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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

\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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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

\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