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# index

## 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 

## Incompresible Flow

Incompresible FlowLecture 1TextbooksIncompressible FlowList of Derivatives Surface Waves and Interfacial PhenomenaIntroduction to waves Continuity EquationLinear dispersive wavesLecture 2Lecture 8: Weakly Nonlinear Waves in Deep Water

### Lecture 1

Class 1: 2020-01-13T15:49:925

Covered course outline and such; assigned first homework

Topics Required to do basic research into FM is the typical coursework of

* Fluid Mechanics
* Viscous Flow & Boundary Layer Theory
* Turbulence
* Compressible flow
* kinetic theory
* continuum theory of fluids
* combustion aerodynamics

part of the big picture this is one of the topics

### Textbooks

Lighthill waves in fluids

#### Incompressible Flow

Density change is negligible which is seen in low 

#### List of Derivatives

* Partial
* Material
* Total

### Surface Waves and Interfacial Phenomena

#### Introduction to waves

Research in fluid dynamics requires the typical coursework

* fluid mechanics
* viscous flow and BL theory
* turbulence
* compressible flow
* kinetic theory
* continuum mechanics for fluids
* speciality courses: aero and combustion

Compressibility  so when  has M <0.3

for water M less than 0.07 much faster speed of sound for compressibility to act

#### Continuity Equation

Conservation of mass:&#x20;

#### Linear dispersive waves

Described by

k is wave number, related to wave length&#x20;

euler identity to show exponent is purely imaginary and is the phase function , the amplitude the real component

using dimensionless NS

### Lecture 2

Major Equations

euler eqns:&#x20;

laplacian or harmonic of potential is 0&#x20;

1d pde&#x20;

apply to momentum potential and integrate : produces Bernoulli incompressible irrotational flow

at FS  for dynamic free surface condition

show dispersion relation: ,&#x20;

so  and for deepwater

shallow water   so  and . y

## Lecture 8: Weakly Nonlinear Waves in Deep Water

due to Rayleigh in a stead motion in coordinate system moving a phase speed c

also

clearly nonlinear

small amplitude but keep high order terms expanded wrt a

substitute into&#x20;

Ch