纳维–斯托克斯方程

from manim import *

class NSDerivation(Scene):
    def construct(self):
        title = Text("纳维–斯托克斯方程", font="Noto Serif CJK SC").to_edge(UP)
        self.play(Write(title), run_time=1.5)

        # Scene 1 · 质量守恒
        cont = MathTex(r"\frac{\partial \rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0")
        lab1 = Text("① 质量守恒", font="Noto Serif CJK SC", color=TEAL).scale(0.7)
        g1 = VGroup(lab1, cont).arrange(DOWN).next_to(title, DOWN, buff=1.0)
        self.play(FadeIn(lab1, shift=UP*0.2), Write(cont))
        self.wait(0.6)

        # Scene 2 · 动量守恒
        mom = MathTex(r"\rho\frac{D u_i}{Dt}=\frac{\partial\sigma_{ij}}{\partial x_j}+\rho f_i")
        lab2 = Text("② 动量守恒 (F = ma)", font="Noto Serif CJK SC", color=BLUE).scale(0.7)
        g2 = VGroup(lab2, mom).arrange(DOWN).next_to(g1, DOWN, buff=0.9)
        self.play(FadeIn(lab2, shift=UP*0.2), Write(mom))
        self.wait(0.6)

        # Scene 3 · 牛顿本构
        cons = MathTex(r"\sigma_{ij}=-p\,\delta_{ij}+\mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)")
        lab3 = Text("③ 牛顿流体本构(不可压)", font="Noto Serif CJK SC", color=GREEN).scale(0.7)
        g3 = VGroup(lab3, cons).arrange(DOWN).next_to(g2, DOWN, buff=0.9)
        self.play(FadeIn(lab3, shift=UP*0.2), Write(cons))
        self.wait(1.0)

        # Transform → 最终形式
        final = MathTex(
            r"\frac{\partial\mathbf{u}}{\partial t}+(\mathbf{u}\cdot\nabla)\mathbf{u}",
            r"=-\frac{1}{\rho}\nabla p+\nu\nabla^{2}\mathbf{u}",
        ).scale(1.25)
        self.play(*[FadeOut(m) for m in [g1, g2, g3]], ReplacementTransform(title.copy(), final))
        box = SurroundingRectangle(final, color=YELLOW, buff=0.25, corner_radius=0.1)
        self.play(Create(box))
        self.wait(2)


class Nondimensionalization(Scene):
    def construct(self):
        sub = MathTex(
            r"\mathbf{x}=L\mathbf{x}^{*},\ \ \mathbf{u}=U\mathbf{u}^{*},\ \ ",
            r"t=\frac{L}{U}t^{*},\ \ p=\rho U^{2}p^{*}"
        ).to_edge(UP)
        eq = MathTex(
            r"\frac{\partial\mathbf{u}^{*}}{\partial t^{*}}+(\mathbf{u}^{*}\cdot\nabla^{*})\mathbf{u}^{*}",
            r"=-\nabla^{*}p^{*}+",
            r"\frac{1}{\mathrm{Re}}",
            r"\nabla^{*2}\mathbf{u}^{*}",
        ).scale(1.2)
        re_def = MathTex(r"\mathrm{Re}=\frac{\rho U L}{\mu}", color=YELLOW).next_to(eq, DOWN, buff=0.9)
        self.play(Write(sub))
        self.wait(0.5)
        self.play(Write(eq))
        self.play(Circumscribe(eq[2], color=YELLOW), FadeIn(re_def, shift=UP*0.3))
        self.wait(2)