Theoretical and experimental study on the motion and shape of viscoelastic falling drops through Newtonian media

Abstract

In this paper, creeping motion of a viscoelastic drop falling through a Newtonian fluid is investigated experimentally and analytically. A polymeric solution of 0.08 % xanthan gum in 80:20 glycerol/water and silicon oil is implemented as the viscoelastic drop and the bulk viscous fluids, respectively. The shape and motion of falling drops are visualized using a high speed camera. The perturbation technique is employed for both interior and exterior flows, and Deborah and capillary numbers are considered as perturbation parameters up to second order. The product of Deborah and capillary numbers is also used as a perturbation parameter to apply the boundary condition on the deformation on the interface of viscoelastic drop and viscous media. We used the Giesekus nonlinear constitutive equation for the drop phase for the first time. It is shown that the obtained analytical solution has better agreement with experimental observations than the previous analytical studies.

Appendices

Appendix

The details of perturbation solution for velocity and pressure fields are presented in this section.

A. 1. The zero order terms

$$ {u}_{r,0}=\frac{1}{2}\left(2-\frac{3k+2}{k+1}\frac{1}{r}+\frac{k}{k+1}\frac{1}{r^3}\right) \cos \theta $$

(A.1)

$$ {\tilde{u}}_{r,0}=\frac{1}{2}\left({r}^2-1\right) \cos \theta $$

(A.2)

$$ {u}_{\theta, 0}=-\frac{1}{4}\left(4-\frac{3k+2}{k+1}\frac{1}{r}-\frac{k}{k+1}\frac{1}{r^3}\right) \sin \theta $$

(A.3)

$$ {\tilde{u}}_{\theta, 0}=-\frac{1}{2}\left(2{r}^2-1\right) \sin \theta $$

(A.4)

$$ {\tau}_0=\left[\begin{array}{ccc}\hfill \frac{\left(3k{r}^2+2{r}^2-3k\right)}{\left(k+1\right){r}^4} \cos \theta \hfill & \hfill -\frac{3}{2}\frac{k}{\left(k+1\right){r}^4} \sin \theta \hfill & \hfill 0\hfill \\ {}\hfill -\frac{3}{2}\frac{k}{\left(k+1\right){r}^4} \sin \theta \hfill & \hfill -\frac{1}{2}\frac{\left(3k{r}^2+2{r}^2-3k\right)}{\left(k+1\right){r}^4} \cos \theta \hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{2}\frac{\left(3k{r}^2+2{r}^2-3k\right)}{\left(k+1\right){r}^4} \cos \theta \hfill \end{array}\right] $$

(A.5)

$$ {\tilde{\tau}}_0=\left[\begin{array}{ccc}\hfill 2r \cos \theta \hfill & \hfill -\frac{3}{2}r \sin \theta \hfill & \hfill 0\hfill \\ {}\hfill -\frac{3}{2}r \sin \theta \hfill & \hfill -r \cos \theta \hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill -r \cos \theta \hfill \end{array}\right] $$

(A.6)

$$ {p}_0=-\frac{1}{2}\frac{ \cos \theta }{r^2}\frac{\left(3k+2\right)}{\left(k+1\right)}+\frac{\rho g{R}^2}{\eta {U}_{\infty }}r \cos \theta $$

(A.7)

$$ \tilde{p}=5r \cos \theta +\frac{\tilde{\rho}g{R}^2}{\tilde{\eta}{U}_0}r \cos \theta +\delta p $$

(A.8)

A. 2. The terms in order of De1

$$ {u}_{r,1}=-\frac{3}{20}\frac{\beta k}{{\left(k+1\right)}^2}\left(\frac{1}{r^2}-\frac{1}{r^4}\right)\left(3{ \cos}^2\theta -1\right)\left(3-\alpha \right) $$

(A.9)

$$ {\tilde{u}}_{r,1}=\frac{3}{20}\frac{\beta k}{k+1}\left(r-{r}^3\right)\left(3{ \cos}^2\theta -1\right)\left(3-\alpha \right) $$

(A.10)

$$ {u}_{\theta, 1}=\frac{3}{10}\left(3-\alpha \right)\frac{k\beta }{{\left(k+1\right)}^2{r}^4} \sin \theta \cos \theta $$

(A.11)

$$ {\tilde{u}}_{\theta, 1}=\frac{3}{20}\frac{k\beta }{k+1}\left(3-\alpha \right)\left(-3r+5{r}^3\right) \sin \theta \cos \theta $$

(A.12)

$$ {p}_1=-\frac{3}{10}\left(3-\alpha \right)\frac{k\beta }{{\left(k+1\right)}^2}\frac{\left(3{ \cos}^2\theta -1\right)}{r^3} $$

(A.13)

$$ {\tilde{p}}_1=-\frac{21}{20}\left(3-\alpha \right)\frac{k\beta }{k+1}r\left(3{ \cos}^2\theta -1\right) $$

