(bubble_shape_deformation)= # Shape deformation {cite:t}`Moore1959-wo` derived a bubble shape model for small perturbation from perfect sphere, that is the bubble shape is approximated as ({numref}`Moore_spheroid`) ```{math} :label: eq:ForceBalance_nonref_19 r = a \left\{ 1 + \epsilon P_{2} (\cos \theta) \right\} ``` where $r$ and $\theta$ are the spherical coordinates, $a$ is the radius of sphere, $\epsilon~(\ll 1)$ is a small perturbation, and $P_{2} (\cos \theta)$ is the second Legendre polynomial defined by ```{toggle} $P_{0}(x) = 1$, $P_{1}(x) = x$, $P_{2}(x) = (3x^{2}-1)/2$, $P_{3}(x) = (35x^{4}-30x^{2}+3)/8$ $\cdots$. The Legendre polynomials are orthogonal within $[-1, 1]$, that is $\int_{-1}^{1} P_{m}(x) P_{n} dx = 2\delta_{mn}/(2n+1)$. ``` ```{math} :label: eq:ForceBalance_nonref_20 P_{2} (\cos \theta) = \frac{1}{2} \left( 3 \cos^{2} \theta - 1 \right) ``` ```{figure} ../fig/Moore_spheroid.png :name: Moore_spheroid Slightly deformed bubble ``` The tangential velocity components in a potential flow about a sphere moving a constant speed $u$ is given by (Appendix {ref}`app_potential_flow_sphere`) ```{math} :label: eq:ForceBalance_nonref_21 v_{\theta} = \frac{u a^{3}}{2 r^{3}} \sin \theta ``` Riding on the sphere, we observe ```{math} :label: eq:ForceBalance_nonref_22 v_{\theta} = \frac{3 u a^{3}}{2 r^{3}} \sin \theta ``` Therefore, ```{math} :label: eq:ForceBalance_nonref_23 p_{s} = p + \frac{1}{2} \rho q^{2} \sim p + \frac{1}{2} \rho v_{\theta}^{2} ``` ```{math} :label: eq:ForceBalance_nonref_24 \begin{split} \frac{1}{2} \rho v_{\theta}^{2} &= \frac{1}{2} \rho \frac{9 u^{2} a^{6}}{4 r^{6}} \sin^{2} \theta \sim \frac{1}{2} \rho \frac{9 u^{2} a^{6}}{4 a^{6} (1 + \epsilon P_{2}(\cos \theta))^{6}} \sin^{2} \theta \sim \rho \frac{9 u^{2}}{8} (1 -6 \epsilon P_{2}(\cos \theta)) \sin^{2} \theta \\ &= \rho \frac{9 u^{2}}{8} \sin^{2} \theta + O(\epsilon \rho u^{2}) \end{split} ``` At the stagnation point ($\theta = 0$), ```{math} :label: eq:shape_eq_stagnation_point p_{g} = p_{s} + \frac{2 \sigma}{a} ``` and for $0 < \theta < \pi$, ```{math} :label: eq:ForceBalance_nonref_25 p_{g} = p_{s} - \frac{1}{2} \rho u_{\theta}^{2} + \sigma \kappa ``` We need to calculate an approximate curvature $\kappa$ at $\theta$. The surface equation is given by ```{math} :label: eq:ForceBalance_nonref_26 f = r - a (1 + \epsilon P_{2}(\cos \theta)) ``` The normal to the isosurface of $f$ is (see Appendix {ref}`app_nseq_in_polar_sys` for differential operators) ```{math} :label: eq:ForceBalance_nonref_27 \mathbf{N} = \left( \frac{\partial f}{\partial n}, \frac{1}{r} \frac{\partial f}{\partial \theta}, \frac{1}{r \sin \theta} \frac{\partial f}{\partial \varphi} \right) = \left( 1, \frac{3 a \epsilon}{r} \sin \theta \cos \theta, 0 \right) ``` The unit normal is therefore ```{math} :label: eq:ForceBalance_nonref_28 \mathbf{n} = \frac{\mathbf{N}}{| \mathbf{N} |} = \frac{ \left( 1, \frac{3 a \epsilon}{r} \sin \theta \cos \theta, 0 \right) }{\left\{ {1 + \frac{9a^{2} \epsilon^{2}}{r^{2}} \sin^{2} \theta \cos^{2} \theta} \right\}^{1/2}} ``` However, for the first order, ```{math} :label: eq:ForceBalance_nonref_29 \mathbf{n} = \left( 1, \frac{3 a \epsilon}{r} \sin \theta \cos \theta, 0 \right) ``` The curvature can be calculated as the divergence of the field of $\mathbf{n}$, that is ```{math} :label: eq:ForceBalance_nonref_30 \begin{split} \kappa &= \nabla \cdot \mathbf{n} = \frac{1}{r^{2}} \frac{\partial r^{2} n_{r} }{\partial r} + \frac{1}{r \sin \theta} \frac{\partial n_{\theta} \sin \theta}{\partial \theta} + \frac{1}{r \sin \theta} \frac{\partial n_{\varphi}}{\partial \varphi} \\ &\sim \frac{2}{r} + \frac{3 a \epsilon}{2} \left(2 - 3 \sin^{2} \theta \right) = \frac{2}{r} + \frac{3 a \epsilon}{r^{2}} P_{2} (\cos \theta) \end{split} ``` Substituting $r = a (1 + \epsilon P_{2}(\cos \theta))$ yields ```{math} :label: eq:ForceBalance_nonref_31 \kappa = \frac{2}{a} (1 - \epsilon P_{2}(\cos \theta)) + \frac{3 \epsilon}{a} (1 - 2 \epsilon P_{2}(\cos \theta)) P_{2} (\cos \theta) \sim \frac{2}{a} + \frac{4 \epsilon}{a} P_{2}(\cos \theta) + O \left( \frac{\epsilon^{2}}{a} \right) ``` Therefore, ```{math} :label: eq:shape_eq_pressure_balance p_{g} = p_{s} - \rho \frac{9 u^{2}}{8} \sin^{2} \theta - O(\epsilon \rho u^{2}) + \sigma \left\{ \frac{2}{a} \left( 1 + 2 \epsilon P_{2} (\cos \theta) \right) + O \left( \frac{\epsilon^{2}}{a} \right) \right\} ``` Subtracting Eq. {eq}`eq:shape_eq_stagnation_point` from Eq. {eq}`eq:shape_eq_pressure_balance` yields ```{math} :label: eq:ForceBalance_nonref_32 \rho \frac{9 u^{2}}{8} \sin^{2} \theta + O(\epsilon \rho u^{2}) = \frac{4 \sigma \epsilon}{a} P_{2} (\cos \theta) + O \left( \frac{\sigma \epsilon^{2}}{a} \right) ``` The Legendre polynomial can be written as ```{math} :label: eq:ForceBalance_nonref_33 P_{2} (\cos^{2} \theta) = \frac{1}{2} (\cos^{2} \theta - 1) = 1 - \frac{3}{2} \sin^{2} \theta ``` Hence, ```{math} :label: eq:ForceBalance_nonref_34 \rho \frac{9 u^{2}}{8} \sin^{2} \theta + O(\epsilon \rho u^{2}) = \frac{4 \sigma \epsilon}{a} \left( 1 - \frac{3}{2} \sin^{2} \theta \right) + O \left( \frac{\sigma \epsilon^{2}}{a} \right) ``` Considering the balance between the terms depending on $\sin^{2} \theta$, we find ```{math} :label: eq:ForceBalance_nonref_35 \epsilon = - \frac{3}{16} \frac{\rho u^{2} a}{\sigma} ``` Using the definition of the Weber number, $We = 2 \rho u^{2} a / \sigma$, we have ```{math} :label: eq:ForceBalance_nonref_36 \epsilon = - \frac{3}{32} We ``` The definition of $We$ used here can be considered as the same as that with the sphere-volume-equivalent bubble diameter since the perturbation from spherical shape is small enough. The coordinates of the surface at $\theta = 0$ and $\pi/2$ are ```{math} :label: eq:ForceBalance_nonref_37 r = a (1 + \epsilon)~~(\theta = 0),~~~~r = a (1 - \epsilon/2)~~(\theta = \pi/2) ``` The aspect ratio (the major axis / the minor axis) of the bubble is therefore given by ```{math} :label: eq:ForceBalance_nonref_38 \chi = \frac{a (1 - \epsilon / 2)}{a (1 + \epsilon)} \sim (1 - \epsilon/2)(1 - \epsilon) \sim 1 - \frac{3}{2} \epsilon = 1 + \frac{9}{64} We ``` The bubble shape is hence oblate spheroid and the deformation increases linearly as the Weber number increases. {cite:t}`Moore1965-go` attempted to exactly satisfy the boundary conditions at the stagnation point and at the bubble equator, which resulted in ```{math} :label: eq:Moore_eq_more_rigorous_model We = \frac{ 4 (\chi^3 + \chi - 2) [\chi^2 \sec^{-1} \chi - (\chi^2 - 1)^{\frac{1}{2}}]^2 }{\chi^{\frac{4}{3}} (\chi^2 - 1)^{3}} ``` The bubble shape models are compared in {numref}`Moore_shape_model`. The