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Let fluid 1 lie above fluid 2, with mass densities ${\rho }_1<{\rho }_2$ and dynamic viscosities ${\mu }_1<{\mu }_2$. Define $\Delta \rho ={\rho }_2-{\rho }_1>0$, $\Delta \mu ={\mu }_2-{\mu }_1>0$, porosity $\varphi$, permeability of a porous medium $k$, and downward interface speed $W$. With Darcy flux $\varphi W$, no capillarity, and two unbounded layers, Darcy's law givesFor a Fourier mode of interface displacement, the two perturbation pressures decay exponentially into the respective layers. The kinematic boundary condition makes their amplitudes proportional to $\varphi {\mu }_1\sigma /(k|\alpha |)$ and $-\varphi {\mu }_2\sigma /(k|\alpha |)$. Linearized pressure continuity adds the base-gradient difference $\varphi W\Delta \mu /k-g\Delta \rho$. Eliminating the amplitudes proves the formula. The critical interface speed is $kg\Delta \rho /(\varphi \Delta \mu )$; the critical Darcy velocity is $kg\Delta \rho /\Delta \mu$. Without surface tension this model has unbounded short-wave growth rate on its unstable side.

A spherical clean bubble with negligible internal dynamic viscosity has zero tangential traction and permits tangential slip. In the creeping-flow, negligible-deformation limit its drag is $-4\pi \mu a\mathbf{U}$, versus $-6\pi \mu a\mathbf{U}$ for a no-slip rigid sphere.

For constant positive $\lambda =p_r/p_g$ and molecular gas constant $\mathcal{R}=k/({\mu }_m{m}_p)$, the gas–radiation equation of state gives $\rho =4\sigma T^3/(3c\lambda \mathcal{R})$. Vertical hydrostatic equilibrium in harmonic central gravity therefore gives $T=T_0(1-z^2/H^2)$ with $T_0={\Omega }^2{H}^2/[8(1+\lambda )\mathcal{R}]$. The mass density and pressure powers are three and four. Integrating both faces yields $\Sigma =(32/35){\rho }_{0}H$ and $\int pdz=\Sigma {\Omega }^2{H}^2/9$. With constant opacity, radiative diffusion produces a linear flux and height-independent dynamic viscosity $\mu =4c\lambda /[9\kappa (1+\lambda )]$. This is an interior formal profile; its low-density surface requires a photospheric treatment beyond diffusion.

Assume a thin disk has height-independent opacity ${\kappa }_{\mathrm{op}}$ and radiation-to-gas pressure ratio $\beta$, with vertical gravity $-{\Omega }_z^{2}z$. Hydrostatic equilibrium and radiative diffusion implyBalancing its derivative against viscous heating $\rho \nu q^2{\Omega }^2$ fixes the dynamic viscosityIt is independent of height under these assumptions. For a spherically symmetric potential, ${\Omega }_z=\Omega$. Integrating $\rho \nu$ through the full surface-to-surface thickness $2H$ gives $\bar{\nu }\Sigma =2H\rho \nu$, which determines the thickness from the radial accretion rate solution.

Combining $F=c{\Omega }^{2}z/\kappa$ with viscous heating $dF/dz=(9/4)\mu {\Omega }^2$ gives a height-independent dynamic viscosity $\mu =4c/(9\kappa )$ under constant opacity. This exact limiting relation differs from an approximate vertically averaged alpha disk closure.

Darcy's law relates the Darcy velocity, the fluid flux per unit total cross-sectional area relative to a solid skeleton, to the driving pressure gradient. With no body force, $\mathbf{u}=-(k/\mu )\nabla p$, where $k$ is permeability of a porous medium and $\mu$ is dynamic viscosity. With gravity, the pressure gradient is replaced by $\nabla p-\rho \mathbf{g}$.

