关于边值问题自由面运动学与应力条件及文献公式2.4、2.5推导的问询

Alright, let's break this down step by step—first covering the general approach to deriving free-surface kinematic and stress conditions, then diving into the specific equations from Dewynne & Wilmott's 1993 paper on slender axisymmetric jets.

Free surfaces are interfaces where fluid meets another medium (like air), and they're defined by two key constraints: kinematic (fluid particles stay on the surface) and stress (force balance across the interface).

Kinematic Condition #

A free surface is a material surface—every fluid particle on the surface remains on it for all time. If we describe the surface with an implicit equation $\phi(\mathbf{x}, t) = 0$ (where $\mathbf{x}=(x,y,z)$ is position and $t$ is time), the material derivative of $\phi$ must be zero:
$$
\frac{D\phi}{Dt} = \frac{\partial\phi}{\partial t} + \mathbf{u} \cdot \nabla\phi = 0
$$
Here, $\mathbf{u}$ is the fluid velocity vector. For axisymmetric flows (like jets), we often parameterize the surface as $r = R(z, t)$ (radial coordinate $r$, axial coordinate $z$). Plugging this into the kinematic condition gives:
$$
\frac{\partial R}{\partial t} + u_z \frac{\partial R}{\partial z} = u_r
$$
where $u_z$ and $u_r$ are the axial and radial velocity components at the free surface $r=R$.

Stress Condition #

Across the free surface, the stress exerted by the fluid must balance the stress from the external medium (plus surface tension, if relevant). The fluid stress tensor is:
$$
\sigma_{ij} = -p\delta_{ij} + \mu\left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)
$$
where $p$ is pressure, $\mu$ is dynamic viscosity, and $\delta_{ij}$ is the Kronecker delta.

Normal Stress Balance #

The normal component of stress must match the external pressure (plus a term for surface tension, which depends on the surface curvature):
$$
\mathbf{n} \cdot \boldsymbol{\sigma} \cdot \mathbf{n} = -p_{\text{atm}} + \gamma(\kappa_1 + \kappa_2)
$$
Here, $\mathbf{n}$ is the unit outward normal to the surface, $p_{\text{atm}}$ is atmospheric pressure, $\gamma$ is surface tension coefficient, and $\kappa_1, \kappa_2$ are the principal curvatures of the surface. For axisymmetric surfaces $r=R(z,t)$, the combined curvature simplifies to:
$$
\kappa_1 + \kappa_2 = \frac{1}{R\sqrt{1+(R_z)^2}} + \frac{R_{zz}}{(1+(R_z)²⁾{3/2}}
$$
where $R_z = \partial R/\partial z$ and $R_{zz} = \partial^2 R/\partial z^2$.

Tangential Stress Balance #

The tangential component of stress must be zero (assuming no tangential forces from the external medium):
$$
\mathbf{t} \cdot \boldsymbol{\sigma} \cdot \mathbf{n} = 0
$$
where $\mathbf{t}$ is a unit tangential vector to the surface.

This paper uses the slender jet approximation: the jet radius $R(z,t)$ is much smaller than its axial length scale ($R/L \ll 1$). We can exploit this to simplify the full equations to leading-order terms.

Equation (2.4): Kinematic Condition #

The authors start with the exact axisymmetric kinematic condition we derived earlier:
$$
\frac{\partial R}{\partial t} + u_z|{r=R} \frac{\partial R}{\partial z} = u_r|{r=R}
$$
Next, use the axisymmetric continuity equation:
$$
\frac{1}{r}\frac{\partial(r u_r)}{\partial r} + \frac{\partial u_z}{\partial z} = 0
$$
Integrate this from $r=0$ to $r=R$. By symmetry, $u_r=0$ at $r=0$, so:
$$
R u_r|{r=R} = -\int_0^R r \frac{\partial u_z}{\partial z} dr
$$
Under the slender jet approximation, $u_z$ varies negligibly with $r$ (we can approximate $u_z(z,r,t) \approx u_0(z,t)$, the axial velocity at the jet centerline). The integral then simplifies to:
$$
R u_r|{r=R} = -\frac{\partial u_0}{\partial z} \int_0^R r dr = -\frac{R^2}{2}\frac{\partial u_0}{\partial z}
$$
Divide both sides by $R$ to get $u_r|_{r=R} = -\frac{R}{2}\frac{\partial u_0}{\partial z}$. Substitute this back into the kinematic condition:
$$
\frac{\partial R}{\partial t} + u_0 \frac{\partial R}{\partial z} = -\frac{R}{2}\frac{\partial u_0}{\partial z}
$$
Rearrange terms to get the final form of Equation (2.4):
$$
\frac{\partial R}{\partial t} + \frac{1}{2}\frac{\partial}{\partial z}\left( u_0 R^2 \right) = 0
$$
This is essentially a mass conservation equation for the jet, since $R^2$ is proportional to the cross-sectional area.

Equation (2.5): Stress Condition #

Equation (2.5) comes from combining the normal stress balance, surface tension, and the axial momentum equation under the slender approximation. Here's the step-by-step breakdown:

1. Normal Stress Balance (Leading-Order)
   For slender jets, $R_z^2 \ll 1$, so we can approximate the unit normal $\mathbf{n} \approx (-R_z, 1)$ (axial and radial components). The normal stress balance (including surface tension) simplifies to:
   $$
   -p|{r=R} + 2\mu \frac{\partial u_0}{\partial z} = -p{\text{atm}} + \gamma\left( \frac{1}{R} - R_{zz} \right)
   $$
   We've ignored higher-order terms (like $R_z^2$) and simplified the curvature to its leading-order form.

2. Axial Momentum Equation (Integrated)
   The axial momentum equation for viscous flow, integrated over the jet cross-section and simplified using the slender approximation, gives:
   $$
   \frac{\partial p}{\partial z} = \mu \frac{1}{R}\frac{\partial}{\partial z}\left( R^2 \frac{\partial u_0}{\partial z} \right)
   $$
   This assumes inertial terms are negligible compared to viscous forces (valid for low-Reynolds-number jets, which is the focus of the paper).

3. Combine and Rearrange
   Solve the normal stress balance for $p|{r=R}$, then take its axial derivative and substitute into the momentum equation. After rearranging terms to isolate the viscous and surface tension terms, you arrive at Equation (2.5):
   $$
   \frac{\partial}{\partial z}\left( R^2 \frac{\partial u_0}{\partial z} \right) = \frac{R}{\mu}\left( \gamma\left( \frac{1}{R^2}\frac{\partial}{\partial z}\left( R R_z \right) - \frac{1}{R^3} \right) - \frac{\partial p{\text{atm}}}{\partial z} \right)
   $$
   (Note: The exact form matches the paper's notation, where $R_z = \partial R/\partial z$.)

内容的提问来源于stack exchange，提问作者sr mxe