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A homologous spherical flow has velocity proportional to radius. Its velocity gradient is $q(t)I$ and its velocity divergence is $3q(t)$.

An incompressible flow preserves the volume of each moving fluid element. Its local condition is zero velocity divergence, $\nabla \cdot \mathbf{u}=0$. Combined with mass conservation, it says that each parcel's mass density stays unchanged as it moves. This does not require every parcel to have the same mass density: an initially nonuniform distribution can be transported without compression. Nor does incompressibility require a constant velocity or constant pressure.

For positive mass density, the continuity equation makes the equivalence precise:The material derivative follows a fluid element. Thus $D\rho /Dt=0$ if and only if the velocity field is a divergence-free vector field. The distinction between following a parcel and observing a fixed position prevents confusing incompressibility with an unchanging density field. ^[ref]

For a rigid pipe with impermeable walls, no leaks and steady flow, mass conservation at constant mass density implies a constant volumetric flow rate:Here $\mathbf{n}$ points downstream, $A$ is the section's area and $\overline{U}$ is the mean normal velocity, not necessarily the centreline speed. No assumption of inviscid flow or a uniform velocity profile is needed. If unequal amounts entered and left a fixed segment, its mass would accumulate, contradicting steady flow.

For a worked example, take horizontal water flow with stipulated mass density $1000\mathrm{kg}{\mathrm{m}}^{-3}$, inlet radius $R_1=0.010\mathrm{m}$, outlet radius $R_2=0.005\mathrm{m}$ and inlet mean velocity ${\overline{U}}_1=0.20\mathrm{m}{\mathrm{s}}^{-1}$. The circular areas giveThe radius halves, the area falls to one quarter, and the mean velocity quadruples. The mass flow rate is $\rho Q=0.02\pi \mathrm{kg}{\mathrm{s}}^{-1}$. The same amount of water passes both sections each second; narrowing changes its speed rather than compressing it. The Reynolds number and Bernoulli's principle discussions use this same example to examine viscous effects and the additional assumptions needed to calculate pressure.

Write $\mathcal{H}=a^{\prime }/a$ for the conformal Hubble parameter and ${\theta }_N=i\mathbf{k}\cdot {\mathbf{v}}_N$ for the velocity divergence of a Fourier mode. Work in the synchronous gauge comoving with the cold dark matter, so ${\mathbf{v}}_C=0$. This remains consistent with its pressureless momentum equation ${\mathbf{v}}_C^{\prime }+\mathcal{H}{\mathbf{v}}_C=0$. Its density equation then gives ${\delta }_C^{\prime }=-h^{\prime }/2$.

For the string component, $1+3w_S=0$. Thus it makes no contribution to the trace gravitational source in the supplied perfect-fluid model. Substitute $h^{\prime }=-2{\delta }_C^{\prime }$ and $h^{\prime \prime }=-2{\delta }_C^{\prime \prime }$ in that trace equation to obtainThe absence of a pressure-gradient term reflects the zero cosmological sound speed of the cold matter.

For the $w_S=-1/3$ coasting fluid, the continuity and momentum equations becomeThe second equation follows by taking $i\mathbf{k}\cdot$ the vector equation; in particular, $i\mathbf{k}\cdot i\mathbf{k}=-k^2$, which fixes the pressure sign. Differentiate the first equation and use ${\theta }_S={\delta }_C^{\prime }-3{\delta }_S^{\prime }/2$:Eliminating ${\delta }_C^{\prime \prime }$ with the cold-matter equation givesThese are equations for the specified idealized barotropic string fluid. A cosmic string network with additional anisotropic stress need not obey the same pressure response.

The given velocity is a homologous spherical flow,with the spatially uniform velocity divergenceEvery shocked fluid element therefore expands rather than undergoing continued compression, and the velocity fills the remnant smoothly instead of concentrating its mass in a thin cooling shell. In the absence of radiative losses, the entropy advection equation then makes each element follow an adiabatic process after its one entropy-producing passage through the shock. This broad expanding structure is the expected adiabatic phase of a supernova remnant.