LHS of the Slender-Jet Axial Equation

The axial velocity of a slender jet is nearly uniform across each section. Radial advection is therefore small at leading order, but it contributes at the same order as the first correction to axial advection. We calculate both terms before truncating the acceleration.

Use dimensional and , with primes denoting . The expansions from continuity are

Here, denotes the omitted Taylor powers with their dimensional coefficients. Under the slender ordering and slow axial variation, is a relative velocity correction.

1. Calculate the local acceleration

Differentiating at fixed position gives

No steady-flow assumption has been made. Both time derivatives remain in their respective coefficient equations.

2. Calculate axial advection

Since

the axial advective term is

The second-order coefficient contains both the advection of by and the product .

3. Calculate radial advection

The radial gradient of axial velocity is

Multiplying by the radial velocity gives

This cancels the term from axial advection.

4. Collect the acceleration

Adding the three contributions gives

Multiplying by the density gives the inertial terms in the momentum equation. The coefficient is , as used in the leading-order model. The coefficient is , as used in (M2) of the second-order hierarchy. Dropping radial advection before collecting this coefficient would leave a spurious term.