced to simplify the velocity components: u=∂ψ/∂r, v=-∂ψ/∂z Substituting ψ into the governing equations yields a reduced form for numerical treatment using Similarity Transformation The dimensionless variables are defined as: η=r/L, f(η)=ψ/√νL Transforming the equations into dimensionless form facilitates numerical computation and physical interpretation.

We combine biological processes and fluid dynamics into a single coupled system of equations by linking the hormonal and follicular growth dynamics with the momentum, energy, and concentration equations governing fluid behavior in the follicular environment. This approach allows us to model the mutual influence of i. Fluid flow, temperature, and mass transport on follicular growth and hormone regulation and ii. Hormonal and follicular dynamics on the fluid properties within the follicular environment. The fluid dynamics equations (1–4) and biological equations (5–8) interact in several ways:

Hormone Transport Affects Follicular Growth

F (FSH) and E_2 (estradiol) are transported in Equation (4) (concentration equation).

Their concentration affects granulosa cell growth in Equation (5).

Fluid Temperature and Flow Influence Hormone Levels

Follicular growth depends on FSH concentration (F) from Equation (6).

F is influenced by concentration transport (Equation 4) and temperature gradients (Equation 3).

Follicular Growth Alters Hormone Feedback Loops

More granulosa cells produce more estradiol (E_2), which regulates ovulation (Equation 8).

Estradiol influences delayed hormonal responses (LH surge) in Equation (8).

Fluid Velocity and Heat Transfer Influence Follicular Development

Increased temperature gradients (Equation 3) affect follicular maturation.

Higher fluid velocity (Equation 2) enhances hormone transport, influencing ovulation timing.

This fully coupled system integrates fluid mechanics and reproductive biology, making it highly suitable for simulating ovulation, hormone regulation, and fluid transport in fertility studies. solving this numerically using Finite Difference Methods

To solve the fully coupled system using the Finite Difference Method (FDM), we will:

The derivatives are approximated using central, forward, and backward differences:

(i) Discretizing the Momentum Equation (2) using the Finite Difference Approximation, we get

\( u_i^{n+1} = u_i^n + \Delta t \left( – u_i^n \frac{u_{i+1}^n – u_{i-1}^n}{2 \Delta r} + \nu \frac{u_{i+1}^n – 2 u_i^n + u_{i-1}^n}{(\Delta r)^2} + g \beta (T_i^n – T_\infty) + S_c (C_i^n – C_\infty) – \frac{1}{\rho} \frac{P_{i+1}^n – P_{i-1}^n}{2 \Delta r} \right) \)

(ii) Discretizing the Momentum Equation (Temperature Evolution) (3) using the Finite Difference Approximation, we get

\( T_i^{n+1} = T_i^n + \Delta t \left( – u_i^n \frac{T_{i+1}^n – T_{i-1}^n}{2 \Delta r} + \alpha \frac{T_{i+1}^n – 2 T_i^n + T_{i-1}^n}{(\Delta r)^2} + Q(T_i^n – T_\infty) \right) \)

(iii) Also, discretizing the concentration Equation (4) using the Finite Difference Approximation, we get

\( C_i^{n+1} = C_i^n + \Delta t \left( – u_i^n \frac{C_{i+1}^n – C_{i-1}^n}{2 \Delta r} + D \frac{C_{i+1}^n – 2 C_i^n + C_{i-1}^n}{(\Delta r)^2} – k C_i^n \right) \)

Similarly, the Biological Equations with FDM. follicular growth, FSH, and estradiol production, we use Euler’s explicit scheme as

(iv) Follicular Growth Equation (5) becomes

\( G^{n+1} = G^n + \Delta t \left( \lambda F^n G^n – \mu (G^n)^2 \right) \)

(v) Hormonal Regulation (FSH) equation (6) reduces to

\( F^{n+1} = F^n + \Delta t \left( k_1 – k_2 F^n + k_3 \sum_i \frac{G_i^n}{1 + G_i^n} \right) \)

(vi) Estradiol Production equation (7) decomposes to

\( E_2^{n+1} = E_2^n + \Delta t \left( \alpha G^n – \beta E_2^n \right) \)

(vii) And the Ovulatory Surge Equation (8) (DDE for LH Response) becomes

\( F(t) = F_0 + \gamma E_2(t – \tau) \)

This requires storing past values using a time delay array. This approach numerically solves the fully coupled system using finite differences for fluid flow and explicit Euler for biological processes.

Numerical computations are carried out for different physical parameters such as mass Grashof number (Gc), thermal Grashof number (Gr), Schmidt number (Sc), and Prandtl number (Pr). The value of Schmidt number (Sc) is taken to be 0.6, which corresponds to water vapor. The value of Prandtl number (Pr) is chosen to represent air (Pr = 0.71). The four fluid equations and four biological equations are transformed into a system of coupled nonlinear ordinary differential equations (ODEs) using similarity transformation, stream function formulation, and non-dimensionalization.

