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Governing equations

Transport equation for the momentum density (Navier-Stokes equations)

$(1)\qquad \varrho\,\frac{d\vec{v}}{dt}\,=\,\nabla\cdot\underline{\tau} + \vec{f}$

where $\vec{v}(x,z)$ and $\vec{f}(x,z)$ are the velocity and a body force field, respectively. We use

$(2)\qquad \vec{v}=\left(\begin{matrix}u\\ w\end{matrix}\right),\qquad \vec{f}=\left(\begin{matrix}f_1\\ f_2\end{matrix}\right) \qquad\mbox{and}\qquad \nabla=\left(\begin{matrix}\partial_x\\ \partial_z\end{matrix}\right).$

In the following we will denote partial derivatives with respect to $i$ either by $\partial_i$ or simply by the subscript $i$ . The body force may be potential, i.e. given by $\vec{f}=-\nabla\phi$ . The stress tensor writes

$(3)\qquad \underline{\tau}\,=\,-p+\eta(\nabla\vec{v}+(\nabla\vec{v})^T).$

where $p (x, z)$ stands for the pressure field. The material derivative is defined by

$(4)\qquad \frac{d}{dt}\,=\,\frac{\partial}{\partial t}+(\vec{v}\cdot\nabla).$

Boundary conditions

At a smooth solid substrate

For the velocity field at the substrate $(z = 0)$ classically one assumes the no-slip and the no-penetration condition.

$(5)\qquad \vec{v}=0.$

At a free surface

At the free surface $(z = h (x))$ one has the

kinematic condition (surface follows flow field)

$(6)\qquad w=\partial_th+u\partial_xh$

and the force equilibrium

$(7)\qquad \underline{\tau}\cdot\vec{n}=-K\gamma\,\vec{n}\,+\,(\partial_s\gamma)\,\vec{t}$

where the surface derivative is defined by $\partial_s=\vec{t}\cdot\nabla$ and we assume that the ambient air does not transmit any force $(\underline{\tau}_{air}=0)$ . The term $p L = - K γ$ corresponds to the Laplace or curvature pressure whereas $\partial_s\gamma$ results from the variation of the surface tension along the surface (resulting, for instance, from solutal or thermal Marangoni effects). The latter is modelled in the simplest case by a linear dependence of the surface tension on temperature; $γ = γ 0 + γ T (T 0 - T)$ , where $γ 0$ is the surface tension at the reference temperature $T 0$ and $γ T = d γ / d T .$

$(8)\qquad \vec{n}=\frac{(-\partial_xh,1)}{\left(1+(\partial_xh)^2\right)^{1/2}}, \quad\vec{t}=\frac{(1,\partial_xh)}{\left(1+(\partial_xh)^2\right)^{1/2}}, \quad K=\frac{\partial_{xx}h}{\left(1+(\partial_xh)^2\right)^{3/2}}$

are the normal vector, tangent vector and curvature of the surface, respectively.

The boundary condition (\ref{th-hddf5}) is of vectorial character, i.e.\ one can derive two scalar conditions by projecting it onto $\vec{n}$ and $\vec{t}$ , respectively.

$(9)\qquad \vec{t}:\quad \eta\,[(u_z+w_x)(1-h_x^2)+2(w_z-u_x)h_x]=\partial_s\gamma(1+h_x^2)$ $(10)\qquad \vec{n}:\quad p+\frac{2\eta}{1+h_x^2}\left[-u_xh_x^2-w_z+h_x(u_z+w_x)\right]=-\frac{\gamma h_{xx}}{(1+h_x^2)^{3/2}}$