# Viscoelastic Slender Jets ### Parity argument (axis regularity + axisymmetry): #### Velocity and pressure Even powers for $v_z, p$ and odd for $v_r$. > [!important] Compact notation: > Primes are $\partial_z$: Expanding $v_z$ in powers of $r$: $ v_z(r,z,t)=v_0(z,t)+\varepsilon^2 r^2 v_2(z,t)+\cdots, $ Using continuity, $v_r$ in the leading order is (see: [[continuity-slender-jet|Continuity equation in slender jet]]) $ 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, $ $ p(r,z,t)=p_0(z,t)+\varepsilon^2 r^2 p_2(z,t)+\cdots. $ #### Stress The Cauchy stress is $ \boldsymbol{\sigma} = -p\boldsymbol{I} + 2\eta \boldsymbol{D} + \boldsymbol{\sigma}_p, $ The momentum balance, free-surface traction balance, and shear-free condition use this $\boldsymbol{\sigma}$. Expanding polymer stresses consistently with [[polymeric-stress-regularity-slender-jet|axisymmetry/regularity]]: $ \sigma_{p,zz}(r,z,t)=\Sigma_{zz}(z,t)+O(\varepsilon^2 r^2),\qquad \sigma_{p,rr}(r,z,t)=\Sigma_{rr}(z,t)+O(\varepsilon^2 r^2), $ $ \sigma_{p,rz}(r,z,t)= r\,S(z,t)+O(\varepsilon^2 r^3). $ > [!note] Hoop stress > Regularity also implies $\Sigma_{rr}=\Sigma_{\theta\theta}$ at leading order. ### Governing equations #### Axial momentum at $O(\varepsilon^0)$ Polymeric stress contribution: $ (\nabla\cdot\boldsymbol{\sigma}_p)_z=\partial_z \sigma_{p,zz}+\frac1r\partial_r(r\sigma_{p,rz}). $ Using $\sigma_{p,zz}\approx \Sigma_{zz}(z,t)$ and $\sigma_{p,rz}\approx r S(z,t)$, $ \frac1r\partial_r(r\sigma_{p,rz})=\frac1r\partial_r(r^2 S)=2S. $ Thus the $r^0$ coefficient of the axial equation becomes (why factor 4? See: [[Laplacian-in-axisymmetric-slender-jet|Laplacian in axisymmetric slender jet]]. For full derivation of LHS, see [[LHS-slender-jet|LHS of slender jet momentum]]) $ \rho\left(\partial_t v_0+v_0 v_0'\right)=-p_0'+\eta(4v_2+v_0'')+\Sigma_{zz}'+2S+O(\varepsilon^2). $ #### Kinematic BC: See [[continuity-slender-jet|Continuity equation in slender jet]] for details $ \partial_t(h^2)+\partial_z(h^2 v_0)=0, $ #### Leading-order stress BCs with polymer normal + shear stresses > [!info] See: [[dynamic-BC-slender-jet|Full free surface stress balance]] for details. Normal traction (leading order, $n\approx e_r$): $ p_0=\frac{\gamma}{h} - \eta v_0' + \Sigma_{rr}+O(\varepsilon^2). $ Tangential traction (leading order, $t\approx e_z$, $n\approx e_r$): $ \underbrace{\eta\Big(-3v_0'h'-\frac12 v_0'' h+2v_2 h\Big)}_{\text{Newtonian part}} +\underbrace{\sigma_{p,rz}(h)}_{\approx hS} +\underbrace{(\Sigma_{rr}-\Sigma_{zz})h'}_{=-\Delta\Sigma\,h'} =O(\varepsilon^2), $ where polymeric “tensile” normal-stress difference is $ \Delta\Sigma \equiv \Sigma_{zz}-\Sigma_{rr}. $ #### Combining Using the tangential stress balance to solve for $v_2$ (substituting $O(\varepsilon^2) \to 0$): $ 2\eta v_2 h =3\eta v_0'h' +\frac{\eta}{2} v_0'' h -\sigma_{p,rz}(h)+\Delta\Sigma\,h'. $ Next use the normal stress BC for $p_0$: $ p_0'=(\gamma/h)'-\eta v_0''+\Sigma_{rr}'. $ Now substitute into the axial momentum equation and using $\sigma_{p,rz}\approx r S(z,t)$: $ \rho\left(\partial_t v_0+v_0 v_0'\right) = -\Big(\frac{\gamma}{h}\Big)' +3\eta\frac{(h^2 v_0')'}{h^2} +\frac{(h^2\Delta\Sigma)'}{h^2} +O(\varepsilon^2), $ with $\Delta\Sigma=\Sigma_{zz}-\Sigma_{rr}$. Here, 3 is the Trouton ratio. $\Sigma_{rr}=\Sigma_{\theta\theta}$ at leading order (so “hoop” is already accounted for). > [!important] > What happened to the shear stress $\sigma_{p,rz}$? It enters both (i) the axial momentum equation through $(\nabla\cdot\sigma_p)_z$, and (ii) the tangential traction condition $t\cdot\sigma\cdot n=0$. Those two appearances will cancel. ### Conservative form Continuity (see: [[continuity-slender-jet|Continuity equation in slender jet]] for details): $ \boxed{\partial_t h^2 + \partial_z(h^2 v_0)=0} $ Momentum equation: $ \rho(\partial_tv_0 + v_0 v_0') = -\Big(\frac{\gamma}{h}\Big)' + 3\eta\frac{(h^2 v_0')'}{h^2} + \frac{(h^2\Delta\Sigma)'}{h^2}. $ Multiply by $h^2$: $ \rho h^2(\partial_tv_0 + v_0 v_0') = -h^2\Big(\frac{\gamma}{h}\Big)' + 3\eta\left(h^2 v_0'\right)' + \left(h^2\Delta\Sigma\right)'. $ Using $h^2(1/h)' = -h'$: $ \rho h^2(\partial_tv_0 + v_0 v_0') = \gamma\Big(h\Big)' + 3\eta\left(h^2 v_0'\right)' + \left(h^2\Delta\Sigma\right)'. $ On the left hand side: $ \partial_t(h^2 v_0) + (h^2 v_0^2)' = h^2(v_t + v v') \quad\text{if (by continuity equation)}\quad \partial_t(h^2)+(v_0h^2)'=0. $ The conservative momentum equation is $ \rho\left(\partial_t(h^2 v_0) + (h^2 v_0^2)'\right) = \left(\gamma h + 3\eta h^2 v_0' + h^2\Delta\Sigma\right)'. $ $ \boxed{ \rho\left(\partial_t(h^2 v_0) + \partial_z(h^2 v_0^2)\right) = \partial_z\left[\gamma h + 3\eta h^2 \partial_zv_0 + h^2\left(\sigma_{p,zz}-\sigma_{p,rr}\right)\right]. } $