Derivation and analysis of a Stokes-transport system in evolving vessels modeling thermoregulation in human skin

We consider a Stokes flow coupled with advective-diffusive transport in an evolving domain with boundary conditions allowing for inflow and outflow. The evolution of the domain is induced by the transport process, leading to a fully coupled problem. Our aim is to model the thermal control of blood flow in human skin. To this end, the model takes into account the temperature-dependent production of biochemical substances, the subsequent dilation and constriction of blood vessels, and the resulting changes in convective heat transfer. We prove existence and uniqueness of weak solutions using a fixed point method that allows us to treat the nonlinear coupling.

💡 Research Summary

The paper presents a comprehensive mathematical model for the thermoregulatory function of human skin, focusing on the interaction between blood flow, heat transport, and biochemical signaling within a single artery or arteriole and its surrounding tissue. The authors formulate a coupled system consisting of a quasi‑steady Stokes flow, an advection‑diffusion equation for temperature in both the fluid (blood) and solid (tissue) domains, and an ordinary differential equation describing the concentration of nitric oxide (NO) at the vessel wall. The geometry is a tubular fluid domain surrounded by a solid domain; both are allowed to deform over time, with the deformation driven by the NO‑mediated dilation or constriction of the vessel wall.

Key modeling features include:

1. Moving domain representation – A deformation map (S(t,\cdot)) transports the reference configuration to the current configuration, thereby turning the free‑boundary problem into a problem on a fixed reference domain. The map depends on a radius function (R(t,x_1)) and a smooth radial scaling function (\rho), ensuring that the evolving interface remains smooth and bounded between prescribed minimal and maximal radii.

2. Stokes flow – Blood is treated as a Newtonian fluid with constant viscosity (\mu). The Stokes equations (-\nabla p + \mu\Delta v = 0) and (\nabla\cdot v = 0) are solved in the fluid region. Boundary conditions consist of a normal‑stress condition at the inflow/outflow ends (allowing both inflow and outflow) and a no‑slip condition on the moving vessel wall, where the wall velocity equals the material velocity (\partial_t S\circ S^{-1}).

3. Heat transport – Temperature in the fluid ((\vartheta_f)) and solid ((\vartheta_s)) satisfies advection‑diffusion equations. In the fluid, advection by the Stokes velocity is present; in the solid, only diffusion occurs, but the solid velocity is taken as the material velocity of the moving domain. At the interface, a Robin‑type transmission condition (-K_f\nabla\vartheta_f\cdot n = -K_s\nabla\vartheta_s\cdot n = \alpha(\vartheta_f-\vartheta_s)) models imperfect heat exchange across the multilayer vessel wall. A Danckwerts condition on the inflow/outflow boundary incorporates convective heat flux together with prescribed external heat flux (f_{\rm in}). The solid exterior is subjected to a Dirichlet condition (\vartheta_s = \vartheta_{\rm ext}).

4. NO dynamics – The concentration (c(t,x_1)) of nitric oxide at the wall follows a linear decay term (-k c) plus a production term (G(x_1,\mathcal{T}(\vartheta_s))). The production depends on a temporally averaged tissue temperature (\mathcal{T}(\vartheta_s)), defined through a two‑step averaging operator: first a spatial average over the solid domain, then a convolution with a smooth kernel (K_\gamma) over a short time window (\gamma). This reflects the physiological delay between temperature sensing by nerves and NO synthesis by endothelial cells.

5. Mathematical analysis – After pulling back all equations to the reference domain, the authors obtain a fully coupled nonlinear system where the deformation gradient (\nabla S) appears as a coefficient in every equation. The Stokes subsystem is treated as a saddle‑point problem; existence and uniqueness follow from the Babuska‑Brezzi condition and standard Lax‑Milgram arguments after eliminating the prescribed boundary data. The temperature equations are handled via a Galerkin approximation that respects the coupled fluid–solid unknowns and the variable coefficients. The ODE for NO is linear and can be solved explicitly once the averaged temperature is known.

The core of the existence proof uses Schaefer’s fixed‑point theorem. The authors define an operator that maps a tentative deformation, velocity, temperature, and NO concentration to the solution of the linearized subproblems. They show that this operator is continuous, compact, and maps bounded sets into bounded sets. A priori energy estimates (based on the kinetic energy of the fluid, the thermal energy of both domains, and the (L^2) norm of NO) provide uniform bounds, allowing the application of Schaefer’s theorem to obtain a fixed point, i.e., a weak solution of the full nonlinear system. Uniqueness is proved by deriving a Grönwall‑type inequality for the difference of two solutions, exploiting the linearity of the NO ODE and the monotonicity of the heat transmission condition.

6. Comparison with prior work – Earlier models either fixed the vessel radius, used Poiseuille flow instead of Stokes, or omitted the biochemical feedback loop. This paper integrates all three mechanisms—fluid dynamics, heat transfer, and NO‑mediated wall motion—within a rigorous analytical framework. The inclusion of a moving interface and the treatment of the coupled PDE‑ODE system on a deforming domain represent a novel contribution to the mathematical modeling of microcirculatory thermoregulation.

In summary, the authors deliver a mathematically sound, physiologically detailed model that captures the bidirectional coupling between blood flow, temperature distribution, and biochemical signaling in skin vessels. The existence‑uniqueness result, based on fixed‑point theory and careful functional‑analytic estimates, provides a solid foundation for future numerical simulations and extensions to more complex vascular networks.