Ticket #1234: Jeffey Orbits.ipynb

File Jeffey Orbits.ipynb, 105.6 KB (added by pkienzle, 5 years ago)
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1{"nbformat":4,"nbformat_minor":0,"metadata":{"colab":{"name":"Jeffey Orbits.ipynb","version":"0.3.2","provenance":[],"collapsed_sections":[]},"kernelspec":{"display_name":"Python 3","language":"python","name":"python3"}},"cells":[{"metadata":{"id":"eNi--SgpRsBA","colab_type":"text"},"cell_type":"markdown","source":["## Jeffery Orbits\n","\n","### Introduction\n","\n","Demonstration of angle for ellipsoids tumbling in shear from using equations from Stover and Cohen (1990).  \n","\n","\n","<table>\n","    <!-- \n","    <tr><td>\n","    <img src=\"Stover1990_Fig1.png\" width=\"320\"/>\n","    </td><td>\n","    <img src=\"Stover1990_Fig2.png\" width=\"300\"/>\n","    </td></tr> \n","    -->\n","    <tr><td>\n","    <img src=\"https://docs.google.com/uc?export=download&id=1KdqU_9-pH5pCShPPZ6-rJJAeZaOFF_EX\" width=\"320\"/>\n","    </td><td>\n","    <img src=\"https://docs.google.com/uc?export=download&id=17MVbd4FUyr3VPgDOVhO52TMlpnkQgiSa\" width=\"300\"/>\n","    </td></tr>\n","    <tr><td colspan=\"2\">Fig. 1 and Fig. 2 from Stover and Cohen (1990) showing the definition of the coodinate system and Jeffery orbits for different values of $C$</td></tr>\n","</table>\n","\n","Tumble period $T$ depends on effective aspect ratio $r_e$ and shear rate $\\dot\\gamma$ as\n","\n","$$\n","T = \\frac{2\\pi}{\\dot\\gamma}\\left(r_e + \\frac{1}{r_e}\\right)\n","$$\n","\n","For ellipsoids, $r_e$ is length/diameter.  For other particles it can be determined experimentally or by simulations, such as those in Ingber (1994).\n","\n","From tumble rate we can get angle as a function of time as\n","\n","$$\n","\\tan \\phi = r_e \\tan\\left(\\frac{2\\pi t}{T} + \\kappa\\right)\n","$$\n","\n","and\n","\n","$$\n","\\tan \\theta = \\frac{C r_e}{\\sqrt{r_e^2 \\cos^2 \\phi + \\sin^2\\phi}}\n","$$\n","\n","where $\\kappa$ is the phase angle determined from the initial orientation of the particle.\n","\n","$C$ is the orbit constant which governs the evolution of $\\theta$ and $\\phi$ over time (Fig. 2 above).  From Stover and Cohen (1990), \"If Jeffery's assumptions are not violated, the value of $C$ is constant and the particle follows one of these trajectories indefinitely. $C$ is zero when the particle is permanently aligned with the vorticity axis and infinity when the particle lies in the $x$-$y$ plane.\"\n","\n","A full treatment requires acknowledging wall effects, particle inertia, medium viscosity, particle interactions, *etc.*, but in the idealized world of the dilute limit we will ignore these.\n","\n","\n","### References\n","\n","[Stover1990]: Stover, C.A., Cohen, C., 1990. *The motion of rodlike particles in the pressure-driven flow between two flat plates.* Rheologica Acta **29**, 192–203. https://doi.org/10.1007/BF01331355\n","\n","[Ingber1994]: Ingber, M.S., Mondy, L.A., 1994. *A numerical study of three‐dimensional Jeffery orbits in shear flow.* Journal of Rheology **38**, 1829–1843. https://doi.org/10.1122/1.550604\n","\n","[Jeffery1922]: Jeffery, G.B., 1922. *The Motion of Ellipsoidal Particles Immersed in a Viscous Fluid.* Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences **102**, 161–179. https://doi.org/10.1098/rspa.1922.0078"]},{"metadata":{"id":"FIufCJ8sRsBC","colab_type":"code","colab":{}},"cell_type":"code","source":["import matplotlib\n","import numpy as np\n","from numpy import pi, sin, cos, tan, arctan, arctan2, sqrt, unwrap, degrees, radians\n","import matplotlib.pyplot as plt\n","%matplotlib inline"],"execution_count":0,"outputs":[]},{"metadata":{"id":"ox6F4_WERsBG","colab_type":"code","colab":{}},"cell_type":"code","source":["def jeffery(t, re, shear, C=1, phase=0, t_is_norm=True):\n","    T = 2*pi/shear*(re + 1./re)\n","    if t_is_norm:\n","        t_norm, t = t, t*T\n","    else:\n","        t_norm, t = t/T, t\n","    x = 2*pi*t_norm + phase\n","    phi = arctan(re*tan(x))\n","    theta = arctan(C*re/sqrt(re**2*cos(phi)**2 + sin(phi)**2))\n","    #theta = arctan(C*sqrt(re**2*sin(x)**2 + cos(x)**2))\n","    #phi, theta = degrees(unwrap(phi)), degrees(unwrap(theta))\n","    return t, phi, theta"],"execution_count":0,"outputs":[]},{"metadata":{"id":"ttmcNVTXRsBI","colab_type":"text"},"cell_type":"markdown","source":["### Evolution of Jeffrey orbits\n","\n","Twiddle orbit constant ```C``` = $C$, ```phase``` = $\\kappa$, aspect ratio ```re``` = $r_e$ and ```shear``` = $\\dot\\gamma$ below.  Using normalized time, ```t_over_T```, we can specify the number of periods to plot.  If using just time then it is harder to see that line shape depends only on aspect ratio."]},{"metadata":{"id":"wMHQuUQ3RsBM","colab_type":"code","outputId":"b8931ee0-01e4-4760-82d3-f9c857267006","executionInfo":{"status":"ok","timestamp":1551187498969,"user_tz":300,"elapsed":721,"user":{"displayName":"Paul Kienzle","photoUrl":"","userId":"06084344534913582643"}},"colab":{"base_uri":"https://localhost:8080/","height":313}},"cell_type":"code","source":["t_over_T = np.linspace(0, 1, 400)\n","C = 0.5\n","phase = 0.\n","re = 8.4\n","shear = 30\n","t_s, phi, theta = jeffery(t_over_T, re, shear, C=C, phase=radians(phase))\n","t = t_over_T if 1 else t_s    \n","plt.plot(t, degrees(phi), label=r'$\\phi$')\n","plt.plot(t, degrees(theta), label=r'$\\theta$')\n","if t is t_over_T:\n","    plt.xlabel('normalized time (t/T)')\n","else:\n","    plt.xlabel(r'time (s)')\n","plt.ylabel(r'angle ($ ^\\circ$)')\n","plt.title(r'$r_e = %g, C = %g$'%(re, C))\n","plt.legend()\n"],"execution_count":4,"outputs":[{"output_type":"execute_result","data":{"text/plain":["<matplotlib.legend.Legend at 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9d3M2VYYt9wMN4WamZ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size 432x288 with 1 Axes>"]},"metadata":{"tags":[]}}]},{"metadata":{"id":"jo-0vhK2VBMg","colab_type":"text"},"cell_type":"markdown","source":["### Simulation of $\\theta_d$ and $\\phi_d$ distribution\n","\n","Generate an initial population of $\\theta$ and $\\phi$ with equal probability in each direction.  In practice, this means that small values of $\\theta$ will be rare because the line of latitude near the pole is short.  Use $\\arccos(U[-1,1])$ where $U[-1,1]$ is the uniform distribution over $[-1, 1]$.  This does covers a half circle corresponding to lines of latitude.\n","$\\phi$ is uniform over the lines of latitude.\n","\n","Next find $C$ and $\\kappa$ from $\\theta$ and $\\phi$ assuming we are at time $t=0$.  This is a simple inversion of the $\\tan \\phi$ and $\\tan \\theta$ expressions above.  Note that $C$ will be negative for $\\theta < 0$.\n","\n","Now forward propagate the system to random time $t$, unifomly covering some multiple of the period.   Can't use a fixed time $t$ since $t$ reverts to its original scrambled state every period.  In practice, there will be random phase shifts over time (and switching to neighbouring $C$ orbits) so the time average should be a reasonable representation.   Besides, except for extremely long periods our measurement should be the time average of the system, and so it doesn't matter where each particle is along its orbit, just the distribution of orbits.