(A.14)

$$ \begin{array}{l}{\tau_1}_{,rr}=-\frac{3}{5}\frac{\beta k\left(6-3{r}^2-2\alpha +\alpha {r}^2\right)}{{\left(k+1\right)}^2{r}^5}\left(3{ \cos}^2\theta -1\right)\\ {}{\tau_1}_{,r\theta }={\tau_1}_{,\theta r}=-\frac{3}{10}\frac{k\beta \left(24-9{r}^2-8\alpha +3\alpha {r}^2\right)}{{\left(k+1\right)}^2{r}^5} \sin \theta \cos \theta \\ {}{\tau_1}_{,r\phi }={\tau_1}_{,\phi r}=0\\ {}{\tau_1}_{,\theta \theta }=\frac{3}{10}\frac{k\beta \left(3-7{ \cos}^2\theta +3{r}^2{ \cos}^2\theta -{r}^2\right)}{{\left(k+1\right)}^2{r}^5}\left(-3+\alpha \right)\\ {}{\tau_1}_{,\theta \phi }={\tau_1}_{,\phi \theta }=0\\ {}{\tau_1}_{,\phi \phi }=\frac{3}{10}k\beta \Big(15{ \cos}^2\theta -9{r}^2{ \cos}^2\theta -5\alpha { \cos}^2\theta +\\ {}\kern2.64em 3\alpha {r}^2{ \cos}^2\theta -3+3{r}^2+\alpha -\alpha {r}^2\Big)/{\left(k+1\right)}^2{r}^5\\ {}\end{array} $$

(A.15)

$$ \begin{array}{l}{\tilde{\tau}}_{1,rr}=\frac{1}{20}\beta \Big(-10+84k{ \cos}^2\theta -92k{r}^2{ \cos}^2\theta -18k\alpha { \cos}^2\theta +\\ {}\kern3em 19k\alpha {r}^2{ \cos}^2\theta +30{ \cos}^2\theta +70{r}^2{ \cos}^2\theta -35\alpha {r}^2{ \cos}^2\theta \\ {}\kern3.12em -28k+44k{r}^2+6k\alpha -63k\alpha {r}^2-10{r}^2-45\alpha {r}^2\Big)/\left(k+1\right)\\ {}{\tilde{\tau}}_{1,r\theta }={\tilde{\tau}}_{1,\theta r}=-\frac{3}{10}\frac{\beta \cos \theta \sin \theta \left(14k-14k{r}^2-3k\alpha +3k\alpha {r}^2+5+10{r}^2-5\alpha {r}^2\right)}{k+1}\\ {}{\tilde{\tau}}_{1,r\phi }={\tilde{\tau}}_{1,\phi r}=0\\ {}{\tilde{\tau}}_{1,\theta \theta }=-\frac{1}{20}\beta \Big(-20-76k{r}^2{ \cos}^2\theta +17k\alpha {r}^2{ \cos}^2\theta +84k{ \cos}^2\theta -\\ {}\kern3.48em 18k\alpha { \cos}^2\theta +30{ \cos}^2\theta +50{r}^2{ \cos}^2\theta -25\alpha {r}^2{ \cos}^2\theta -\\ {}\kern3.48em 56k+12k\alpha -80{r}^2-8k{r}^2+21k\alpha {r}^2+45\alpha {r}^2\;\Big)/\left(k+1\right)\\ {}{\tilde{\tau}}_{1,\theta \phi }={\tilde{\tau}}_{1,\phi \theta }=0\\ {}{\tilde{\tau}}_{1,\phi \phi }=-\frac{1}{10}\beta \Big(-23k{r}^2{ \cos}^2\theta +16k\alpha {r}^2{ \cos}^2\theta -19k{r}^2+14k-3k\alpha +\\ {}3k\alpha {r}^2+5-10{r}^2-5{r}^2{ \cos}^2\theta +10\alpha {r}^2{ \cos}^2\theta \Big)/\left(k+1\right)\\ {}\end{array} $$

(A.16)

A. 3. The terms in order of DeCa

$$ \begin{array}{l}{u_r}^{(1)}=\frac{3}{140}\frac{\alpha_2}{k+1}\left(\frac{21k+18}{r}-\frac{21k+4}{r^3}\right)\left(10{ \cos}^2\theta -6\right) \sin \theta \cos \theta \\ {}-\frac{\alpha_2}{10{\left(k+1\right)}^2}\left(\frac{3{k}^2-k+8}{r}-\frac{3{k}^2-3k+6}{r^3}\right) \cos \theta \end{array} $$

(A.17)

$$ {{\tilde{u}}_r}^{(1)}=\frac{-{\alpha}_2}{5k+5}\left(\left(4-2k\right){r}^2+\left(3k-3\right)\right) \cos \theta -\frac{3{\alpha}_2}{70}\left(5{r}^4-12{r}^2\right)\left(10{ \cos}^2\theta -6\right) \sin \theta \cos \theta $$

(A.18)

$$ \begin{array}{l}{u_{\theta}}^{(1)}=\frac{1}{20}\frac{\alpha_2}{{\left(k+1\right)}^2}\left(\frac{3{k}^2-k+8}{r}+\frac{3{k}^2-3k+6}{r^3}\right) \sin \theta \\ {}-\frac{3}{280}\frac{\alpha_2}{\left(k+1\right)}\left(-\frac{21k+18}{r^3}+\frac{3\left(21k+4\right)}{r^5}\right) \sin \theta \left(5{ \cos}^2\theta -1\right)\end{array} $$

