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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

  • $$

    $$

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:

Linear dispersive waves

Described by

k is wave number, related to wave length

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:

laplacian or harmonic of potential is 0

1d pde

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

at FS for dynamic free surface condition

show dispersion relation: ,

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

Ch