\n","\n","Histogram the population of the forward propagated $\\theta$ and $\\phi$ showing both correlation and independent histograms.  As we see below, regardless of what $r_e$ we select we still get $\\theta$ and $\\phi$ acting independently.\n","\n","Note that the histograms from the simulation represent the true weights of $theta$ and $phi$.  When using this within *sasmodels*, which does a correction for latitude in the distribution, need to set $\\theta_d = (90-\\theta)$ and modify the weight of the $\\Delta\\theta_d$ distribution by $1/\\cos(\\theta_d)$ compared to whatever analytic form the distributions below exhibit.\n","\n","The plots below use $\\phi$ instead of $\\phi_d$ to show that there is a $90^\\circ$ shift when the long axis switches between polar and equatorial radius and $\\theta_d$ instead of $\\theta$ because $\\theta = 0^\\circ$ is perpendicular to the flow in this coordinate system, and it makes the plots confusing having $P(\\theta) = 0$ at the center.  The preferred orientation for theta is along the flow regardless of aspect ratio."]},{"metadata":{"id":"CoN2Puw8Zm1c","colab_type":"code","outputId":"562eb7d0-4715-4cd0-8ecd-1532a40c03bd","executionInfo":{"status":"ok","timestamp":1551203357776,"user_tz":300,"elapsed":2183,"user":{"displayName":"Paul Kienzle","photoUrl":"","userId":"06084344534913582643"}},"colab":{"base_uri":"https://localhost:8080/","height":283}},"cell_type":"code","source":["re = 3\n","n_pop = 1000000\n","theta_pop = np.arccos(np.random.rand(n_pop)*2-1)\n","phi_pop = np.random.rand(n_pop)*2*pi\n","C_pop = tan(theta_pop)*sqrt(cos(phi_pop)**2 + sin(phi_pop)**2/re**2)\n","k_pop = arctan(tan(phi_pop)/re)\n","t_pop = np.random.rand(n_pop)\n","#_ = plt.hist(np.log10(C_pop), bins=200, density=True)\n","#_ = plt.hist(k_pop, bins=200, density=True)\n","t, phi_next, theta_next = jeffery(t_pop, re, shear=1, C=C_pop, phase=k_pop)\n","if re >= 1: phi_next[phi_next<0]+=pi # phase unwrap\n","#if re >= 1: phi_next -= pi/2  # phi => phi_d\n","theta_next = pi/2 - theta_next  # theta => theta_d\n","theta_next[theta_next >= pi/2] -= pi # phase unwrap\n","weights = np.ones_like(theta_next)#/cos(theta_next)\n","plt.subplot(221)\n","plt.hist(degrees(phi_next), bins=200, density=True, weights=weights)\n","plt.xticks([])\n","plt.ylabel(r'$P(\\phi)$')\n","plt.subplot(224)\n","plt.hist(degrees(theta_next), bins=200, density=True, weights=weights, orientation='horizontal')\n","plt.yticks([])\n","plt.xlabel(r'$P(\\theta_d)$')\n","plt.subplot(223)\n","# Use log density to show more detail\n","from matplotlib import colors\n","plt.hist2d(degrees(phi_next), degrees(theta_next), bins=200, weights=weights, norm=colors.LogNorm())\n","#plt.xlabel(r'$\\phi_d$ for $\\phi$ = %d ($^\\circ$)'%(90 if re > 1.0 else 0))\n","plt.xlabel(r'$\\phi$ ($^\\circ$)')\n","plt.ylabel(r'$\\theta_d$ ($^\\circ$)')\n","_ = None"],"execution_count":235,"outputs":[{"output_type":"display_data","data":{"image/png":"iVBORw0KGgoAAAANSUhEUgAAAY8AAAEKCAYAAADq59mMAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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pb0Gr1WJycpKNjQ1GRkY4c+aMJrrj8TiVSkWNhOQDxMuQXIIYQ4BQKES1WqVWq+kiXyqV6HQ6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\nWHho/P+2tsfK09q1fgAAAABJRU5ErkJggg==\n","text/plain":["<Figure size 432x288 with 3 Axes>"]},"metadata":{"tags":[]}}]},{"metadata":{"id":"shokLKwv1g9m","colab_type":"text"},"cell_type":"markdown","source":["\n","### Analytic forms for $\\Delta\\theta_d$ and $\\Delta\\phi_d$\n","\n","**Work in Progress: This section is incomplete and incorrect.**  \n","\n","See simulation cell below for the empirical forms of the distribution.