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Chapter 2: Problem 43

Crude oil, with specific gravity \(\mathrm{SG}=0.85\) and viscosity \(\mu=2.15 \times 10^{-3}\) lbf \(\cdot \mathrm{s} / \mathrm{ft}^{2},\) flows steadily down a surface inclined \(\theta=45\) degrees below the horizontal in a film of thickness \(h=0.1\) in. The velocity profile is given by \\[u=\frac{\rho g}{\mu}\left(h y-\frac{y^{2}}{2}\right) \sin \theta\\] (Coordinate \(x\) is along the surface and \(y\) is normal to the surface.) Plot the velocity profile. Determine the magnitude and direction of the shear stress that acts on the surface.

Short Answer

Expert verified
The procedure involves calculating the different velocities at different film thicknesses to establish the velocity profile. By differentiating the velocity equation in respect to the \( y \) coordinate and substitizing the coordinates into the shear stress equation, the magnitude and direction of the shear stress at the surface can be found.

Step by step solution

01

Define Variables

First, define the given variables like: specific gravity \( \mathrm{SG} = 0.85 \), viscosity \( \mu = 2.15 \times 10^{-3} \, \mathrm{lbf \cdot s / ft^{2}} \), thickness of the oil film \( h = 0.1 \, \mathrm{in} \), and the angle of the surface \( \theta = 45^\circ \). The velocity profile is given by the equation \( u = \frac{\rho g}{\mu}(hy - \frac{y^{2}}{2})\sin\theta \).
02

Calculate and Plot the Profile

Using the given equation, calculate different velocities for different depth (\(y\)) values and plot the profile. For this, substitute the known values into the velocity equation obtaining the velocity for different points in the film's thickness. This plot gives the velocity profile.
03

Determine the Shear Stress

The shear stress can be found using the equation \( \tau_{xy} = -\mu \frac{du}{dy} \). To get \( du/dy \), differentiate the velocity equation with respect to \( y \). Then substitute the given values into the shear stress formula to obtain the magnitude of the shear stress at the surface (\(y=0\)). Additionally, the direction of the shear stress is opposite the direction of fluid motion since it acts on the surface.

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Most popular questions from this chapter

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