- #1
member 428835
Hi PF!
I have been on and off working on a fluids problem for 2 years. I am SO close but the answer isn't coming out clean. I'll highlight the equations I solve and the technique. If you can help me finish this, I'll not only be incredibly grateful but I'll either thank you in the paper acknowledgments or if you're interested we can discuss authorship with you. I know I am SO close! Here's the problem: given an inviscid liquid in a cylindrical tank of radius 1 and height h, what are the resonant frequencies of the liquid? Gravity is off, so we also must be given the contact angle ##\alpha## of the liquid (##\alpha = 90^\circ## is very simple, and is the benchmark with which to compare for general ##\alpha##). Before continuing, since the fluid is assumed inviscid, let's call the potential field ##\phi##. Also, let's call the cylindrical container ##\Sigma##, the equilibrium interface ##\Gamma## (which will be a spherical cap, since zero gravity, unless ##\alpha = 90^\circ##, which it's then perfectly flat, which is parameterized in terms of it's surface coordinate ##s##), and let ##\gamma## define the equilibrium gas-liquid-solid line of contact, or the contact-line. Denote subscript ##n## as the normal derivative to the surface for which it is prescribed over.
The equations of motion are: $$\nabla^2 \phi = 0 \,\,\, (\Omega) $$
$$ \phi_n = 0 \,\,\, (\Sigma) $$
$$\int_\Gamma \phi_n = 0 \,\,\, (\Gamma) $$
$$\phi_n'(s) + \left( k \cot \alpha - \frac{ \overline k}{\sin\alpha} \right)\phi_n = 0 \,\,\, (\gamma) $$
$$-\Delta_\Gamma \phi_n - (k_1^{-2} + k_2^{-2})\phi_n = \lambda^2 \phi \,\,\, (\Gamma)$$
Equation 1 is continuity. Equation 2 is no penetration through the cylinder walls (sides and bottom). Equation 3 is conservation of volume. Equation 4 is a constant contact angle condition (as the surface is disturbed, the contact angle ##\alpha## never changes). Equation 5 is an energy balance between kinetic and potential energy (capillary and inertial), where ##\lambda## is the eigenvalue of the system. All ##k## values are curvatures, which I'll explain in detail for the interested reader. ##\Delta_\Gamma## is the Laplace-Beltrami operator.
The solution approach: analytically solve equations 1-3 for the ##\alpha = 90^\circ## case. This is simple to do via separation of variables. Build equation 4 into the an inverse operator (more on this for those interested). Use these solutions as basis functions for equation 5, which will be solved via the Rayleigh-Ritz variational approach.
Now there's a few tricks along the way, which I can describe in detail, and there's a paper on sessile drops which outlines the process for drops on a flat surface. Our problem behaves identically to the sessile drop.
In short, if you're good with math, familiar with spectral techniques, and like fluid dynamics, please let me know. I would seriously appreciate any help you can offer! All code I've used is in Mathematica, and I can implement any suggestions you have so you don't even have to code.
I have been on and off working on a fluids problem for 2 years. I am SO close but the answer isn't coming out clean. I'll highlight the equations I solve and the technique. If you can help me finish this, I'll not only be incredibly grateful but I'll either thank you in the paper acknowledgments or if you're interested we can discuss authorship with you. I know I am SO close! Here's the problem: given an inviscid liquid in a cylindrical tank of radius 1 and height h, what are the resonant frequencies of the liquid? Gravity is off, so we also must be given the contact angle ##\alpha## of the liquid (##\alpha = 90^\circ## is very simple, and is the benchmark with which to compare for general ##\alpha##). Before continuing, since the fluid is assumed inviscid, let's call the potential field ##\phi##. Also, let's call the cylindrical container ##\Sigma##, the equilibrium interface ##\Gamma## (which will be a spherical cap, since zero gravity, unless ##\alpha = 90^\circ##, which it's then perfectly flat, which is parameterized in terms of it's surface coordinate ##s##), and let ##\gamma## define the equilibrium gas-liquid-solid line of contact, or the contact-line. Denote subscript ##n## as the normal derivative to the surface for which it is prescribed over.
The equations of motion are: $$\nabla^2 \phi = 0 \,\,\, (\Omega) $$
$$ \phi_n = 0 \,\,\, (\Sigma) $$
$$\int_\Gamma \phi_n = 0 \,\,\, (\Gamma) $$
$$\phi_n'(s) + \left( k \cot \alpha - \frac{ \overline k}{\sin\alpha} \right)\phi_n = 0 \,\,\, (\gamma) $$
$$-\Delta_\Gamma \phi_n - (k_1^{-2} + k_2^{-2})\phi_n = \lambda^2 \phi \,\,\, (\Gamma)$$
Equation 1 is continuity. Equation 2 is no penetration through the cylinder walls (sides and bottom). Equation 3 is conservation of volume. Equation 4 is a constant contact angle condition (as the surface is disturbed, the contact angle ##\alpha## never changes). Equation 5 is an energy balance between kinetic and potential energy (capillary and inertial), where ##\lambda## is the eigenvalue of the system. All ##k## values are curvatures, which I'll explain in detail for the interested reader. ##\Delta_\Gamma## is the Laplace-Beltrami operator.
The solution approach: analytically solve equations 1-3 for the ##\alpha = 90^\circ## case. This is simple to do via separation of variables. Build equation 4 into the an inverse operator (more on this for those interested). Use these solutions as basis functions for equation 5, which will be solved via the Rayleigh-Ritz variational approach.
Now there's a few tricks along the way, which I can describe in detail, and there's a paper on sessile drops which outlines the process for drops on a flat surface. Our problem behaves identically to the sessile drop.
In short, if you're good with math, familiar with spectral techniques, and like fluid dynamics, please let me know. I would seriously appreciate any help you can offer! All code I've used is in Mathematica, and I can implement any suggestions you have so you don't even have to code.
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