A dilute suspension of identical rigid spheres raises the dynamic viscosity by a fraction $5\varphi /2$, where $\varphi$ is the particle volume fraction. At imposed trace-free rate-of-strain tensor $E$, a sphere of radius $a$ adds viscous dissipation $20\pi \mu a^3(E:E)/3$. Multiplication by the number density $3\varphi /(4\pi a^3)$ gives $5\mu \varphi E:E$, which added to $2\mu E:E$ proves the formula. This is the leading noninteracting-sphere term; hydrodynamic interactions enter at higher concentration.

For constant positive order-parameter mobility $M$, dynamic viscosity $\eta$ and mass density $\rho$, deterministic Model H dynamics dissipates the sum of composition free energy and kinetic energy:Assume smooth fields and periodic or closed, no-work boundaries. Using the chemical potential of a composition field, integration by parts and incompressible flow gives $\dot{F}=\int \varphi \mathbf{v}\cdot \nabla \mu -M\int |\nabla \mu {|}^2$. Dotting the momentum equation with velocity gives ${\dot{E}}_{\mathrm{kin}}=-\int \varphi \mathbf{v}\cdot \nabla \mu -\eta \int |\nabla \mathbf{v}{|}^2$. Pressure and convective work vanish under the boundary assumptions; the two capillary transfer terms cancel. This proves both the sign of the Korteweg force and its reversible exchange of energy with the composition field.

For a bed depth $b=\alpha x$, a thin subglacial till layer of thickness $l$ is idealized by Newtonian till lubrication, with dynamic viscosity $\lambda \mu$ and basal drag $\lambda \mu u/l$. Neglecting longitudinal stress divergence in the bulk yields $u=-(l/\lambda )(g/\nu )H{h}_x$. Retaining the unbuttressed Newtonian grounding-line stress condition as boundary data and differentiating ice-sheet flotation along the moving grounding line givesIndeed, mass conservation gives $H_t=-u{H}_x-H{u}_x$, $H_x=h_x+\alpha$, and ice-sheet flotation gives $H_t=(\alpha {\rho }_w/\rho -H_x){\dot{x}}_G$. This is a formal local friction closure, not a uniformly valid removal of membrane stress near every grounding line. If the coefficient of ${\dot{x}}_G$ vanishes, it is an implicit compatibility condition and cannot be divided to obtain a finite speed.

Under single-length dynamical scaling of binary-fluid coarsening, mass density $\rho$, dynamic viscosity $\eta$ and surface tension $\sigma$ determine the displayed unique length and time scales by dimensional analysis. Writing $L=L_{0}f(t/t_0)$ reduces the force-density estimate to $\alpha f^{\prime \prime }+\beta f^{\prime 2}/f=\gamma f^{\prime }/f^2+\delta /f^2$. Resistive and driving terms need opposite signed geometric coefficients. Viscous hydrodynamic coarsening has $f\sim u$ and inertial hydrodynamic coarsening has $f\sim u^{2/3}$. These follow respectively from density-independent and viscosity-independent powers in $L\propto {\eta }^{2-3p}{\rho }^{p-1}{\sigma }^{2p-1}{t}^p$. They require thin interfaces, statistically self-similar bicontinuous domains, dominant advection and no additional relevant macroscopic scales. A degeneracy in the signed inertial coefficients can invalidate the pure-power force balance without invalidating the dimensional calculation.

Magma is molten or partially molten rock within a planet. Its dynamic viscosity, cooling and interaction with the surrounding elasticity influence transport and storage.

For isotropic incompressible flow of a Newtonian fluid, the stress tensor is the displayed expression, with pressure $p$, dynamic viscosity $\mu$, velocity $\mathbf{u}$, and identity matrix $I$. In Couette flow, its off-diagonal component is the shear stress.

A thin layer of subglacial till idealized as a Newtonian fluid of dynamic viscosity $\lambda \mu$ and thickness $l$ undergoes Couette flow between stationary bedrock and ice moving at speed $u$. Its resisting basal shear stress has magnitude ${\tau }_b=\lambda \mu u/l$ and acts opposite to the sliding motion. Combining this drag with the hydrostatic driving force gives the bulk approximation used in the friction-dominated grounding-line evolution law.