Conservation Vector Review Mass Balance General Statement Simplifications The Control Volume A Useable Form Problems

Vector calculus--review a b  dot product between two vectors Magnitude Force and velocity are vectors and important quantities in fluid mechanics v n Unit outward normal Vector with unit magnitude (N =1) in direction normal to a surface A dot product So dot-product gives Magnitude of vector u in direction of outward normal

Some General Concepts Imagine a wire ring With cross section A m 2 A Place this ring in so that it area is normal to a fluid flow of average speed V m/s V Fluid Volume passing through the ring per second Q = VA m 3 /s The Discharge n If the ring is placed at an angle to the flow Mass Rate crossing the surface is m =  Q v

The Mass Continuity Equation Consider a volume FIXED in a general flow field with velocity v(x,t) and density  (x,t) that change in space and time The mass in the volume V at time t is (essentially density x Volume BUT since density varies in V We need to integrate) The net rate at which mass enters V across its surface A is And the rate of change of this mass with time is Rate of mass accumulation in V = Net rate of mass into V = Mass is conserved in the volume so (again we need to integrate over the surface since rho and velocity vary everywhere Net rate of mass into VRate of mass accumulation in V = OR =

Simplifications = IF Steady Sate = 0 Net flow of mass out of volume is zero If also Incompressible (  = 0) Volume conservation Q1Q1 Q2Q2 Q3Q3 If we have Specific entrances and exits into and out of a steady state system Which can be characterized with average velocities Or Q in = Q out -(sum of negative Q’s) - Q 1 (sum of positive Q’s) Q 2 + Q 3 + ( accumulation ) If not steady accumulation AND

Q in = Q out Assume Steady State--Incompressible

Control volume

Problems 5.28, 5.77, 5.39, 5.61