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Generalization of the Hele-Shaw approach to flow in thin curved layers

2 Generalization of the Hele-Shaw approach to flow in thin curved layers [Pg.175]

The Christoffel symbols (Spiegel, 1974), appearing in the second term on the right-hand side of Equation (5.57), are defined as [Pg.177]

Here /, r and, v are unequal integers in the set 1, 2, 3. As already mentioned, in the thin-layer approach the fluid is assumed to be non-elastic and hence the stress tensor here is given in ternis of the rate of deforaiation tensor as r(p) = riD(ij), where, in the present analysis, viscosity p is defined using the power law equation. The model equations are non-dimensionalized using [Pg.177]

Non-dimensionalization of the stress is achieved via the components of the rate of deformation tensor which depend on the defined non-dimensional velocity and length variables. The selected scaling for the pressure is such that the pressure gradient balances the viscous shear stre.ss. After substitution of the non-dimensional variables into the equation of continuity it can be divided through by ieLr U). Note that in the following for simplicity of writing the broken over bar on tire non-dimensional variables is dropped. [Pg.177]




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Flow curve

General Approach

Generalization to

Hele-Shaw

Hele-Shaw approach

In approaches

In general

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Layered flow

The -Curve

The general approach

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