Yeah, if ##b=0,a\neq 0## then the solution is ##y=\frac{x^2}{4a}+C_1\ln{x}+C_2##. If ##a=0,b\neq 0##, then it becomes the modified Bessel equation of order zero and the solution is ##y=C_1I_0\left(\frac{x}{\sqrt{b}}\right)+C_2K_0\left(\frac{x}{\sqrt{b}}\right)##.
It can be transformed if...
I'm trying to solve the following nonlinear second order ODE where ##a## and ##b## are constants: $$\frac{d^2y}{dx^2}+\frac{1}{x}\frac{dy}{dx}-\frac{y}{ay+b}=0$$ It looks somewhat like the modified Bessel equation, except the third term on the left makes it nonlinear. I've been trying to...
This all looks sound to me. I think the second to last term in the first equation should read ##\frac{1}{r}\frac{\partial(r \tau_{rz})}{\partial r}## if I'm doing my tensor divergence correctly, but it's being neglected later anyway. For step two, would it also be necessary to assume ##u_z=0##...
Well I have multiple sources now which give the following equation for the normal force balance as well as my own analysis that gives me that equation so I'm just going to run with it. If need be, I'll discount the hydrostatic terms later if they're already involved in the pressure calculations...
My only guess is they substituted the sum of the hydrostatic and dynamic pressures in. So in your equation, letting ##p=p_H+p_D## where ##p_H=p_0-\rho gz## and ##p_0## is the hydrostatic pressure at the undisturbed interface. Dropping the ##D## subscript from the dynamic terms gives their...
Doing some more reading, I found a book and journal paper by the same author on interfacial hydrodynamics which gives a balance of normal forces for a quasi-flat interface in cartesian coordinates. It gives the following description in the book:
"The second boundary condition, the so-called...
Starting from the top, I'm going to change some notations to make everything clear. The top, less dense, fluid will be designated with a superscript ##T## and the bottom, more dense, fluid will be designated using the superscript ##B##. The interfacial tension will be designated with ##\gamma##...
I realized I made a mistake in the derivation of the original equation. Instead it should read: $$ \frac{\sigma_I}{r} \frac{\partial}{\partial r} \left(r \frac{\partial z(r,t)}{\partial r} \right)=\rho_D gz(r,t)-p(r,t)-\frac{2\mu_T}{r} \frac{\partial}{\partial r} \left(r \frac{\partial...
No, the viscosity is different on either side as is density. In the above equation, I am only considering a free surface (such as a water-air interface) and saying the viscosity and density on the L (low density or fluid on top) side are very small. I assumed that would reduce the dynamic term...
Would it be valid to consider ##p## as the sum of the hydrostatic and hydrodynamic pressures so ##p=p_{\infty}-\rho gz+p_{dynamic}##? That seems to be what they did in that paper to derive their equation 1, I’m just wondering what assumptions are made to allow that. It seems like they use that...
Could an approximation or assumption be made that if the interface is not moving too quickly, the problem can be treated as quasi-static and the hydrostatic term is dominant and can thus be representative of the pressure ##p## at the interface? Otherwise, I'll need some other way to determine ##p##.
And is that also true for your derivation of ##n \cdot \sigma \cdot n## that the hydrostatic term is not valid for a deforming fluid? Or is it still there in the ##r## and ##z## coordinate system?
Instead of representing the viscous stress tensor in ##r## and ##z##, would it still be valid to use the normal, ##n##, and tangent, ##t##? So instead, the following could be written: $$ n \cdot \sigma \cdot n=-p_\infty+\rho gz+\tau_{nn}$$ where $$\tau_{nn}=2\eta \frac{\partial u_n}{\partial...