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MANE 6560: Homework
From: Chris Nkinthorn, 2020-01-23
For: Prof. A. Hirsa for Incompressible Flow
Problem 1: Water Jet Stability
Consider the temporal instability of a water jet in the absence of gravity the jet diameter $2R$ is $1 \text{ [mm]}$ in his flowing at $1 \text{ [m/s]}$ based on linear theory
Begin first by linearizing the pressure and velocity fields
$\mathbf{u}=\mathbf{u}_{s}+\mathbf{u}^{\prime}$
$p=p_{s}+p^{\prime}$
These linearized disturbance equations are subsituted into the equations of
Continuity ... $\nabla\cdot{\bf{u}} = 0 $
Momentum ... $\dot{\bf{u}} + \frac{\nabla p'}{\rho} = \alpha T'g \bf{k} +\nu\nabla^2\bf{u} $
the gradient of the static pressure field is identically zero
Use the relationship of energy
Part a: Fundamental Wavelength
use the NS and continuity to find a 6th order ODE describing the viscous momentum balance in terms of position which varies only along the length of the jet, and time
Use the definitions of radii of curvature from calculus
$R_1 = r$
$R{2}=\frac{-\left[1+\left(\frac{\partial r{0}}{\partial x}\right)^{2}\right]^{3 / 2}}{\left(\frac{\partial^{2} r_{0}}{\partial x^{2}}\right)}$
$\left(\frac{\partial}{\partial t}+u \frac{\partial}{\partial x}\right)^{2} r=-\frac{\sigma R}{2 \rho}\left(\frac{1}{R^{2}} \frac{\partial^{2} r}{\partial x^{2}}+\frac{\partial^{4} r}{\partial x^{4}}\right)$
an initial disturbance of the form
$r = a \text{ e}^{kx -\omega t}$ much smaller than the nozzle diameter $10^{-3}R$
Part b: Fundamental Mode
Part c: Volume of Diameter
diameter of a drop is the length of the unstable wavelength found in part a
Part d: Time to pinch off
initial disturbance ... 10E-3 fundamental wavelength
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Problem 2: Thermal Convection for Rotating Gap
Rayleigh-Bernard convection in couette flow between rotating cylinders in the narrow gap approximation can be described by similar sets of equations. In the stress free condition, a $6^\text{th}$ order ODE suffices
Taylor Number … $T=\frac{4 d^{4} \Omega_{1} A}{\nu^{2}}$, which relates the rotational centrifugal force to the viscous force in a fluid
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