For the steady flow of an unspecified fluid through a cylindrical conduit (pipe, cylindrical airduct, blood vessel, etc.) the velocity profile has been determined from a differential equation relationship based upon the fluid contact force per unit area as:
Fluid Velocity Distance from Pipe Wall
y’ y’ < 5
u’ = – 3.051 + 5 ln y’ 5 < y’ < 30
5.49 + 2.52 ln y’ y’ > 30
where u’ is the dimensionless velocity in the x-direction and y’ is the dimensionless y-direction coordinate and:
u’ = u √o /
y’ = y √o / / v
where u is the fluid velocity (m/s) in the x-direction, o is the fluid shear stress (Pa), is the fluid density (kg/m3), y is the distance (m) from the pipe wall, and v is the kinematic viscosity (m2/sec). If the x-direction velocity depends only upon the vertical distance (y) from the pipe wall, please determine the following.
a.) Using only the flow system description provided (without the velocity distribution information) with the definitions of the fluid velocity vector ( V ) and the acceleration field vector ( a ), determine general expressions for the differential continuity equation acceleration and vector fields for this flow system (50 POINTS).
b.) If the fluid is air at the conditions of 70.1 oF and 2.25 atm and flows through a 2.35 foot inner diameter pipe, determine the Reynolds number and the flow regime. Also determine the general continuity expression for this flow system and its boundary conditions. Assume that absolute fluid viscosity is a weak function of pressure (35 POINTS).
c.) Determine the shear stress distribution of the fluid in this system using the information from Parts (a) and (b). Please state all assumptions. (15 POINTS)
d.) Formulate the simplified Navier-Stokes Equations for the system described. Please state all assumptions (15 Points – BONUS)
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