Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 75 / 3

For a slender viscous thread, leading axial velocity is uniform over a section. Incompressibility gives $u_r=-r{w}_z/2$. The radial and azimuthal strain rates are $-w_z/2$, and the axial one is $w_z$. The Newtonian fluid stress tensor then has ${\sigma }_{rr}=-p-\mu w_z$. Matching the inviscid exterior normal stress requires $p=p_a-\mu w_z$, so ${\sigma }_{zz}=-p_a+3\mu w_z$. The coefficient three is the Trouton ratio.

A short section has end-stress resultant ${\partial }_z(A{\sigma }_{zz})dz$. The side's hydrostatic traction adds $p_a{A}_{z}dz$, and gravity adds $\rho gAdz$. Neglecting inertia and capillarity, their sum is zero:Since $(p_a{)}_z={\rho }_{a}g$, the balances areThe second is cross-sectional volume conservation, equivalently $D_{t}A=-A{w}_z$. Small interface slope justifies the sectionwise extensional approximation; the slow-flow assumption is additionally needed to neglect inertia.

For steady nonzero flux, $Aw=Q$, so $A=Q/w$. Substitution in the slender viscous thread force balance givesIf $f=w_z/w$, thenHence $\boxed{\frac{1}{2}(w_z/w{)}^2=U/w+c}$. This argument does not divide by $w_z$, so it also applies at turning points of the axial velocity. The symbol $U$ here has dimensions inverse length-time; it is not itself a speed.

With no net body force, the tensile resultant is constant along the thread: $3\mu A{w}_z=F(t)$. In material coordinates $z_0$, volume conservation gives $D_{t}A=-F/(3\mu )$. ThusThe material Jacobian satisfies $A\partial z/\partial z_0=A_0$. Integrating with $z(0,t)=0$ givesThis remains valid while $\Delta <\min A_0$. The even profile now specified ensures symmetric endpoints $\pm L(t)$.

For $A_0=C(1+k^2{z}_0^2)$ and $k\ne 0$, direct integration yieldsFor $k=0$, instead $L=C{L}_0/(C-\Delta )$. A constant tensile $F>0$ gives $\Delta =Ft/(3\mu )$ and the finite blow-up timePut $\alpha =F/(3\mu )$ and $\delta =C-\Delta =\alpha (t^{\ast }-t)$. For fixed nonzero $k$, the arctangent tends to $\pi /2$, whereas for the uniform profile the whole length has the same thinning. Consequently the localized-neck and uniform-thread stretching exponents areA nonuniform thread's divergent integral is concentrated in a material interval $|z_0|\sim \sqrt{\delta /C}/|k|$ near its quadratic minimum. Its width shrinks like ${\delta }^{1/2}$ while the Jacobian grows like ${\delta }^{-1}$, yielding ${\delta }^{-1/2}$. The remote ends no longer affect this leading coefficient, explaining independence from $L_0$. For $k=0$ every element contributes the same ${\delta }^{-1}$ stretch. The limits $k\to 0$ and $t\to t^{\ast }$ do not commute.