# Leading-Order Dynamic BC in a Slender Jet
Leading-order free-surface stress balances (normal and tangential) used in [[polymeric-stress-regularity-slender-jet|Slender jet momentum equations]].
Use cylindrical coordinates $(r,z)$, axisymmetry, no swirl, interface $r=h(z,t)$.
Total liquid stress:
$
\boldsymbol{\sigma}=-p\,\boldsymbol{I}+2\eta\,\boldsymbol{D}+\boldsymbol{\sigma}_p.
$
With passive gas outside, $\boldsymbol{\sigma}^{\mathrm{out}}=-p_a\boldsymbol{I}$, and constant surface tension $\gamma$,
$
(\boldsymbol{\sigma}-\boldsymbol{\sigma}^{\mathrm{out}})\cdot\boldsymbol{n}=-\gamma\kappa\,\boldsymbol{n}.
$
#### Projections:
$
\boldsymbol{n}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{n}=-p_a-\gamma\kappa,
\qquad
\boldsymbol{t}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{n}=0.
$
#### Geometry:
$
\boldsymbol{n}=\frac{\boldsymbol{e}_r-h_z\boldsymbol{e}_z}{\sqrt{1+h_z^2}},
\qquad
\boldsymbol{t}=\frac{h_z\boldsymbol{e}_r+\boldsymbol{e}_z}{\sqrt{1+h_z^2}},
\qquad
\kappa=\frac{1}{h}+O(\varepsilon^2),
\qquad
h_z=O(\varepsilon).
$
#### Long-wave fields:
$
v_z(r,z,t)=v_0(z,t)+\varepsilon^2 r^2 v_2(z,t)+\cdots,
$
$
v_r(r,z,t)=-\frac{1}{2}v_0'(z,t)\,r-\frac{1}{4}\varepsilon^2 v_2'(z,t)\,r^3+\cdots,
$
See: [[continuity-slender-jet|Continuity equation in slender jet]]
$
\sigma_{p,rr}=\Sigma_{rr}(z,t)+O(\varepsilon^2 r^2),
\quad
\sigma_{p,zz}=\Sigma_{zz}(z,t)+O(\varepsilon^2 r^2),
\quad
\sigma_{p,rz}=r\,S(z,t)+O(\varepsilon^2 r^3).
$
See: [[polymeric-stress-regularity-slender-jet|Polymeric stress in slender jet]]
Needed derivatives at $r=h$:
$
\partial_r v_r=-\frac{1}{2}v_0'+O(\varepsilon^2),
\qquad
\partial_z v_z=v_0'+O(\varepsilon^2),
$
$
\partial_r v_z+\partial_z v_r=-\frac{1}{2}h\,v_0''+2h\,v_2+O(\varepsilon^2).
$
## 1) Leading-order normal stress balance
At small slope, $\boldsymbol{n}\approx\boldsymbol{e}_r$, so
$
\boldsymbol{n}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{n}
=-p+2\eta\,\partial_r v_r+\sigma_{p,rr}+O(\varepsilon^2).
$
Using $\boldsymbol{n}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{n}=-p_a-\gamma\kappa$:
$
p=p_a+\gamma\kappa+2\eta\,\partial_r v_r+\sigma_{p,rr}+O(\varepsilon^2).
$
Therefore
$
p_0(z,t)=p_a+\frac{\gamma}{h}-\eta v_0'+\Sigma_{rr}+O(\varepsilon^2).
$
If pressure is gauged by $p_a=0$:
$
\boxed{p_0=\frac{\gamma}{h}-\eta v_0'+\Sigma_{rr}+O(\varepsilon^2).}
$
## 2) Leading-order tangential stress balance
At small slope, expand $\boldsymbol{t}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{n}=0$ to first nontrivial order:
$
\eta\left[(\partial_r v_z+\partial_z v_r)+2h_z(\partial_r v_r-\partial_z v_z)\right]
+\boldsymbol{t}\cdot\boldsymbol{\sigma}_p\cdot\boldsymbol{n}
=O(\varepsilon^2).
$
Newtonian contribution:
$
\partial_r v_z+\partial_z v_r=-\frac{1}{2}h v_0''+2h v_2+O(\varepsilon^2),
$
$
2h_z(\partial_r v_r-\partial_z v_z)=-3v_0'h'+O(\varepsilon^2),
$
so
$
\eta\left(-3v_0'h'-\frac{1}{2}h v_0''+2h v_2\right)+\boldsymbol{t}\cdot\boldsymbol{\sigma}_p\cdot\boldsymbol{n}
=O(\varepsilon^2).
$
Polymer projection at small slope:
$
\boldsymbol{t}\cdot\boldsymbol{\sigma}_p\cdot\boldsymbol{n}
=\sigma_{p,rz}(h)+h'\big(\Sigma_{rr}-\Sigma_{zz}\big)+O(\varepsilon^2).
$
Hence
$
\eta\left(-3v_0'h'-\frac{1}{2}h v_0''+2h v_2\right)
+\sigma_{p,rz}(h)
+h'\big(\Sigma_{rr}-\Sigma_{zz}\big)
=O(\varepsilon^2).
$
Define $\Delta\Sigma\equiv\Sigma_{zz}-\Sigma_{rr}$, then equivalently
$
\eta\left(-3v_0'h'-\frac{1}{2}h v_0''+2h v_2\right)
+\sigma_{p,rz}(h)-\Delta\Sigma\,h'
=O(\varepsilon^2).
$