Pressure-Driven Flow in a Planar Channel (LO-6.4, 6.5) The sketch shows the steady flow of an inc...1 answer below »

Pressure-Driven Flow in a Planar Channel (LO-6.4,
6.5)

The sketch shows the steady flow of an incompressible,
Newtonian fluid between parallel planar walls. Since
P2<P1,
the pressure difference causes the fluid to flow in the x-
direction. Gravity acts in the negative y-direction.
(a) Draw your own diagram with coordinate axes. List all
simplifying assumptions required to develop a model
assuming one-dimensional flow.
(b) Based on your assumptions, most of the terms in the
Continuity Equation and the Navier-Stokes Equation are zero.
Write the simplified forms of the Continuity Equation and the
Navier-Stokes Equation for this problem. Also write out the
boundary conditions for velocity and pressure.
(c) Solve these equations to determine the pressure and velocity as
functions of position inside the channel.
(d) Integrate the velocity profile to determine the average
velocity of the flow. Also determine the volume flow rate (assume
the channel width in the z-direction is w).
(e) What is the relationship between pressure drop and flow rate
(the Hagen-Poiseuille Law) for planar channel flow?
(f) What are the strain rate and shear stress as functions of
position in the fluid?
(g) What is the shear force (magnitude and direction) exerted by
the fluid on the bottom wall? On the top wall?

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