MANE 6560: Homework 4

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

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

(d2dy2(kd)2)3Vθ=4κ2d4Ω1Aν2(1+αy)Vθ\left(\frac{d^{2}}{d y^{2}}-(k d)^{2}\right)^{3} V_{\theta}=\frac{4 \kappa^{2} d^{4} \Omega_{1} A}{\nu^{2}}(1+\alpha y) V_{\theta}
  • 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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