(A.19)

$$ \begin{array}{l}{{\tilde{u}}_{\theta}}^{(1)}=\frac{\alpha_2}{5\left(k+1\right)}\left(\left(8-4k\right){r}^2+\left(3k-3\right)\right) \sin \theta +\\ {}\frac{3{\alpha}_2}{140}\left(30{r}^4-48{r}^2\right) \sin \theta \left(5{ \cos}^2\theta -1\right)\end{array} $$

(A.20)

$$ \begin{array}{l}{p}^{(1)}=\frac{1}{140}\frac{\alpha^2{ \cos}^2\theta }{{\left(k+1\right)}^2{r}^4}\Big(-112{r}^2+14{r}^2k-42{r}^2{k}^2+1350{ \cos}^2\theta +2925k{ \cos}^2\theta \\ {}+1575{k}^2{ \cos}^2\theta -810-945{k}^2-1755k\Big)\end{array} $$

(A.21)

$$ {\tilde{p}}^{(1)}=-\frac{2}{7}\frac{\alpha_2r \cos \theta }{k+1}\left(45{r}^2k{ \cos}^2\theta +45{r}^2{ \cos}^2\theta +28-27{r}^2-14k-27{r}^2k\right) $$

(A.22)

$$ \begin{array}{l}{\tau_{rr}}^{(1)}=-\frac{1}{35}{\alpha}_2 \cos \theta \Big(180+810{r}^2{ \cos}^2\theta -360{r}^2-56{r}^4+945{r}^2{k}^2{ \cos}^2\theta \\ {}-1116k{r}^2-504{r}^2{k}^2+1125k+945{k}^2+7k{r}^4-21{k}^2{r}^4-300{ \cos}^2\theta \\ {}-1557{k}^2{ \cos}^2\theta -1875k{ \cos}^2\theta +1755k{r}^2{ \cos}^2\theta \Big)/\left({r}^6{\left(k+1\right)}^2\right)\\ {}{\tau_{r\theta}}^{(1)}={\tau_{\theta r}}^{(1)}=-\frac{9}{140}{\alpha}_2 \sin \theta \Big(-20{r}^2+520k{r}^2{ \cos}^2\theta +280{r}^2{k}^2{ \cos}^2\theta \\ {}-118k{r}^2+240{r}^2{ \cos}^2\theta -525{k}^2{ \cos}^2\theta -625k{ \cos}^2\theta -42{r}^2{k}^2+125k\\ {}+105{k}^2+20-100{ \cos}^2\theta \Big)/\left({r}^6{\left(k+1\right)}^2\right)\\ {}{\tau_{r\phi}}^{(1)}={\tau_{\phi r}}^{(1)}=0\\ {}{\tau_{\theta \theta}}^{(1)}=\frac{1}{140}\Big(540-990{r}^2-112{r}^4-2817k{r}^2-1323{r}^2{k}^2+3375k\\ {}+2835{k}^2+1890{r}^2{ \cos}^2\theta -4095{k}^2{ \cos}^2\theta -4875k{ \cos}^2\theta +4095k{r}^2{ \cos}^2\theta \\ {}+2205{r}^2{k}^2{ \cos}^2\theta -780{ \cos}^2\theta +14{r}^4k-42{r}^4{k}^2\Big){\alpha}_2 \cos \theta /\left({r}^6{\left(k+1\right)}^2\right)\\ {}{\tau_{\theta \phi}}^{(1)}={\tau_{\phi \theta}}^{(1)}=0\\ {}{\tau_{\phi \phi}}^{(1)}=\frac{1}{140}{\alpha}_2 \cos \theta \Big(180-420{ \cos}^2\theta +1575{r}^2{k}^2{ \cos}^2\theta +14k{r}^4-42{k}^2{r}^4\\ {}+2925k{r}^2{ \cos}^2\theta -693{r}^2{k}^2-1647k{r}^2-112{r}^4+1125k+945{k}^2\\ {}+1350{r}^2{ \cos}^2\theta -450{r}^2-2205{k}^2{ \cos}^2\theta -2625k{ \cos}^2\theta \left)/\right({r}^6{\left(k+1\right)}^2\end{array} $$

(A.23)

