progressively-moving-surface.tex (4717B)
1 \subsubsection{Progressively moving surface} 2 3 When a particle touches a parametric surface given by 4 \begin{equation*} 5 \vec{S}=\vec{S}(a,b,t); 6 \qquad 7 a,b\in{}A=[0,1]; 8 \qquad 9 \vec{n}=\frac{\partial\vec S}{\partial a} \times \frac{\partial\vec S}{\partial b} 10 \end{equation*} 11 where \(a\) and \(b\) are parameters, boundary condition is written as 12 \begin{equation*} 13 \frac{d}{d t} \vec\nabla w \cdot\vec{n} = \frac{d}{d t} \vec S \cdot\vec{n}; 14 \qquad 15 \vec\zeta = \vec S 16 . 17 \end{equation*} 18 We seek solutions of the form 19 \begin{equation*} 20 \begin{aligned} 21 & w\left(\alpha,\beta,\delta,t\right) 22 = C_1\exp\left( \left(i\vec{d}_i+\vec{d}_{k}\right)\cdot\vec\zeta - i\omega t\right) + \\ 23 & 24 \iint\limits_{a,b\,\in{}A} 25 C_2\exp\left( \left(i\vec{d}_r+\vec{d}_{k}\right)\cdot\vec\zeta - i\omega t\right) 26 da\,db\,+ \\ 27 & 28 \iint\limits_{a,b\,\in{}A} 29 C_3\exp\left( \left(i\vec{d}_i+i\vec{d}_\upsilon+\vec{d}_{k_3}\right)\cdot\vec\zeta - i\omega_3 t\right) 30 da\,db, 31 \end{aligned} 32 \end{equation*} 33 where \(\vec{d}_\upsilon=\vec{n}\left(\DerivativeT{\vec{S}}\cdot\vec{n}\right)\) 34 is a projection of surface velocity to the surface normal. 35 Here \(C_2\), \(C_3\) and \(\vec{n}\) depend on both \(a\) and \(b\), 36 i.e.~we have one reflection for each point of the surface. 37 We plug this expression into the boundary condition and get 38 \begin{equation*} 39 \begin{aligned} 40 & 41 C_1 f_1 42 \exp\left( \left(i\vec{d}_i+\vec{d}_{k}\right)\cdot\vec{S}-i\omega{}t \right) + \\ 43 & 44 \iint\limits_{a,b\,\in{}A} 45 C_2 f_2 46 \exp\left( \left(i\vec{d}_r+\vec{d}_{k}\right) \cdot \vec{S} -i\omega{}t \right) 47 da\,db\, 48 + \\ 49 & 50 \iint\limits_{a,b\,\in{}A} 51 C_3 f_3 52 \exp\left( 53 \left(i\vec{d}_i+i\vec{d}_\upsilon+\vec{d}_{k_3}\right)\cdot\vec{S}-i\omega_3{}t\right) 54 da\,db 55 = \frac{\partial\vec{S}}{\partial t} \cdot \vec{n}, 56 \end{aligned} 57 \end{equation*} 58 where 59 \begin{equation*} 60 \begin{aligned} 61 & 62 f_1 = \omega \left(\vec{d}_i\cdot\vec{n}\right); 63 \qquad 64 f_2 = -\omega \left(\vec{d}_i\cdot\vec{n}\right); \\ 65 & 66 f_3 = \left( 67 i\DerivativeT{\vec{d}_\upsilon} + 68 \left(\vec{d}_i + \vec{d}_\upsilon\right) \circ 69 \left(\omega_3-\DerivativeT{\vec{d}_\upsilon}\right) 70 \right)\cdot\vec{n}. 71 \end{aligned} 72 \end{equation*} 73 Here \(\circ\) denotes Hadamard (element-wise) vector product. 74 Take double integral over \(a\) and \(b\) of each side of the equation, 75 assume that expressions under integral sign are equal (i.e.