indexhw3

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

  • $$

    $$

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

$$\left(\frac{d^{2}}{d y^{2}}-(k d)^{2}\right)^{3} V_{\theta}=\frac{4 \kappa^{2} d^{4} \Omega_{1} A}{\nu^{2}}(1+\alpha y) V_{\theta}$$

• 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