$$ \begin{array}{l}{{\tilde{\tau}}_{rr}}^{(1)}=-\frac{8}{35}{\alpha}_2r \cos \theta \Big(-45{r}^2+75{r}^2{ \cos}^2\theta -90k{ \cos}^2\theta -90{ \cos}^2\theta \\ {}+75k{r}^2{ \cos}^2\theta +47k+68-45{r}^2k\Big)/\left(k+1\right)\\ {}{{\tilde{\tau}}_{r\theta}}^{(1)}={{\tilde{\tau}}_{\theta r}}^{(1)}=\frac{3}{70}r{\alpha}_2 \sin \theta \Big(152+68k+375k{r}^2{ \cos}^2\theta +375{r}^2{ \cos}^2\theta \\ {}-75k{r}^2-75{r}^2-480k{ \cos}^2\theta -480{ \cos}^2\theta \Big)/\left(k+1\right)\\ {}{{\tilde{\tau}}_{r\phi}}^{(1)}={{\tilde{\tau}}_{\phi r}}^{(1)}=0\\ {}{{\tilde{\tau}}_{\theta \theta}}^{(1)}=\frac{1}{35}r{\alpha}_2 \cos \theta \Big(525k{r}^2{ \cos}^2\theta +525{r}^2{ \cos}^2\theta -405{r}^2-405{r}^2k\\ {}+548k+632-720k{ \cos}^2\theta -720{ \cos}^2\theta \Big)/\left(k+1\right)\\ {}{{\tilde{\tau}}_{\theta \phi}}^{(1)}={{\tilde{\tau}}_{\phi \theta}}^{(1)}=0\\ {}{{\tilde{\tau}}_{\phi \phi}}^{(1)}=\frac{1}{35}r{\alpha}_2 \cos \theta \left(45{r}^2+75{r}^2{ \cos}^2\theta +75k{r}^2{ \cos}^2\theta -172k-88+45k{r}^2\right)/\left(k+1\right)\end{array} $$

(A.24)

A. 4. The terms in order of De2

$$ {u}_{r,2}=\frac{1}{2}\frac{\left(-5{A}_3{ \cos}^2\theta -5{B}_3{r}^2{ \cos}^2\theta +3{A}_3+3{B}_3{r}^2-2{A}_1{r}^2+2{A}_1{r}^4\right)}{r^5} cos\theta $$

(A.25)

$$ {\tilde{u}}_{r,2}=\frac{1}{2}\left({C}_3{r}^2\left(-5{ \cos}^2\theta +3\right)+2a\left(1+{r}^2\right)-2{C}_1\right)\left(1-{r}^2\right) \cos \theta $$

(A.26)

$$ {u}_{\theta, 2}=-\frac{1}{2}{A}_1\left(\frac{1}{r^3}+\frac{1}{r}\right) \sin \theta -\frac{1}{8}{A}_3\left(\frac{3}{r^5}-\frac{1}{r^3}\right) \sin \theta \left(5{ \cos}^2\theta -1\right) $$

(A.27)

$$ {\tilde{u}}_{\theta, 2}=\frac{1}{2}\left(a\left(6{r}^4-2\right)+{C}_1\left(2+4{r}^2\right)\right) \sin \theta +\frac{1}{8}{C}_3\left(4{r}^2-6{r}^4\right) \sin \theta \left(5{ \cos}^2\theta -1\right) $$

(A.28)

$$ \begin{array}{l}{p}_2=-\frac{1}{1400}\frac{\beta k}{{\left(k+1\right)}^3{r}^4}\Big(8550-6525\alpha +1350k{\alpha}^2+8550k-14250{ \cos}^2\theta -\\ {}6525k\alpha +1350{\alpha}^2-1440\beta k{\alpha}^2-15930\beta k-238\beta k{\alpha}^2{r}^2+798\beta k\alpha {r}^2-\\ {}500{r}^2+9630\beta k\alpha -252\beta k{r}^2-5355k\alpha {r}^2+3465k{\alpha}^2{r}^2+3465{\alpha}^2{r}^2-\\ {}5355\alpha {r}^2-500k{r}^2+10875\alpha { \cos}^2\theta -2250{\alpha}^2{ \cos}^2\theta -14250k{ \cos}^2\theta +\\ {}26550\beta k{ \cos}^2\theta -2250k{\alpha}^2{ \cos}^2\theta +10875k\alpha { \cos}^2\theta -16050\beta k\alpha { \cos}^2\theta +\\ {}2400\beta k{\alpha}^2{ \cos}^2\theta \Big) \cos \theta \end{array} $$

(A.29)

$$ \begin{array}{l}{\tilde{p}}_2=\frac{1}{2800}\frac{\beta r \cos \theta }{{\left(k+1\right)}^2}\Big(-1700-1680{r}^2+6090\beta {k}^2{\alpha}^2+3400k+13580\beta k\alpha {r}^2{ \cos}^2\theta -\\ {}1540\beta k{\alpha}^2{r}^2{ \cos}^2\theta -3150\beta k\alpha -7560\beta k+19656\beta k{r}^2+7700{r}^2{ \cos}^2\theta +\\ {}42750k\alpha {r}^2{ \cos}^2\theta +37680k{r}^2+60200k\alpha -60700{k}^2{r}^2{ \cos}^2\theta -41050k\alpha {r}^2-\\ {}53000k{r}^2{ \cos}^2\theta +3150{\alpha}^2-17150\alpha -4074\beta k{\alpha}^2{r}^2-4865\alpha {r}^2+11835{k}^2{\alpha}^2{r}^2-\\ {}56808\beta {k}^2{r}^2-36185\;{k}^2\alpha {r}^2-4725\alpha {r}^2{ \cos}^2\theta -63460\beta \alpha {k}^2{r}^2{ \cos}^2\theta +17190k{\alpha}^2{r}^2+\\ {}9980\beta {k}^2{\alpha}^2{r}^2{ \cos}^2\theta +1890\beta k{\alpha}^2+77350\alpha {k}^2-16310\beta \alpha {k}^2-58800k{\alpha}^2+5670\beta k\alpha {r}^2-\\ {}61950{k}^2{\alpha}^2+875{\alpha}^2{r}^2{ \cos}^2\theta -9925{k}^2{\alpha}^2{r}^2{ \cos}^2\theta +47475\alpha {k}^2{r}^2{ \cos}^2\theta +5100{k}^2+\\ {}100560\beta {k}^2{r}^2{ \cos}^2\theta -5880\beta {k}^2-26880\beta k{r}^2{ \cos}^2\theta -9050k{\alpha}^2{r}^2{ \cos}^2\theta +5355{\alpha}^2{r}^2-\\ {}10986\beta {k}^2{\alpha}^2{r}^2+51894\beta \alpha {k}^2{r}^2+39360{r}^2{k}^2\Big)\end{array} $$

