@article{10.11648/j.ijamtp.20261202.12,
  author = {Kande Dickson Kinyua and Karimi Kennedy John Mwangi},
  title = {A Convective Heat and Mass Transfer Model: A Differential Geometry Approach},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {12},
  number = {2},
  pages = {70-78},
  doi = {10.11648/j.ijamtp.20261202.12},
  url = {https://doi.org/10.11648/j.ijamtp.20261202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20261202.12},
  abstract = {This paper uses the mathematical formalism of differential geometry to present a geometry-consistent framework for modeling convective heat and mass transfer. Conventional methods are usually limited to Euclidean spaces and frequently ignore how curvature and torsion affect scalar transport. On the other hand, we develop the advection-diffusion equations for a general Riemannian manifold (M, g), utilizing curvature-aware operators to capture the impact of geometric complexity on thermal and concentration fields, specifically the Laplace--Beltrami operator and invariant advection terms. For canonical curved geometries, such as annular, helical, and torsional domains, analytical solutions are obtained. These disclose modified Nusselt and Sherwood number scalings and other curvature-induced corrections to classical transport laws. Specifically, we demonstrate that the Nusselt number scales as Nu ∝ Ra1/4 for natural convection in curved annuli, consistent with prior asymptotic analyses. We recover Dean-number-dependent transport behavior in torsional duct flows, which is consistent with empirical findings from nanofluidic heat exchangers. We use mesh-based and meshless discretizations designed for curved surfaces for numerical validation, such as Lattice Boltzmann methods for porous or irregular domains, weighted finite volume schemes, and DEC. Even in highly non-Euclidean environments, these methods maintain geometric fidelity and exhibit second-order convergence. We're bringing performance analysis under one roof by introducing curvature-adjusted, dimensionless groups-- mainly, geometric Peclet, Nusselt, and Sherwood numbers. These capture how the shape of a system affects the way things like heat and mass move through it. What really stands out is how field synergy -- basically, the angle between the way stuff moves and the direction gradients point -- changes with curvature. This opens up fresh ways to boost transport in systems that aren't flat. With this approach, we lay out a unified platform, both in theory and computation, to look at heat and mass transfer in all kinds of complex shapes. Think microfluidic devices, porous media reactors, cutting-edge heat exchangers, or even thermal systems inspired by nature. Next up, we're tackling multi-phase and reactive transport, plus real-world tests in truly three-dimensional curved spaces.},
 year = {2026}
}

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TY  - JOUR
T1  - A Convective Heat and Mass Transfer Model: A Differential Geometry Approach
AU  - Kande Dickson Kinyua
AU  - Karimi Kennedy John Mwangi
Y1  - 2026/07/07
PY  - 2026
N1  - https://doi.org/10.11648/j.ijamtp.20261202.12
DO  - 10.11648/j.ijamtp.20261202.12
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 70
EP  - 78
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20261202.12
AB  - This paper uses the mathematical formalism of differential geometry to present a geometry-consistent framework for modeling convective heat and mass transfer. Conventional methods are usually limited to Euclidean spaces and frequently ignore how curvature and torsion affect scalar transport. On the other hand, we develop the advection-diffusion equations for a general Riemannian manifold (M, g), utilizing curvature-aware operators to capture the impact of geometric complexity on thermal and concentration fields, specifically the Laplace--Beltrami operator and invariant advection terms. For canonical curved geometries, such as annular, helical, and torsional domains, analytical solutions are obtained. These disclose modified Nusselt and Sherwood number scalings and other curvature-induced corrections to classical transport laws. Specifically, we demonstrate that the Nusselt number scales as Nu ∝ Ra1/4 for natural convection in curved annuli, consistent with prior asymptotic analyses. We recover Dean-number-dependent transport behavior in torsional duct flows, which is consistent with empirical findings from nanofluidic heat exchangers. We use mesh-based and meshless discretizations designed for curved surfaces for numerical validation, such as Lattice Boltzmann methods for porous or irregular domains, weighted finite volume schemes, and DEC. Even in highly non-Euclidean environments, these methods maintain geometric fidelity and exhibit second-order convergence. We're bringing performance analysis under one roof by introducing curvature-adjusted, dimensionless groups-- mainly, geometric Peclet, Nusselt, and Sherwood numbers. These capture how the shape of a system affects the way things like heat and mass move through it. What really stands out is how field synergy -- basically, the angle between the way stuff moves and the direction gradients point -- changes with curvature. This opens up fresh ways to boost transport in systems that aren't flat. With this approach, we lay out a unified platform, both in theory and computation, to look at heat and mass transfer in all kinds of complex shapes. Think microfluidic devices, porous media reactors, cutting-edge heat exchangers, or even thermal systems inspired by nature. Next up, we're tackling multi-phase and reactive transport, plus real-world tests in truly three-dimensional curved spaces.
VL  - 12
IS  - 2
ER  - 

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