Question
Download Solution PDFIn incompressible fluid flows over a flat plate with zero pressure gradient. The boundary layer thickness is 1 mm at a location where the Reynolds number is 1000. If the velocity of the fluid alone is increased by a factor of 4, then the boundary layer thickness at the same location, in mm, will be
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFExplanation:
Boundary Layer Thickness in Incompressible Flow
Definition: In fluid dynamics, the boundary layer is the thin region adjacent to a solid surface where the effects of viscosity are significant. For incompressible flow over a flat plate with zero pressure gradient, the boundary layer thickness depends on the Reynolds number and the velocity of the fluid. The Reynolds number (Re) is a dimensionless quantity that represents the ratio of inertial forces to viscous forces in the flow.
For laminar flow over a flat plate with zero pressure gradient, boundary layer thickness is inversely proportional to the square root of the Reynolds number, i.e.,
\( \delta \propto \frac{1}{\sqrt{Re}} \)
Calculation:
Given:
Initial Reynolds number, \( Re_1 = 1000 \)
Initial boundary layer thickness, \( \delta_1 = 1 \text{ mm} \)
Velocity increased by a factor of 4 ⇒ \( U_2 = 4U_1 \)
Reynolds number is directly proportional to velocity, so \( Re_2 = 4 \times Re_1 = 4000 \)
Now,
\( \frac{\delta_2}{\delta_1} = \sqrt{\frac{Re_1}{Re_2}} = \sqrt{\frac{1000}{4000}} = \frac{1}{2} \)
\( \delta_2 = 1 \times \frac{1}{2} = 0.5~\text{mm} \)
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