The Shell Momentum Balance:Step-by-Step Method

CBE 320, DJK, Fall 2013

Step 1:Draw a (crude) picture and list assumptions

• What velocity components are nonzero?

• What spatial variables does the velocity depend on?

• Choose coordinate system.

• Start a list of assumptions.

Step 2:Select system

• Draw a second picture, showing the “shell” in detail.

• “Shell”:

• a ‘box’ with sides‖ or⊥ to the velocityv.

• box faces should lie on coordinate surfaces (you choose yourcoordinate system to guarantee this).

• Make shell thin in the direction the velocity is varying.

• You will develop intuition about how to do this as you work prob-lems.

Step 3: Apply conservation of momentum (compo-nent of interest)

Rate at whichmomentum istransported

intothe system

{ {Rate at whichmomentum istransported

out ofthe system

{ { Force of gravityacting on

the system{ {+ = 0-

f d t rij ij ij i j= + +p v v add other bodyforces here (e.g.,electrical, magnetic,etc.)

if unsteady, thenthe RHS is the timerate of change ofmomentum

• Add arrows to second picture indicating where flux componentstransport momentum in and out.

• Write out the conservation of momentum equation (momentumbalance) for the system.

Step 4:Simplify/eliminate terms if possible

• Insert appropriate definitions for theφij components (i.e.,φij =pδij + τij + ρvivj).

• Use assumptions to eliminate terms that are zero, or terms thatcancel.

• Simplify here as much as possible; you will have more opportu-nities to simplify later.

Step 5:Let Shell thickness→ 0

• Let the small dimension of the ‘box’→ 0.

• This will give a differential equation for the momentum flux (therelevant component ofτij or φij , depending how much simplify-ing you did in the previous step).

• If you haven’t already, make sure you have substituted for allcomponents ofφij , and simplified as much as possible.

Step 6:Integrate once, if possible

• This will give a constant of integration.

• If you have a boundary condition for the momentum flux compo-nentτij , apply the boundary and solve for the constant.

• You will not evaluate the constant here if both of your boundaryconditions are for the velocity.

Step 7:Insert Newton’s Law of Viscosity

• Use Appendix B.1 (pp. 843–844) to replace the components ofτij with the appropriate expressions in terms of velocity compo-nents.

• Simplify as much as possible:

• Appendix B.1 is general; some velocity components andderivatives will be zero.

• Make use of your assumptions, and add to the list of assump-tions if new ones occur to you.

• This will lead to a differential equation for the desired velocitycomponent (First order if you completed step 6, second orderifyou skipped step 6).

Step 8:Solve the differential equation

• This will produce another unknown constant.

• Use a second boundary condition to get this constant.

• The result of this step will be the velocity profile (for example,vz(r)).

Step 9:Use the velocity profile to engineer

• Using the velocity (or stress) profile, you can calculate a varietyof quantities of interest, e.g.,

• force on a surface,

• torque on an object,

• volumetric flow rate,

• maximum flow rate,

• etc.

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