# For reynolds numbers re see table 16 and example 14 2

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4. For Reynolds numbers Re (see Table 1.6 and Example 1.4-2) below 2100, steady, pressure-driven flow through a cylindrical tube is “laminar”. In steady, laminar tube flow, the volumetric flow rate Q depends only on the tube radius R, the fluid viscosity μ, and the pressure drop per unit tube length dp/dx. Derive an appropriate dimensionless relationship between these variables.
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Π=μR4dp / dx()Q. Since there are no other dimensionless groups, a dimensionless relationship takes the form Π=μR4dp / dx()Q=Cwhere C is a constant. The flow rate is then given by Q=CR4μdpdx. We will show later this quarter that C=p/8, and the resulting expression is then known as the Hagen-Poiseuille equation. Note that the fluid density plays no role—why do you think that is? 5. a) Evaluate the dot product of a and b if a=xex+1xy()ey+etezand b=2ex+3ey+ez. b) Find the magnitude of the vector a if a=cosθex+sinθey+ez. Solution
6. Find the component of the vector field a in the direction of the unit vector n, if a=xex+1xy()ey+etezand n=22ex+22ey.
Solution