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06_loc_full_flow

# 06_loc_full_flow - 1 Criteria for locally fully developed...

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1 Criteria for locally fully developed viscous flow Ain A. Sonin , MIT October 2002 Contents 1. Locally fully developed flow ………………………………………. 2 2. Criteria for locally fully developed flow ……………………………. 3 3. Criteria for constant pressure across abrupt cross-section changes .… 8

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2 1. Locally fully developed flow Fig. 1: Locally fully developed flow (left) and fully developed flow (right) Consider (as an example) a two-dimensional, laminar, incompressible, viscous flow in a diverging channel, as shown at left in Fig. 1. Let x be the coordinate in the primary flow direction and y the transverse coordinate. The flow is bounded below by a wall and above by either a wall or a free surface, and it may be steady or unsteady, either because the volume flow rate changes with time or because the upper boundary not only depends x but also moves up and down with time, that is, h=h(x,t) . The velocity and pressure fields in the channel are determined by the Navier-Stokes equation, u u u P + µ 2 u + y u 2 t + u x + v y = x x 2 2 (1) v v v P + µ 2 v + y v 2 = (2) t + u x + v y y x 2 2 , the mass conservation equation ± u ± v + = 0 , (3) ± x ± y and the appropriate boundary and initial conditions. In (1) and (2) P = p + ± gz (4) is a modified pressure in which p is the ordinary static pressure and z the distance measured up against gravity from some chosen reference level (it is
3 not the third Cartesian coordinate that goes with x and y ). The term ± gz in (4) accounts for the gravitational body force. The simultaneous presence of the nonlinear inertial terms on the left and the second order viscous terms on the right makes it difficult to solve the Navier-Stokes equation (1)-(2) in the general case. Under certain circumstances, however, all the inertial terms on the left hand sides of (1) and (2), while not zero, are small enough compared with the viscous term to be neglected, and the y-derivative in the viscous term dominates over the x- derivative. Under these conditions (1) and (2) simplify to ² P ² 2 u 0 ³ + µ (5) 2 ² x ² y ² P 0 ³ (6) ² y These equations are similar in form to the equations for a truly fully developed flow . The velocity profile at a station x in this diverging and possibly unsteady flow will thus be identical to the profile in a fully developed flow with the same height, the same boundary conditions at y =0 and y = h , and the same pressure gradient. The flow can be said to be locally fully developed , that is, having at every station x essentially the same velocity profile as a fully developed flow with the same cross-sectional geometry and boundary conditions. For example, if the flow is bounded by solid, immobile walls such that u =0 at y =0 and y=h , the local solution is the familiar parabolic one h 2 y 1 y ´ ² P u ³ . (7) 2 µ h µ h ² x A dependence on x and t enters implicitly, however, through h=h(x,t) and through the (as yet unknown) axial modified pressure distribution P(x,t).

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4 2. Criteria for locally fully developed flow Under what conditions can a flow be considered locally fully developed? If
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06_loc_full_flow - 1 Criteria for locally fully developed...

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