~remove 76 integral sign from the equation), and make substitutions for derivatives that 77 were made in the previous section. Then, the equation reduces to 78 the form, similar to moving and rotating panel, but with different terms for the 79 boundary: 80 \begin{equation*} 81 \begin{aligned} 82 & 83 C_1 f_1 84 \exp\left( \left(i\vec{d}_i+\vec{d}_{k}\right)\cdot\vec{S}\right) + 85 S C_2 f_2 86 \exp\left( \left(i\vec{d}_r+\vec{d}_{k}\right)\cdot\vec{S}\right) 87 + \\ 88 & 89 S C_3 f_3 90 \exp\left( 91 \left(i\vec{d}_i+i\vec{d}_\upsilon+\vec{d}_{k_3}\right)\cdot\vec{S}-i\omega_3{}t+i\omega{}t\right) 92 = 93 \left(\DerivativeT{\vec{S}}\cdot\vec{n}\right)\exp\left(i\omega{}t\right), 94 \end{aligned} 95 \end{equation*} 96 Here \(S\) without vector arrow denotes surface area. Choose \(C_1\) and \(C_2\) 97 so that the first and the second term cancel each other out, and calculate 98 \(C_3\) from the resulting equation: 99 \begin{equation*} 100 \begin{aligned} 101 & 102 C_1 = \frac{1}{f_1} 103 \exp\left( 104 -\left(i\vec{d}_i+\vec{d}_{k}\right)\cdot\vec{S} 105 \right); \\ 106 & 107 C_2 = \frac{1}{S f_2} 108 \exp\left( 109 -\left(i\vec{d}_r+\vec{d}_{k}\right)\cdot\vec{S} 110 \right); \\ 111 & 112 C_3 = \frac{\DerivativeT{\vec{S}}\cdot\vec{n}}{ S f_3 } 113 \exp\left(- 114 \left(i\vec{d}_i+i\vec{d}_\upsilon+\vec{d}_{k_3}\right)\cdot\vec{S}+i\omega_3{}t\right) 115 \end{aligned} 116 \end{equation*} 117 118 Plug the solution into continuity equation and get 119 \begin{equation*} 120 \begin{aligned} 121 & 122 - \left(u - n_1 \left(\vec{d}_i\cdot\vec{n}\right)\right)^2 123 - \left(v - n_2 \left(\vec{d}_i\cdot\vec{n}\right)\right)^2 124 + k^2 = 0 \\ 125 & 126 - \left(u + n_1 \left(\vec{d}_\upsilon\cdot\vec{n}\right)\right)^2 127 - \left(v + n_2 \left(\vec{d}_\upsilon\cdot\vec{n}\right)\right)^2 128 + k_3^2 = 0 \\ 129 & 130 k = \pm\sqrt{u^2+v^2} \\ 131 & 132 k_3 = \pm\sqrt{\smash[b]{u^2+v^2 + 133 \left(\vec{d}_\upsilon\cdot\vec{n}\right) \left( 134 \vec{d}_\upsilon\cdot\vec{n} + \vec{d}_i\cdot\vec{n} \right)}} 135 \end{aligned} 136 \end{equation*} 137 This equation makes \(k_3\) depend on time, which does not cause trouble, 138 because time derivative is always multiplied by \(n\) and the third component 139 of \(n\) is always nought. 140 141 When \(\vec{n}\) depends on time, i.e.~the surface rotates, the 142 coefficients we write coefficients as 143 \begin{equation*} 144 \begin{aligned} 145 & 146 f_1 = \left(\omega\vec{d}_i\right)\cdot\vec{n} 147 \\ 148 & 149 f_2 = 150 \left(i\vec{d}_r\circ\left(\DerivativeT{\vec{d}_s}-i\omega\right)\right)\cdot\vec{n}; 151 \\ 152 & 153 f_3 = 154 \end{aligned} 155 \end{equation*} 156