Stationary Flows in Viscous Fluid Bodies

Josef Bemelmans

A chapter in Variational Methods for Free Surface Interfaces, 1987, pp 173-178 from Springer

Abstract: Abstract Consider a drop of a viscous, incompressible fluid under the influence of some exterior force density f. A stationary flow inside the fluid body can be described by the Navier-Stokes system 1 $$\begin{array}{rclr}-v\Delta\upsilon + Dp + \upsilon \cdot D\upsilon = f\quad\quad\quad\\ \text{in}\,\Omega, \\\text{div}\,\upsilon = 0 \\\end{array}$$ together with the boundary conditions 2 $$\upsilon \cdot n = 0, t_{k} \cdot T \cdot n = 0\,\,\,\,\,\,\text{on}\, \Sigma, k = 1,\, 2,$$ 3 $$ n \cdot T \cdot n = p_{0} \quad\text{on}\,\Sigma$$ As usual, $$v=(v^1,v^2,v^3)=v(x), x=(x^1,x^2,x^3)$$ denotes the velocity, p = p(x) the pressure, and v > 0 is the kinematical viscosity. The unknown domain occupied by the fluid is denoted by Ω, its boundary by Σ; n is the outer normal to Σ, and t 1, t 2 span the tangent plane.

Date: 1987
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DOI: 10.1007/978-1-4612-4656-5_20

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