# 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: #### 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 --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. 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