(A.30)

$$ \begin{array}{c}\hfill \begin{array}{l}{\tau_2}_{,rr}=-\frac{1}{175}\beta k \cos \theta \Big(-17100-21750k\alpha { \cos}^2\theta -19872k\beta {r}^2-17100k-\\ {}17100k{r}^2{ \cos}^2\theta +8760{r}^2+500{r}^4-2700{\alpha}^2{r}^2{ \cos}^2\theta -2700k{\alpha}^2-19260k\beta \alpha -\\ {}53100k\beta { \cos}^2\theta +12015k{\alpha}^2{r}^2+4500{\alpha}^2{ \cos}^2\theta +5355k\alpha {r}^4-3465{\alpha}^2{r}^4+500k{r}^4+\\ {}13050\alpha {r}^2{ \cos}^2\theta -23895k\alpha {r}^2+2880k\beta {\alpha}^2+28500{ \cos}^2\theta -23895\alpha {r}^2+\\ {}8760k{r}^2+13050k\alpha +28500k{ \cos}^2\theta +31860k\beta -21750\alpha { \cos}^2\theta -17100{r}^2{ \cos}^2\theta +\\ {}12015{\alpha}^2{r}^2+13050\alpha +4500k{\alpha}^2{ \cos}^2\theta -7981k\beta \alpha {r}^4+238k\beta {\alpha}^2{r}^4-2700{\alpha}^2-\\ {}19260k\beta \alpha {r}^2{ \cos}^2\theta +2880k\beta {\alpha}^2{r}^2{ \cos}^2\theta -2442k\beta {\alpha}^2{r}^2+31860k\beta {r}^2{ \cos}^2\theta -\\ {}2700k{\alpha}^2{r}^2{ \cos}^2\theta +13950k\beta \alpha {r}^2+32100k\beta \alpha { \cos}^2\theta -4800k\beta {\alpha}^2{ \cos}^2\theta +\\ {}13050k\alpha {r}^2{ \cos}^2\theta +252k\beta {r}^4+5355\alpha {r}^4-3465k{\alpha}^2{r}^4\Big)/\left({r}^6{\left(k+1\right)}^3\right)\end{array}\hfill \\ {}\hfill \begin{array}{l}{\tau_2}_{,r\theta }={\tau_2}_{,\theta r}=-\frac{3}{1400}\beta k \sin \theta \Big(-2850k-3084\beta k{r}^2-2850-10875k\alpha { \cos}^2\theta -\\ {}7600k{r}^2{ \cos}^2\theta +1020{r}^2-1200{\alpha}^2{r}^2{ \cos}^2\theta -450k{\alpha}^2-3210k\beta \alpha +480k\beta {\alpha}^2-\\ {}26550k\beta { \cos}^2\theta +3705k{\alpha}^2{r}^2+2250{\alpha}^2{ \cos}^2\theta +5800\alpha {r}^2{ \cos}^2\theta -6515k\alpha {r}^2+\\ {}14250{ \cos}^2\theta -6515\alpha {r}^2+1020k{r}^2+2175k\alpha +14250k{ \cos}^2\theta +5310k\beta -\\ {}10875\alpha { \cos}^2\theta -7600{r}^2{ \cos}^2\theta +3705{\alpha}^2{r}^2+2175\alpha +2250k{\alpha}^2{ \cos}^2\theta -450{\alpha}^2-\\ {}8560k\beta \alpha {r}^2{ \cos}^2\theta +1280k\beta {\alpha}^2{r}^2{ \cos}^2\theta -494k\beta {\alpha}^2{r}^2+14160k\beta {r}^2{ \cos}^2\theta -\\ {}1200k{\alpha}^2{r}^2{ \cos}^2\theta +2510k\beta \alpha {r}^2+16050k\beta \alpha { \cos}^2\theta -2400k\beta {\alpha}^2{ \cos}^2\theta +\\ {}5800k\alpha {r}^2{ \cos}^2\theta \Big)/\left({r}^6{\left(k+1\right)}^3\right)\\ {}{\tau_2}_{,r\phi }={\tau_2}_{,\phi r}=0\\ {}{\tau_2}_{,\theta \theta }=\frac{\beta k \cos \theta }{1400}\Big(-25650-25650k-25182\beta k{r}^2+47790\beta k+11610k{r}^2+\\ {}500{r}^4-28275k\alpha { \cos}^2\theta -19950k{r}^2{ \cos}^2\theta +11610{r}^2-3150{\alpha}^2{r}^2{ \cos}^2\theta -\\ {}4050k{\alpha}^2+500k{r}^4-28890k\beta \alpha -69030k\beta { \cos}^2\theta +12465k{\alpha}^2{r}^2+5850{\alpha}^2{ \cos}^2\theta +\\ {}5355k\alpha {r}^4-3465{\alpha}^2{r}^4+15225\alpha {r}^2{ \cos}^2\theta -26070k\alpha {r}^2+4320k\beta {\alpha}^2+37050{ \cos}^2\theta -\\ {}26070\alpha {r}^2+19575k\alpha +37050k{ \cos}^2\theta -28275\alpha { \cos}^2\theta -19950{r}^2{ \cos}^2\theta +12465{\alpha}^2{r}^2+\\ {}19575\alpha +5850k{\alpha}^2{ \cos}^2\theta -798k\beta \alpha {r}^4+238k\beta {\alpha}^2{r}^4-4050{\alpha}^2-22470k\beta \alpha {r}^2{ \cos}^2\theta +\\ {}3360k\beta {\alpha}^2{r}^2{ \cos}^2\theta +17160k\beta {r}^4-2922k\beta {\alpha}^2{r}^2+37170k\beta {\alpha}^2{ \cos}^2\theta -3150k{\alpha}^2{r}^2{ \cos}^2\theta +\\ {}41730k\beta \alpha { \cos}^2\theta -6240k\beta {\alpha}^2{ \cos}^2\theta +15225k\alpha {r}^2{ \cos}^2\theta +252k\beta {r}^4+5355\alpha {r}^4-\\ {}3465k{\alpha}^2{r}^4\Big)/\left({r}^6{\left(k+1\right)}^3\right)\\ {}{\tau_2}_{,\theta \phi }={\tau_2}_{,\phi \theta }=0\\ {}{\tau_2}_{,\phi \phi }=\frac{1}{700}\beta k \cos \theta \Big(-8550-15225k\alpha { \cos}^2\theta -14562k\beta {r}^2-8550k-14250k{r}^2{ \cos}^2\theta +\\ {}5910{r}^2+500{r}^4-2250{\alpha}^2{r}^2{ \cos}^2\theta -1350k{\alpha}^2-9630k\beta \alpha -37170\beta k{ \cos}^2\theta +\\ {}11565k{\alpha}^2{r}^2+3150{\alpha}^2{ \cos}^2\theta +5355k\alpha {r}^4-3465{\alpha}^2{r}^4+500k{r}^4+10875\alpha {r}^2{ \cos}^2\theta -\\ {}21720k\alpha {r}^2+1440k\beta {\alpha}^2+19950{ \cos}^2\theta -21720\alpha {r}^2+5910k{r}^2+6525k\alpha +19950k{ \cos}^2\theta +\\ {}15930k\beta -15225\alpha { \cos}^2\theta -14250{r}^2{ \cos}^2\theta +11565{\alpha}^2{r}^2+6525\alpha +3150k{\alpha}^2{ \cos}^2\theta -\\ {}798k\beta \alpha {r}^4+238k\beta {\alpha}^2{r}^4-1350{\alpha}^2-16050k\beta \alpha {r}^2{ \cos}^2\theta +2400k\beta {\alpha}^2{r}^2{ \cos}^2\theta -\\ {}1962k\beta {\alpha}^2{r}^2+26550k\beta {r}^2{ \cos}^2\theta -2250k{\alpha}^2{r}^2{ \cos}^2\theta +10740k\beta \alpha {r}^2+22470k\beta \alpha { \cos}^2\theta -\\ {}3360k\beta {\alpha}^2{ \cos}^2\theta +10875k\alpha {r}^2{ \cos}^2\theta +252k\beta {r}^4+5355\alpha {r}^4-3465k{\alpha}^2{r}^4\Big)/\left({r}^6{\left(k+1\right)}^3\right)\end{array}\hfill \end{array} $$