linearized model agrees with the more rigorous model when the bubble deformation is weak (small $\chi$), while the deviation becomes very large for $We > 1$. The model of {cite:t}`Moore1965-go` shows a saturation of the Weber number, $We = 3.745 \dots$, showing a limitation of the model assumption. In reality, larger Weber numbers are possible, but the bubble rise behavior changes from rectilinear to oscillating leading to a different $We$-$\chi$ relationship. For deformed bubbles in oscillating motion {cite:t}`Hayashi2021-tl` proposed the following empirical equation: ```{math} :label: eq:ForceBalance_nonref_39 \chi = 1 + 0.62 We^{0.376} ``` A similar correlation was derived by {cite:t}`Puncochar2022-mp` from a force balance in the bubble detachment from a nozzle tip. ```{figure} ../python/MooreWechi.png :name: Moore_shape_model Moore's bubble shape models ``` Various bubble shape correlations have been proposed so far. Most of them are extensions of Moore's model or fitting of the following functional form to experimental data: ```{math} :label: eq:Shape_eq_chi_Eo_type \chi = 1 + \alpha Eo^{\beta} ``` where $\alpha$ and $\beta$ are constants. Some examples are given in Table {ref}`Shape_tab_chi_Eo_type`. ```{table} Constants in aspect ratio correlation :name: Shape_tab_chi_Eo_type | Source | $\alpha$ | $\beta$ | Data | | :--- | :--- | :--- | :--- | | {cite:t}`Wellek1966-vc` | 0.163 | 0.757 | drops in liquids | | {cite:t}`Lee2020-uo` | 0.21 | 0.58 | deformed bubbles in water flow | | {cite:t}`Okawa2003-mg` | 1.97 | 1.3 | small bubbles in water | | {cite:t}`Sugihara2007` | 6.5 | 1.925 | small bubbles in water | | {cite:t}`Hessenkemper2021-hc` | 0.94 | 0.0875 | deformed bubbles in water flow | ``` Neither $We$ nor $Eo$ accounts for the viscous effect on shape deformation. {cite:t}`Tadaki1961` proposed to use a combination of $Re$ and $M$ in correlating the bubble aspect ration, i.e., ```{math} :label: eq:eq_E_tadaki_maeda \frac{1}{\chi^{1/3}} = \left\{ \begin{array}{ll} 0.62 & \text{for } 16.5 < T_a \\ 1.36 T_a^{-0.28} & \text{for } 6 < T_a \leq 16.5 \\ 1.14 T_a^{-0.176} & \text{for } 2 < T_a \leq 6 \\ 1 & \text{for } T_a \leq 2 \end{array} \right. ``` where $Ta$ is the Tadaki number defined by ```{math} :label: eq:eq_tadaki_number Ta = Re M^{0.23} ``` {cite:t}`Fan1990-qr` extended a correlation proposed by Vakrushev and Efremov (1970) as ```{math} :label: eq:eq_E_fan_tsuchiya \frac{1}{\chi} = \left\{ \begin{array}{ll} 1 & Ta < Ta_1 \\ \{c_1 + c_2 \tanh[c_3 (c_4 - \log_{10} Ta)]\}^m & Ta_1 \leq Ta < Ta_2 \\ 0.24 & Ta_2 \leq Ta \end{array} \right. ``` where $m = 3$, $Ta_{1} = 1.0, 0.3$, $Ta_{2} = 40, 20$, $c_{1} = 0.81$, $c_{2} = 0.20$, $c_{3} = 2.0, 1.8$, and $c_{4} = 0.80, 0.40$, and the former and latter values are for contaminated and clean bubbles, respectively. {cite:t}`Aoyama2016-rs` proposed an alternative way to take into account the viscous effect; they used ```{math} :label: eq:ForceBalance_nonref_40 Ao = Eo^{1.12} Re ``` to correlate $\chi$ rather than $Ta$: ```{math} :label: eq:eq_chi_Aoyama_clean \chi = \left( 1 + 0.016 Ao \right)^{0.388} ``` Large bubbles cannot maintain an ellipsoidal shape and exhibit the so-called spherical cap shape. The shape and rise velocity of spherical cap bubble will be discussed in {ref}`spherical_cap`.