The system of seven coupled nonlinear ODEs describes fluid flow (velocity, temperature, concentration) and biological processes (hormone regulation, follicular growth, ovulation). Validation of the numerical results is conducted by comparing simulated outcomes with experimental data from follicular thermodynamics studies (Tanveer et al., 2023) and hormone diffusion measurements in reproductive fluids (Yasmin, 2024). Benchmark comparisons with existing mathematical models are also performed to ensure model consistency. These methodological steps ensure the robustness of the model and its applicability in reproductive endocrinology and assisted reproductive technologies.

Figure 2: The nonlinear behavior of the fluid and biological system

The graph generated from the Maple code provides a comprehensive visualization of the fluid dynamics described by the momentum, energy, and concentration equations. The stream function ff(η) profile illustrates the velocity distribution within the fluid, showing how the flow evolves from the boundary (e.g., no-slip condition at η=0) to the free stream (as ηη approaches infinity). The temperature θ(η) profile demonstrates the thermal boundary layer, indicating how heat is transferred within the fluid, with the temperature decaying from the boundary to the ambient condition. The concentration ϕ(η) profile represents the distribution of a scalar quantity (e.g., hormone concentration), showing how diffusion and advection influence its transport. The interplay between these profiles highlights the coupling between fluid flow, heat transfer, and mass transport, governed by parameters such as the Grashof number (Gr), Prandtl number (Pr), and Schmidt number (Sc). The graph effectively captures the nonlinear behavior of the system, providing insights into the boundary layer dynamics and the influence of buoyancy, thermal effects, and concentration gradients. This visualization is crucial for understanding the physical phenomena and validating theoretical models against experimental or numerical results

Figure 3: Interplay between follicular growth, hormonal regulation, and estradiol production

The graph provides a clear visualization of the dynamic interplay between follicular growth, hormonal regulation, and estradiol production. The follicular growth G(τ) curve typically shows an initial increase due to the stimulating effect of FSH, followed by a plateau or decline as negative feedback mechanisms (e.g., μG2) come into play. The FSH, F(τ) profile reflects the balance between production (k1), decay (k2), and feedback from follicular growth (k3).

Figure 4: The nonlinear and coupled nature of the biological system

The estradiol E2(τ) curve demonstrates how estradiol levels rise in response to follicular growth and decay over time due to metabolic clearance (β). The graph highlights the nonlinear and coupled nature of the biological system, providing insights into the regulatory mechanisms governing follicular development and hormonal dynamics. This visualization is valuable for understanding the biological processes and can guide further analysis or experimental validation

Figure 5: The thermal and convective properties affecting follicular dynamics

The graphs generated from the Maple code provide a visual representation of the interplay between fluid dynamics and biological processes. For the fluid equations, the stream function f(η), temperature θ(η), and concentration ϕ(η) profiles demonstrate how buoyancy effects (Grashof number Gr), heat transfer (Prandtl number Pr), and concentration gradients (Schmidt number Sc) influence the system. The biological plots illustrate the dynamics of follicular growth G(τ), hormonal regulation F(τ), and estradiol production E2(τ), showcasing how these variables evolve over time due to growth rates, feedback mechanisms, and hormonal interactions. The graphs highlight the nonlinear nature of the system, with potential coupling between fluid and biological processes, such as hormone transport affecting follicular growth or temperature influencing estradiol production. Overall, the plots provide valuable insights into the system’s behavior and can guide further analysis or experimental validation.

CONCLUSION

This study presents a novel computational approach to modeling follicular dynamics by integrating fluid mechanics, nanofluid effects, and hormonal regulation through a system of coupled nonlinear equations. The methodology follows a multi-step process comprising mathematical formulation, numerical solution, and validation against existing literature. The findings provide significant insights into the interplay between fluid dynamics and biological processes, particularly in the context of follicular growth, hormone regulation, and ovulation.

The key contributions of this study include:

1. Integration of Nanofluid Effects: The incorporation of nanofluid dynamics into follicular fluid modeling offers a deeper understanding of heat and mass transfer processes within ovarian follicles.
2. Delay Differential Equations (DDEs): The use of DDEs captures delayed hormonal responses, providing a more accurate representation of follicular wave interactions and feedback mechanisms.
3. Interdisciplinary Approach: By combining fluid mechanics with reproductive biology, this study bridges the gap between computational modeling and clinical applications in fertility treatments and endocrine therapies.

Future Directions:

1. Experimental Validation: Future work should focus on experimental validation to confirm the model’s predictions against real-world data, particularly in follicular thermodynamics and hormone diffusion.
2. Real-World Applications: The model has potential applications in assisted reproductive technologies (ART) and endocrine therapies, where it can be used to optimize fertility treatments by simulating different hormonal and thermal conditions.
3. Computational Efficiency: Improvements in computational efficiency, such as the use of parallel computing techniques, adaptive mesh refinement, and machine learning algorithms, could enhance the model’s scalability and predictive accuracy for complex biological systems.

This study establishes a foundation for future advancements in reproductive medicine, offering a robust framework for understanding and optimizing follicular dynamics in both clinical and research settings.

REFERENCES