(A.31)

$$ \begin{array}{l}{\tilde{\tau}}_{2,rr}=\frac{1}{700}\beta r \cos \theta \Big(-1510-12900k\alpha { \cos}^2\theta +8568k\beta {r}^2-8860k-\\ {}12300k{r}^2{ \cos}^2\theta -3390{r}^2+1750{\alpha}^2{r}^2{ \cos}^2\theta -8640k{\alpha}^2-3780k\beta \alpha -\\ {}1260k\beta { \cos}^2\theta -8325{k}^2{\alpha}^2+21690k{\alpha}^2{r}^2-17550{k}^2{r}^2{ \cos}^2\theta +\\ {}13375{k}^2\alpha -6125\alpha {r}^2{ \cos}^2\theta -1850{k}^2{\alpha}^2{r}^2{ \cos}^2\theta -22480k\alpha {r}^2+\\ {}1092k\beta {\alpha}^2+5250{ \cos}^2\theta -6020\alpha {r}^2+6900k{r}^2+10820k\alpha +1512k\beta +\\ {}21900k{ \cos}^2\theta -2100\alpha { \cos}^2\theta +5250{r}^2{ \cos}^2\theta +9765{\alpha}^2{r}^2-2555\alpha +\\ {}1800k{\alpha}^2{ \cos}^2\theta +11925{k}^2{\alpha}^2{r}^2-315{\alpha}^2+13860k\alpha \beta {r}^2{ \cos}^2\theta -\\ {}2520k\beta {\alpha}^2{r}^2{ \cos}^2\theta -2562k\beta {\alpha}^2{r}^2-18900k\beta {r}^2{ \cos}^2\theta -100k{\alpha}^2{r}^2{ \cos}^2\theta +\\ {}4830k\beta \alpha {r}^2-1470k\beta \alpha { \cos}^2\theta +630k\beta {\alpha}^2{ \cos}^2\theta +5150k\alpha {r}^2{ \cos}^2\theta -\\ {}10800{k}^2\alpha { \cos}^2\theta +1800{k}^2{\alpha}^2{ \cos}^2\theta +16650{k}^2{ \cos}^2\theta -16920{k}^2\beta {r}^2+\\ {}10290{r}^2{k}^2-13080{k}^2\beta \alpha +2720{k}^2\beta {\alpha}^2-22500{k}^2\beta { \cos}^2\theta -\\ {}11820{k}^2\beta \alpha {r}^2{ \cos}^2\theta +1320\beta {k}^2{\alpha}^2{r}^2{ \cos}^2\theta +20238\beta \alpha {k}^2{r}^2-7350{k}^2-\\ {}4866{k}^2\beta {\alpha}^2{r}^2+11275{k}^2\alpha {r}^2{ \cos}^2\theta +23580\beta {k}^2{r}^2{ \cos}^2\theta -16460\alpha {k}^2{r}^2+\\ {}11370{k}^2\beta \alpha { \cos}^2\theta -1290{k}^2\beta {\alpha}^2{ \cos}^2\theta +14760{k}^2\beta \Big)/{\left(k+1\right)}^2\\ {}{\tilde{\tau}}_{2,r\theta }={\tilde{\tau}}_{2,\theta r}=-\frac{3}{2800}\beta r \sin \theta \Big(-1860-17200k\alpha { \cos}^2\theta +5880k\beta {r}^2-5760k-\\ {}15900k{r}^2{ \cos}^2\theta -1800{r}^2+2100{\alpha}^2{r}^2{ \cos}^2\theta -8040k{\alpha}^2-910k\beta \alpha -455\alpha -\\ {}1680k\beta { \cos}^2\theta -7725{k}^2{\alpha}^2+14550k{\alpha}^2{r}^2-22200{k}^2{r}^2{ \cos}^2\theta +12575{k}^2\alpha -\\ {}7350\alpha {r}^2{ \cos}^2\theta -2400{k}^2{\alpha}^2{r}^2{ \cos}^2\theta -20800k\alpha {r}^2+462k\beta {\alpha}^2+700{ \cos}^2\theta -\\ {}8225\alpha {r}^2+2100k{r}^2+12120k\alpha +29200k{ \cos}^2\theta -1428k\beta -2800\alpha { \cos}^2\theta +\\ {}6300{r}^2{ \cos}^2\theta +6825{\alpha}^2{r}^2+2400k{\alpha}^2{ \cos}^2\theta +7725{k}^2{\alpha}^2{r}^2-315{\alpha}^2-3900{k}^2+\\ {}16940k\beta \alpha {r}^2{ \cos}^2\theta -3080k\beta {\alpha}^2{r}^2{ \cos}^2\theta -490k\beta {\alpha}^2{r}^2-23100k\beta {r}^2{ \cos}^2\theta -\\ {}300k{\alpha}^2{r}^2{ \cos}^2\theta -490k\beta \alpha {r}^2-1960k\beta \alpha { \cos}^2\theta +840k\beta {\alpha}^2{ \cos}^2\theta +\\ {}7050\alpha k{r}^2{ \cos}^2\theta -14400{k}^2\alpha { \cos}^2\theta +2400{k}^2{\alpha}^2{ \cos}^2\theta +22200{k}^2{ \cos}^2\theta -\\ {}4740\beta {k}^2{r}^2+3900{r}^2{k}^2-5930{k}^2\beta \alpha +1450{k}^2\beta {\alpha}^2-30000{k}^2\beta { \cos}^2\theta -\\ {}15160{k}^2\beta \alpha {r}^2{ \cos}^2\theta +1720\beta {k}^2{\alpha}^2{r}^2{ \cos}^2\theta +5930\beta \alpha {k}^2{r}^2-1450{k}^2\beta {\alpha}^2{r}^2+\\ {}14400{k}^2\alpha {r}^2{ \cos}^2\theta +30000{k}^2\beta {r}^2{ \cos}^2\theta -12575{k}^2\alpha {r}^2+15160{k}^2\beta \alpha { \cos}^2\theta -\\ {}1720{k}^2\beta {\alpha}^2{ \cos}^2\theta +4740{k}^2\beta \Big)/{\left(k+1\right)}^2\\ {}{\tilde{\tau}}_{2,r\phi }={\tilde{\tau}}_{2,\phi r}=0\end{array} $$

