Mechanical Equations
Ordinary fluids may be treated as a continuous media over a wide range of pressures and temperatures. In this chapter we present the fundamentals such as the deformation and the stress tensor required to understand any deformable fluid medium. This is fol
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42
Chapter 3
MECHANICAL EQUATIONS
Ordinary fluids may be treated as a continuous media over a wide range of pressures and temperatures. In this chapter we present the fundamentals such as the deformation and the stress tensor required to understand any deformable fluid medium. This is followed by the iniroduction of the conservation laws for mass and momentum in a general setting. 1\vo good texts in fluid mechanics, where the concepts given here are discussed in great detail, are [1 ], [2]. A separate section, following the argument of Landau & Lifshitz [3], considers the important problem of determining the electrical forces in dielectric fluids, with either a linear or a nonlinear polarization, in the context of thermodynamics. The boundary and jump conditions are stated for electrodes and interfaces, paying special attention to the capillary phenomena. Finally, some important nondimensional numbers found in EHD are discussed in the last section.
3.1
Mechanical equations
· A fluid, liquid or gas, is composed of an enormous number of molecules. Under standard conditions (1 atmosphere, 273 K) the typical separation between molecules is of the order of do ~ 3.5 x 10- 10m in liquids and 10d0 in gases. Measuring instruments probe local volumes with linear dimensions several orders of magnitude larger than d0 , but still quite small by any macroscopic standard. Denoting this volume by 8V, the mass density p, and the velocity u, are given by: p=
L:~=l mk 8V
'
(3.1)
where N is the number of molecules in volume 8V, and mk, uk, are the mass and velocity of each molecule respectively.
A. Castellanos (ed.), Electrohydrodynamics © Springer-Verlag Wien 1998
43
Basic Concepts and Equations in Electrohydrodynamics
3.1.1
Continuum hypothesis
The fluid may be modeled as a continuum, where density and velocity are smooth functions of position, if the following inequalities hold:
(6V/N) 113
«
(6V) 1/ 3 « l,
(3.2)
with l a typical scale of variation of any macroscopic quantity. In particular, if l is defined either as pj\l;p, or as ujj\l;uj where i, j = 1, 2, 3, for any arbitrary chosen point in the fluid at a given time, the above inequality must be true for all points and over all time.
3.1.2
Kinematics of fluids
There are two ways to specify the position of a given element of fluid. In the first, called the Lagrangian description, the position, x, of a particle at time t is specified as a function of its initial position, x(O) = x 0 , at t = 0, i.e., x = x(x 0 , t). In the second, called Eulerian and used in this text, the position of the particle is described as a function of the independent variables x, and t. Any physical magnitude is also a function of these variables. Consider how a given magnitude, f(x, t), varies in time for a fixed particle of fluid. This is defined as the material derivative of J, given by
df = f(x dt
+ dx, t + dt)- f(x, t)
= aj
dt
at
+
u. \lf
'
(3.3)
where we have used the fact that dx(x, t) = u(x, t)dt is the displacement of the particle during the time interval dt. In
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