Mathematical Modeling of Viscous Fluid Dynamics
Summary
Mathematical modelling of viscous fluid dynamics centres on the formulation and analysis of the Navier–Stokes equations, which describe the motion of fluid elements under the influence of viscous stresses, pressure gradients and external forces. These nonlinear partial differential equations embody the conservation of mass and momentum for Newtonian fluids, while extensions for non-Newtonian and viscoelastic fluids introduce additional constitutive relations that capture memory effects and rate‐dependent behaviour. Analytical approaches, including exact and asymptotic solutions, complement numerical methods such as finite‐element and spectral discretisations, enabling insight into flow stability, transition and turbulence. Boundary conditions—ranging from no‐slip to Navier‐type slip or mixed regimes—play a crucial role in determining near‐wall dynamics, while coupling with heat transfer and magnetohydrodynamics extends the framework to non‐isothermal and electromagnetic flows. Applications span microfluidics, polymer processing, biological transport and geophysical circulation, where predictive models guide design optimisation and hazard mitigation. Recent advances have emphasised rigorous solution concepts, high‐fidelity simulation and unified treatment of mechanical, thermal and electromagnetic interactions in complex geometries.
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Mathematical Modeling of Viscous Fluid Dynamics publication trend
The graph below shows the total number of articles in mathematical modeling of viscous fluid dynamics across all publications each year (not limited to Nature Index journals).
Technical terms
Navier–Stokes equations: Governing equations for viscous fluid flow expressing conservation of mass and momentum.
Viscosity: Measure of a fluid’s resistance to deformation under shear or extensional stress.
Second grade fluid: A non‐Newtonian continuum model with constitutive terms accounting for normal stress differences and material memory.
Slip boundary condition: Boundary specification permitting finite relative velocity between fluid and solid surface proportional to shear stress.
Bifurcation: The mathematical phenomenon where small changes in system parameters lead to qualitative changes in solution structure or stability.
References
- Analytical Solutions to the Unsteady Poiseuille Flow of a Second Grade Fluid with Slip Boundary Conditions. Polymers (2024).
- Exact Solutions for Non-Isothermal Flows of Second Grade Fluid between Parallel Plates. Nanomaterials (2023).
- Uniqueness and Bifurcation Branches for Planar Steady Navier–Stokes Equations Under Navier Boundary Conditions. Journal of Mathematical Fluid Mechanics (2021).
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