(A.32)

$$ \begin{array}{l}{\tilde{\tau}}_{2,\theta \theta }=-\frac{1}{1400}\beta r \cos \theta \Big(-13060-25800k\alpha { \cos}^2\theta +43848k\beta {r}^2-42360k-\\ {}23100k{r}^2{ \cos}^2\theta -4440{r}^2+2800{\alpha}^2{r}^2{ \cos}^2\theta -10440k{\alpha}^2+7770k\beta \alpha -\\ {}2520k\beta { \cos}^2\theta -10125{k}^2{\alpha}^2+5490k{\alpha}^2{r}^2-31500{k}^2{r}^2{ \cos}^2\theta +32575{k}^2\alpha -\\ {}9800\alpha {r}^2{ \cos}^2\theta -3500{k}^2{\alpha}^2{r}^2{ \cos}^2\theta -19780k\alpha {r}^2-2058k\beta {\alpha}^2+10500{ \cos}^2\theta +\\ {}1855\alpha {r}^2+21900k{r}^2+40520k\alpha +43800k{ \cos}^2\theta -4788k\beta -4200\alpha { \cos}^2\theta +\\ {}8400{r}^2{ \cos}^2\theta +315{\alpha}^2{r}^2+7945\alpha +3600k{\alpha}^2{ \cos}^2\theta +5175{k}^2{\alpha}^2{r}^2-315{\alpha}^2+\\ {}23100k\beta \alpha {r}^2{ \cos}^2\theta -4200k\beta {\alpha}^2{r}^2{ \cos}^2\theta +6678k\beta {\alpha}^2{r}^2-31500k\beta {r}^2{ \cos}^2\theta -\\ {}700k{\alpha}^2{r}^2{ \cos}^2\theta -34650k\beta \alpha {r}^2-2940k\beta \alpha { \cos}^2\theta -1260k\beta {\alpha}^2{ \cos}^2\theta +\\ {}10850k\alpha {r}^2{ \cos}^2\theta -21600{k}^2\alpha { \cos}^2\theta +3600{k}^2{\alpha}^2{ \cos}^2\theta +33300{k}^2{ \cos}^2\theta -\\ {}13500\beta {k}^2{r}^2+26340{r}^2{k}^2-14370\beta \alpha {k}^2+1490\beta {\alpha}^2{k}^2-4500\beta {k}^2{ \cos}^2\theta -\\ {}21840{k}^2\beta \alpha {r}^2+2520\beta {\alpha}^2{r}^2{ \cos}^2\theta +18\beta \alpha {k}^2{r}^2+1494\beta {k}^2{\alpha}^2{r}^2-30300{k}^2+\\ {}20650{k}^2\alpha {r}^2{ \cos}^2\theta +42840\beta {k}^2{r}^2{ \cos}^2\theta -21635{k}^2\alpha {r}^2+22740\alpha \beta {k}^2{ \cos}^2\theta -\\ {}2580\beta {k}^2{\alpha}^2{ \cos}^2\theta +29700\beta {k}^2\Big)/{\left(k+1\right)}^2\\ {}{\tilde{\tau}}_{2,\theta \phi }={\tilde{\tau}}_{2,\phi \theta }=0\\ {}{\tilde{\tau}}_{2,\phi \phi }=-\frac{1}{1400}\beta r \cos \theta \Big(-3560+11088k\beta {r}^2+440k-5700k{r}^2{ \cos}^2\theta +3960{r}^2+\\ {}2800{\alpha}^2{r}^2{ \cos}^2\theta -6840k{\alpha}^2+4830k\beta \alpha -6525{k}^2{\alpha}^2+90k{\alpha}^2{r}^2-5700{k}^2{r}^2{ \cos}^2\theta +\\ {}10975{k}^2\alpha -1400\alpha {r}^2{ \cos}^2\theta +1900{k}^2{\alpha}^2{r}^2{ \cos}^2\theta -10480k\alpha {r}^2-798k\beta {\alpha}^2-6545\alpha {r}^2+\\ {}4500k{r}^2+14720k\alpha -7308k\beta +315{\alpha}^2{r}^2+3745\alpha -225{\alpha}^2{k}^2{r}^2-315{\alpha}^2+798k\beta {\alpha}^2{r}^2-\\ {}5460k\beta \alpha {r}^2{ \cos}^2\theta +1680k\beta {\alpha}^2{r}^2{ \cos}^2\theta +1260k\beta {r}^2{ \cos}^2\theta +4700k{\alpha}^2{r}^2{ \cos}^2\theta -6090k\beta \alpha {r}^2+\\ {}1550k\alpha {r}^2{ \cos}^2\theta +17460{k}^2\beta {r}^2+540{r}^2{k}^2+8370\beta \alpha {k}^2-1090{k}^2\beta {\alpha}^2-11880{k}^2\beta \alpha {r}^2{ \cos}^2\theta +\\ {}2640{k}^2\beta {\alpha}^2{r}^2{ \cos}^2\theta -9942{k}^2\beta \alpha {r}^2+1374{k}^2\beta {\alpha}^2{r}^2+2950{k}^2\alpha {r}^2{ \cos}^2\theta -15300{k}^2\beta +\\ {}11880{k}^2\beta {r}^2{ \cos}^2\theta -3935{k}^2\alpha {r}^2+3000{k}^2\Big)/{\left(k+1\right)}^2\end{array} $$