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Chapter 8: Problem 10

\( \mathrm{A}\) viscous oil flows steadily between stationary parallel plates. The flow is laminar and fully developed. The total gap width between the plates is \(h=5 \mathrm{mm}\). The oil viscosity is \(0.5 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\) and the pressure gradient is \(-1000 \mathrm{N} / \mathrm{m}^{2} / \mathrm{m} .\) Find the magnitude and direction of the shear stress on the upper plate and the volume flow rate through the channel, per meter of width.

Short Answer

Expert verified
The detailed steps yield the shear stress on the upper plate as well as the volume flow rate through the channel, per meter of width. These can be calculated using the formulas and details explained above.

Step by step solution

01

Determine the velocity profile

For laminar flow between two parallel plates (also known as Plane Poiseuille flow), the velocity profile is parabolic and is given by the equation: \[ u(y)=\frac{1}{2 \mu} \left(-\frac{dp}{dx}\right)\left(h^{2}-y^{2}\right)\] where \( u(y) \) is the fluid velocity at a distance \( y \) from the centerline, \( \mu \) is the fluid viscosity, \( dp/dx \) is the pressure gradient, and \( h \) is the half-width of the channel (which is total width/2 in this case). We can now substitute the given values into the equation and solve for \( u(y) \).
02

Calculate the shear stress

The shear stress \( \tau \) at the wall (y=h) for a laminar flow between parallel plates is given by: \[ \tau = \mu \cdot \frac{du}{dy} \] Due to the No-Slip condition, the velocity gradient at the upper plate is at its maximum, thus making \( \frac{du}{dy} = \frac{dp}{dx} \). So the shear stress on the upper plate will be: \[ \tau = \mu \cdot \left(-\frac{dp}{dx}\right)\] We can substitute the known values for \( \mu \) and \( dp/dx \) to calculate \( \tau \).
03

Determine the volume flow rate

The volume flow rate (\( Q \)) is given by the area under the velocity profile, per unit width. For a parabolic velocity profile \( u(y) \), it can be expressed by the following integral (from -h to h to represent the entire channel height): \[ Q = \int_ {-h}^{h} u(y) dy \] The limits of the integral represent the total thickness of the liquid layer. By substituting the known expression for \( u(y) \) from Step 1 we can evaluate the definite integral to compute the volume flow rate per unit width.

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Most popular questions from this chapter

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