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We have a narrow slit that is oriented vertically. The thickness of the slit is 2B....

We have a narrow slit that is oriented vertically. The thickness of the slit is 2B. The width, W, and the length, L, of the slit are both much larger than the thickness. A fluid is being forced through the top of the slit and runs out of the bottom. Determine the shear stress and velocity distribution through the slit. Sketch the shape of the velocity profile based on the velocity distribution you obtained. Determine the maximum velocity of the fluid within the slit and where within the slit it occurs. Find the analog of the Hagen-Poiseuille equation for this system.

Expert Solution

velocity profile for narrow slit at x and y direction are zero

From continuity equation

Momentum balance

momentum in (z=0) =

momentum out ( z=L) =

Momentum in through viscous fluid flow (x=x) =

Momentum out through viscous fluid ( x=x+Δx) =

Force due to Pressure (z=0) =

Force due to Pressure ( z=L) =

Force due to Gravity =

From the basic momentum balance equation

From the Newton’s law of viscosity

Integrate the equation

Apply the boundary conditions

At x=0

At x=B

velocity profile in narrow slit

Now calculate the mass flow rate of the fluid

Mass flow rate = density x Volumetric flow rate

Average velocity = (Volumetric flow rate) / (Area)

the analog of the Hagen-Poiseuille equation for this system.

Amanda K answered 2 years ago

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