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ENSC3003 – Fluid Mechanics

Assignment 3

Equations of MotionDimensional Analysis

Questions 4, 5, and 8 are to be submitted for Marking

Assignment Due : 4:00 pm on Monday, April 15th, 2013;The assignment must be submitted to LMS in pdf form

1. Using the Navier-Stokes Equations (in Cartesian coordinates), derive thedifferential equation form of the basic equation of hydrostatics. You may attacha marked equation sheet showing the manner in which you simplified theNavier-Stokes Equations.

2. Using the Navier-Stokes equations in Cylindrical coordinates, derive a generalversion of the Hagen-Poiseuille equation giving the Volume flow rate for laminar flow in a straight circular pipe that holds for a straight pipe aligned inthe horizontal plane (ie at right angle to gravity). You may attach a markedequation sheet showing the manner in which you simplified the Navier-Stokes

Equations.

3. During the derivation of the Navier-Stokes Equation, it is initially shown that

Showing all working, demonstrate how the left hand side of the equation canbe simplified to

Note – This will hold regardless of whether the fluid is incompressible or compressible. So you may not assume the fluid is incompressible.

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4. A high-viscosity oil is transported through a wide rectangular duct via

pressure-driven flow, as illustrated below. The duct is aligned an angle θ above the horizontal, and is sufficiently broad that edge effects may beneglected in the x2 direction. The pressure at the upstream end of the duct(x1=0) is Po, and at the downstream end (x1=L) is PL. The flow may be

regarded as laminar and isothermal, and the oil is a Newtonian liquid; the ductmay be regarded as stationary.

(a) Starting with the appropriate form of the Continuity Equation andEquations of Motion derive an equation for determining the x1 component of velocity at steady state. Be sure to identify allassumptions and boundary conditions. You may attach a markedequation sheet showing the manner in which you simplified the Navier-Stokes Equations.

(b) Derive an equation for the volume flow rate of oil through the duct at

steady state under these conditions.

(c) Derive an expression for the shear stress T13 at the upper surface of theduct (x3 = B) at steady state.

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5. A centrifugal mixing device, as illustrated in the diagram below, is beingdeveloped to stir small quantities of blood for experimental procedures. Theproposed device is confined within a cylinder 2 cm in diameter and 10 cmtall, and uses a stirrer 1 cm in diameter. The proposed operating speed for the stirrer is 20 rpm.

Blood is very viscous, and significant surface vortex can develop. If thisvortex becomes large enough to reach the stirrer, the delicate cells within theblood will be severely damaged by the extreme shear gradients arising. It istherefore proposed to build a large scale model using an alternative fluid inorder to investigate the phenomenon. Due to material restrictions, it has beenestablished that the model cylinder diameter (DCM) must be 10 cm.

If the specific gravity of blood is 1.1, and the viscosity of blood at roomtemperature is 2.0 cP, determine

(i) The other dimensions (height HM and stirrer diameter DSM) of the model(ii) The kinematic viscosity required of the model fluid (in m2/s)(iii) The speed (in rpm) of the stirrer in the model system

DS

DC

H

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 6. “Vortexing" is a common problem in the design of tank outlets. As liquid is

drawn into an outlet, a vortex will often form. If the tank level is low enough, or the flow rate high enough, the vortex may reach into the outlet pipe. This mayresult in air entrainment and/or NPSH problems, both of which may lead to

inefficient operation of a downstream process.

 A group of engineers are interested in investigating this problem for water pumping systems. In particular, they would like to establish operatingparameters (safe levels, safe flow rates, etc) for a tank of the design shown

below. The real system uses water (µ=0.001 Pas, ρ=1000 kg/m3), and gravitymay be taken to be 9.81 m/s

2for both the real and model systems

To accomplish this, they are considering building a model system in which thetank diameter is 1/10th the diameter of the real system.

(a) Determine the dimensions of the model tank (tank diameter Dt, outletdiameter D0, tank height H and bellmouth radius R) if the model tank isto provide dynamically similar behaviour.

(b) It is sought to model the behaviour of the real system at a flow of 2200litres per minute. For a model system yielding behaviour that isdynamically similar to that of the real tank, determine(i) The kinematic viscosity required in the model fluid (in m2/s)(ii) The volume flow rate in the model system (in litres/minute)

Dt = 20 m

H = 6.5 m

D0

= 600 mm

R = 300 mm

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7. The drag force F exerted on an object in a flowing fluid is a function of;

Far Field Flow Velocity UCharacteristic Length of Body L

Fluid Density ρ 

Fluid Viscosity µ 

Using the Buckingham Pi theorem, determine the important dimensionlessgroups that characterize this problem.

8 To model the performance of a centrifugal pump, it may be assumed that the

pump power consumption! is a function of:

Volume flow rate through the pump QThe effective diameter of the Impeller D

The angular velocity of the impeller  ω 

Fluid Density ρ Fluid Viscosity µ 

Using the Buckingham Pi theorem, determine the dimensionless groups thatcan be used to develop equations to model the pump power consumption.