\n","\n","In order to develop a orientation distribution for use with *sasmodels*, assume we start with a set of particles in random orientation. The scattering pattern is integrating over the population of particles over time, so we need a distribution that captures the proportion of time spent at each orientation.  From Fig 2 in Stover and Cohen (1990), we can assume that each initial orientation will determine a unique $C$ and $\\kappa$ pair, with $C$ remaining constant and $\\kappa$ moving along the orbit as the system evolves.\n","\n","\n","We can ignore the initial phase $\\kappa$ since it only serves to shift the curves uniformly, and will vanish in the integration. So we need to determine the distribution of $C$ values from the initial orientation and for each $C$, determine the joint distribution of $\\theta$ and $\\phi$, then marginalize.  That is, find the form of $P(\\theta, \\phi, C)$ then integrate to get $P(\\theta) = \\int_{\\phi, C} P(\\theta | \\phi, C) d\\theta dC$ and $P(\\phi) = \\int_{\\theta, C} P(\\phi | \\theta, C) d\\phi dC$.  Should cross check by random sampling of evolved systems, and verify that $\\int_C P(\\theta, \\phi | C) dC$ is separable.\n","\n","Using\n","\n","$$ \\frac{d}{dy} f^{-1}(y)\\Big|_{y=y_o} = \\left(\\frac{d}{dx}f(x)\\right)^{-1}\\Big|_{x=f^{-1}(y_o)}$$\n","\n","density should be inversely related to the slope, so for $\\phi$ with $x = \\pi\\,t/T \\in [0, 1]$ covering $[-\\pi/2, \\pi/2]$ with\n","\n","$$ \\phi(x) = \\tan^{-1}(r_e\\tan(x))$$\n","\n","then\n","\n","$$\n","\\frac{d\\phi}{dx} = (r_e \\sin^2(x) + (1/r_e)\\cos^2(x))^{-1}\n","$$\n","\n","and so\n","\n","$$\n","\\frac{dx}{d\\phi} = r_e \\sin^2(\\phi) + (1/r_e)\\cos^2(\\phi)\n","$$\n","\n","Normalize by the integral to make it into a probability\n","\n","$$\n","\\int_{-\\pi/2}^{\\pi/2} \\frac{dx}{d\\phi}\\,d\\phi = \\tfrac{\\pi}{2} (r_e + 1/r_e)\n","$$\n","\n","Some more trig:\n","\n","\\begin{align}\n","\\cos(\\arctan(x)) &= 1\\big/ \\sqrt{1 + x^2} \\\\\n","\\sin(\\arctan(x)) &= x \\big/ \\sqrt{1 + x^2}\n","\\end{align}\n","\n","so\n","\n","\\begin{align}\n","\\tan \\theta &= \\frac{C r_e}{\\sqrt{r_e^2 \\cos^2 \\phi + \\sin^2\\phi}} \\\\\n","                   &= \\frac{C r_e}{\\sqrt{r_e^2 (1 + \\tan^2 x)/(1 + r_e^2\\tan^2 x)}} \\\\\n","                   &= C\\sqrt\\frac{1 + r_e^2\\tan^2 x}{1 + \\tan^2 x} \\\\\n","                   &= C\\sqrt{r_e^2\\sin^2 x + \\cos^2 x}\n","{}\\end{align}\n","\n","and\n","\n","$$\n","\\frac{d\\theta}{dx} = \\frac{C\\,(r_e^2 - 1)\\,\\sin x \\cos x}{\\sqrt{r_e^2\\sin^2 x + \\cos^2 x}\\, \n","(C^2 (r_e^2 \\sin^2 x + \\cos^2 x) + 1)}\n","$$\n"]},{"metadata":{"id":"2bTDi7VcYmhC","colab_type":"code","outputId":"78868af1-44cf-4b85-cdcf-58e20aca513d","executionInfo":{"status":"ok","timestamp":1551187508329,"user_tz":300,"elapsed":561,"user":{"displayName":"Paul Kienzle","photoUrl":"","userId":"06084344534913582643"}},"colab":{"base_uri":"https://localhost:8080/","height":300}},"cell_type":"code","source":["re=8.4\n","#re=1e10\n","phi = 180*t_over_T - 90\n","dx_dphi = re*sin(pi*t_over_T)**2 + cos(pi*t_over_T)**2/re\n","dx_dphi_norm = 90*(re+1/re)\n","p_phi = dx_dphi / dx_dphi_norm\n","plt.plot(phi, p_phi)\n","plt.xlabel(r'$\\phi ( ^\\circ)$')\n","plt.ylabel(r'$P(\\phi)$')\n","plt.ylim([0, max(p_phi)])"],"execution_count":5,"outputs":[{"output_type":"execute_result","data":{"text/plain":["(0, 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NqVULVAL/LCnJ05MjMLhB0NopzPW7Ag+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\nf8y+I42IsBA+PFBKe4e0LOkPUjxEj9ZK23Xhh6IiHMy+I5W6xjb2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size 432x288 with 1 Axes>"]},"metadata":{"tags":[]}}]}]}