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Take the curl of the Navier-Stokes equation. The pressure term disappears, curl commutes with the Laplacian, and the incompressible vector identitygivesTherefore the material derivative form is

Let $D/Dt={\partial }_t-Sx{\partial }_y$ be the material derivative along the background linear shear flow ${\mathbf{u}}_0=-Sx{\mathbf{e}}_y$. Linearizing about constant $\Sigma$ givesThe coefficient $2\Omega -S$ includes the contribution $\mathbf{v}\cdot \nabla {\mathbf{u}}_0=-S{v}_x{\mathbf{e}}_y$ from perturbation advection of the background shear.

For zero forcing, the given equation for $v_y$ isUse the shearing-wave ansatzThe explicit $x$ dependence cancels from the material derivative whenThe amplitude then obeys the time-dependent harmonic oscillator equationThis evolving wavevector is the characteristic signature of a shearing wave.

An objective time derivative transforms as a tensor under every superposed time-dependent rigid translation and rotation of the observer. A constitutive prediction using it is therefore frame-indifferent. The upper-convected derivative, lower-convected derivative, and Jaumann derivative are objective; the componentwise partial time derivative, and generally the uncorrected material derivative of a tensor, are not objective under rotating observers.

The Johnson--Segalman--Oldroyd model uses the Gordon--Schowalter derivativeHere $\eta$ is the zero-shear viscosity, ${\lambda }_1$ is the stress-relaxation time, and ${\lambda }_2$ is the retardation time associated with the more rapidly responding viscous contribution. Passive Oldroyd-type materials normally have $0\le {\lambda }_2\le {\lambda }_1$.

A low-rate steady-shear experiment measures $\eta$. A step strain followed by stress relaxation determines ${\lambda }_1$ from the exponential decay. Creep and recovery, or a frequency sweep in small-amplitude oscillatory shear, then separates ${\lambda }_2$ from ${\lambda }_1$; fitting both storage modulus and loss modulus is especially direct.

For the homologous spherical flow $\mathbf{u}=q\mathbf{r}$, the ideal magnetohydrodynamic induction equation and $\nabla \cdot \mathbf{B}=0$ give the material derivativeLet $x=r/R(t)$ and use the self-similar ansatzPart (b) gives $q=3\dot{R}/(4R)$, and thereforeThe radial induction equation becomes $x{f}^{\prime }=6f$. The boundary value $f(1)=1$ gives $f=x^6$. The solenoidal vector field condition requireswhich also matches the tangential shock value in part (c). Hence the interior field is

Outside the shock, the uniform-field lines obey $r\sin \theta =\text{constant}$. Inside, the magnetic-field-line equation givessoA sketch therefore shows straight exterior lines refracting at the spherical shock into north-south symmetric curves that bow toward the equatorial interior before leaving through the opposite hemisphere. The field strength falls as $r^6$ toward the centre.

Write the density as a hydrostatic reference profile $\hat{\rho }(z)$ plus a small perturbation ${\rho }^{\prime }$, and define buoyancy and buoyancy frequency byThe Boussinesq approximation to the Navier-Stokes equation, together with mass conservation, iswhere $D/Dt$ is the material derivative, $\nu$ is kinematic viscosity, and $\kappa$ is mass diffusivity. Linearizing about rest and eliminating pressure and buoyancy givesFor a plane wave proportional to $e^{i(kx+mz-\omega t)}$, with $K^2=k^2+m^2$, the viscous-diffusive dispersion relation isIn the inviscid limit this becomesFor weak diffusion the two oscillatory roots areso a freely evolving Fourier mode decays.

A single plane wave is also an exact solution of the nonlinear equations. Every field depends only on its phase $\Theta =\mathbf{k}\cdot \mathbf{x}-\omega t$, while incompressible flow gives $\mathbf{u}\cdot \mathbf{k}=0$. Therefore $\mathbf{u}\cdot \nabla$ annihilates both $\mathbf{u}(\Theta )$ and $b(\Theta )$, and all nonlinear advection terms vanish.

For a boundary-forced wave with real $\omega$, nonzero $\nu$ or $\kappa$ instead makes the bulk vertical wavenumber complex, attenuating the propagating beam. Because diffusion raises the spatial order of the equations, additional short vertical-wavenumber roots form viscous and scalar boundary layers; they allow a no-slip velocity condition and a scalar no-flux condition to accompany impermeability. These layers and bulk attenuation become essential near critical internal-wave reflection, where the inviscid reflected wavelength collapses.

The boundary is material precisely when its material derivative vanishes on $f=0$. Direct differentiation givesFor this to vanish at every point of the ellipsoidal free surface, each coefficient must vanish:With these relations the expression vanishes for every interior point as well, sothroughout the affine stellar model. Thus $f$ is a Lagrangian label carried by each fluid element.

A plain material derivative of the conformation tensor is not an objective time derivative: an observer undergoing a time-dependent rigid rotation would infer a different constitutive response, and even rigid-body rotation could appear to change polymer deformation. The upper-convected derivative subtracts deformation and rotation carried by the velocity gradient and is frame indifferent.

The Oldroyd-B model is often inadequate because its Hookean dumbbells are infinitely extensible. It predicts constant shear viscosity rather than shear thinning, zero second normal-stress difference, and an unbounded extensional viscosity at a finite extension rate. Real polymer chains have finite extensibility and commonly exhibit shear thinning, bounded extensional stress, multiple relaxation times, and nonlinear solvent or concentration effects.

A potential vorticity budget has the formThe material derivative follows the fluid motion and $S_q$ collects forcing or dissipation. With $S_q=0$ it becomes potential-vorticity conservation. For a quasi-geostrophic streamfunction, horizontal advection is $J(\psi ,q)={\psi }_x{q}_y-{\psi }_y{q}_x$.

The amplitude-dependent extra potential in the Madelung equations. Taking the gradient of their phase equation and using the guidance equation yields $mD\mathbf{v}/Dt=-\nabla (V+Q)$, where $D/Dt$ is the material derivative. At wavefunction nodes this local expression may be singular.

The material derivative of surface concentration combines tangential and normal area change, surface diffusion, and relaxation towards a reservoir concentration $C_0$. The rate $k$ represents exchange; setting $k=0$ recovers transport without exchange. The concentration is measured per interfacial area, so stretching dilutes it.

For equal-depth layers coupled by an interface,The Laplacian terms are relative vorticity; the streamfunction differences encode opposite layer-thickness changes. Each material derivative follows that layer's own velocity. The sum has a barotropic mode without interface displacement, while the difference contains a baroclinic mode.