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Chapter 9: Problem 33

For a Tuid of specific gravity \(S G=0.86\) flowing past a flat plate with an upstream velocity of \(U=5 \mathrm{m} / \mathrm{s}\), the wall shear stress on the flat plate was determined to be as indicated in the table below, Use the momentum integral equation to determine the boundary layer momentum thickness, \(\Theta=\Theta(x)\). Assume \(\Theta=0\) at the leading edge, \(x=0\)

Short Answer

Expert verified
To find the boundary layer momentum thickness, the density of the fluid needs to first be calculated using the specific gravity. Then, using the constant wall shear stress assumption, you can integrate over the control volume to calculate the momentum thickness. However, if the shear stress changes, additional data or knowledge would be required to solve the integral. Without specific given wall shear stress values, this solution is incomplete.

Step by step solution

01

Expressing the Momentum Integral Equation

The Momentum Integral Equation can be expressed generally as: \[\int_{0}^{\delta} u\left(1 - \frac{u}{U}\right) du = \frac{\tau_{w}}{\rho U^{2}} x]\ Where \(u\) is the velocity of the fluid at an arbitrary location in the boundary layer, \(\delta\) is the boundary-layer thickness, \(U\) is the free-stream velocity, \(\tau_{w}\) is the wall shear stress, \(\rho\) is the fluid density, and \(x\) is the distance from the leading edge.
02

Determine the Density of the Fluid

We know the specific gravity (SG) of the fluid, which is defined as the ratio of the density of the fluid to the density of water at 4 degrees Celsius. As the density of water is 1000 kg/m³, the fluid density \(\rho\) can be calculated as: \[ρ= SG\times ρ_{water} = 0.86\times 1000 = 860 kg/m^3\]
03

Evaluate Momentum Thickness

Now that we have the density of fluid, we can solve the momentum integral equation. Since the equation given in the problem statement does not specify any wall shear stress \(\tau_{w}\), we need to assume for the purpose of this solution that the wall shear stress is constant. The integration can then be performed over the control volume to find the momentum thickness \(\Theta\). The equation now should be: \[\Theta = \int_0^Δ (1 - \frac{u}{U})dx\], The solution requires advanced integral knowledge and likely additional data, especially if the shear stress \(τ_w\) changes.

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