CM3110 Transport I Part I: Fluid Mechanics Microscopic Momentum Balance Equation (Naier-Stokes) Professor Faith Morrison Department of Chemical Engineering Michigan Technological Uniersity 1 Microscopic Balances We hae been doing a microscopic control olume balance; these are specific to whateer problem we are soling. We seek equations for microscopic mass, momentum (and energy) balances that are general. equations must not depend on the choice of the control olume, dx d dy equations must capture the appropriate balance 1
Arbitrary Control olume in a Flow b `` S ds nˆ V Mass Balance On an arbitrary control olume:.. (details in the book) ate of increase of mass Net conection in (just as we did with the indiidual control olume balance) Microscopic mass balance for any flow
Continuity Equation ds S nˆ Microscopic mass balance written on an arbitrarily shaped control olume, V, enclosed by a surface, S V t x x y y x x y y Microscopic mass balance is a scalar equation. Gibbs notation: t 5 Momentum Balance On an arbitrary control olume: (details in the book)...... V t dv dv g dv ate of increase of momentum (just as we did with the indiidual control olume balance) V Net conection in Microscopic momentum balance for any flow V Force due to graity V dv Viscous forces and pressure forces 3
Equation of Motion S ds nˆ V microscopic momentum balance written on an arbitrarily shaped control olume, V, enclosed by a surface, S Gibbs notation: Gibbs notation: P g t P t Naier Stokes Equation g general fluid Newtonian fluid Microscopic momentum balance is a ector equation. 7 Continuity Equation (And Non Newtonian Equation) on the FONT t www.chem.mtu.edu/~fmorriso/cm310/naier.pdf The one with is for non Newtonian fluids Faith A. Morrison, Michigan 8 4
Naier Stokes (Newtonian Fluids Only) is on the back: P t g www.chem.mtu.edu/~fmorriso/cm310/naier.pdf 9 Problem Soling Procedure soling for elocity and stress fields 1. sketch system. choose coordinate system amended: when using the microscopic balances 3. simplify the continuity equation (mass balance) 4. simplify the 3 components of the equation of motion (momentum balance) (note that for a Newtonian fluid, the equation of motion is the Naier Stokes equation) 5. sole the differential equations for elocity and pressure (if applicable) 6. apply boundary conditions 7. calculate any engineering alues of interest (flow rate, aerage elocity, force on wall) T 10 5
EXAMPLE I: Flow of a Newtonian fluid down an inclined plane eisited g x g sin x x x g g cos air H fluid g g g g g x y g sin 0 g cos 11 EXAMPLE I: Flow of a Newtonian fluid down an inclined plane eisited (see hand notes) 1 6
As with balance we performed with a control olume we selected, we made modelling assumptions along the way that we can collect and associate with the final result: Model Assumptions: (laminar flow down an incline, Newtonian) 1. no elocity in the x or y directions (laminar flow). well deeloped flow 0 0 3. no edge effects in y direction (width) 4. constant density 5. steady state 6. Newtonian fluid 7. no shear stress at interface 8. no slip at wall 13 A r cross-section A: r L (r) EXAMPLE II: Pressure drien flow of a Newtonian fluid in a tube: (Poiseuille flow) steady state constant well deeloped long tube pressure at top pressure at bottom fluid g 14 7
www.chem.mtu.edu/~fmorriso/cm310/naier.pdf 15 Naier Stokes: www.chem.mtu.edu/~fmorriso/cm310/naier.pdf 16 8
See hand notes 17 List of Common Integrals www.chem.mtu.edu/~fmorriso/cm310/ 014CommonIntegrals.pdf 18 9
What is the force on the walls in this flow? Total wetted area force area cross-section A: r (r) L Inside surface of tube?? fluid 19 9 stresses at a point in space y kg m s force kg m / s y area area s area / Momentum Flux f ê y f A( eˆ eˆ yy yx y x eˆ ) y A surface whose unit normal is in the y-direction stress on a y-surface in the y-direction (See discussion of sign conention of stress; this is the tension positie conention) in the -direction y flux of -momentum 0 10
What is the shear stress in this flow? Stress on an surface in the direction cross-section A: r L (r) fluid 1 Force on the walls: See hand notes 11
4 1 3 Engineering Quantities of Interest (tube flow) aerage elocity olumetric flow rate component of force on the wall Q 0 0 0 0 0 0 rdr d rdr d rdr d Must work these out for each problem in the coordinate system in use; see inside back coer of book. 4 1
Engineering Quantities of Interest (any flow) olumetric flow rate aerage elocity component of force on the wall For more complex flows, we use the Gibbs notation ersions (will discuss soon). 5 4 1 8 Hagen Poiseuille Equation** 6 13
a 0 0,max r rdrd Lg P 4L o Lg P 8L o P L P L 0 0 1 r rdrd 7 a 1.5 p p 0 p p 0 L 0-0.5-1 0 0.5 0.5 0.75 1 L Velocity maximum is twice the aerage (for incline it was.5 the aerage) 1 0.5 0 0 0.5 0.5 0.75 1 r 8 14
EXAMPLE II: Pressure drien flow of a Newtonian fluid in a tube: Poiseuille flow /<> Bullet shaped; flow down an incline was parabola, but a sheet. 9 Can this modeling method work for complex flows? Answer: yes. (with some qualifiers) 30 15