Korteweg stress

In fluid mechanics and continuum mechanics, the Korteweg stress (or Korteweg stress tensor) is an additional component of the total Cauchy stress tensor, depending on the density gradients or concentration gradient. The stress tensor can be used to model the macroscopic effect of capillarity and cohesive forces via continuous local variations in density. Unlike classical hydrodynamics, where surface tension is treated as a sharp interfacial boundary condition, Korteweg's theory allows surface tension effects to be smoothly distributed across a diffuse interface where sharp gradients of density exist.

The tensor was first proposed by the Dutch mathematician Diederik Korteweg in 1901. It has significant applications in modeling liquid-vapor phase transitions, miscible fluid mixing, geodynamical flows. The Korteweg stress tensor can be derived from kinetic theories, that accounts for van der Waals' gradient energy. Besides the molecular origin, the Korteweg stress also can emerge from pure continuum transport, provided the internal flow is kinematically coupled to the scalar gradient.

The Korteweg stress tensor ${𝐓}_{\text{K}}$, for the scalar field $\rho$, is defined by

${𝐓}_{\text{K}}=({\alpha }_1\Delta \rho +{\alpha }_2|\mathrm{\nabla }\rho {|}^2)𝐈+{\alpha }_3\mathrm{\nabla }\rho \otimes \mathrm{\nabla }\rho +{\alpha }_4\mathrm{\nabla }\mathrm{\nabla }\rho$

where

* $|\mathrm{\nabla }\rho {|}^2$ is the gradient magnitude squared,

• $\Delta \rho$ is the Laplacian of the density ($\mathrm{\nabla }\cdot \mathrm{\nabla }\rho$),
• $\mathrm{\nabla }\rho \otimes \mathrm{\nabla }\rho$ denotes the dyadic product (tensor product), * $\mathrm{\nabla }\mathrm{\nabla }\rho$ is the Hessian matrix of the density,
• ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ are phenomenological Korteweg coefficients (material properties) that typically depend on $\rho$.

In 1985, J. E. Dunn and James Serrin rigorously examined the mathematical validity of Korteweg's proposed relation through the lens of rational thermodynamics. They proved that to satisfy the second law of thermodynamics, the classical balance of energy must be modified to include a non-classical energy supply term known as the "interstitial work flux" or "interstitial working". This energy flux ensures that the density-gradient terms do not violate local entropy production constraints. They also derived specific conditions for the Korteweg coefficients if the Korteweg stress is used to model diffuse interfaces in colloidal suspensions and miscible liquid systems (where two liquids mix without a sharp interface and temporary, off-equilibrium localized surface tension exists). However, in the emergent Korteweg stress due to hydrodynamic origin, the Dunn–Serrin condition does not hold.

See also

• Cauchy stress tensor
• Navier–Stokes equations
• Surface tension
• Capillary action

References

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