Microscopic Momentum Balance Equation (Navier-Stokes)

Transcription

1 CM3110 Transport I Part I: Fluid Mechanics Microscopic Momentum Balance Equation (Navier-Stokes) Professor Faith Morrison Department of Chemical Engineering Michigan Technological University 1 Microscopic Balances We have been doing a microscopic control volume balance; these are specific to whatever problem we are solving. We seek equations for microscopic mass, momentum (and energy) balances that are general. equations must not depend on the choice of the control volume, dx d dy equations must capture the appropriate balance 2 1

2 Arbitrary Control volume in a Flow b `` S ds nˆ V Mass Balance On an arbitrary control volume:.. (details in the book) ate of increase of mass Net convection in (just as we did with the individual control volume balance) Microscopic mass balance for any flow 2

3 Continuity Equation ds S nˆ Microscopic mass balance written on an arbitrarily shaped control volume, V, enclosed by a surface, S V v t x v x y v y vx x vy y v Microscopic mass balance is a scalar equation. Gibbs notation: 5 Momentum Balance On an arbitrary control volume: (details in the book) ate of increase of momentum (just as we did with the individual control volume balance) Π Net convection in Force due to gravity Microscopic momentum balance for any flow Viscous forces and pressure forces 3

4 Equation of Motion S ds nˆ V Microscopic momentum balance written on an arbitrarily shaped control volume, V, enclosed by a surface, S Gibbs notation: general fluid Gibbs notation: Newtonian fluid Navier Stokes Equation Microscopic momentum balance is a vector equation. 7 Continuity Equation (And Non Newtonian Equation) on the FONT The one with is for non Newtonian fluids Faith A. Morrison, Michigan 8 4

5 Navier Stokes (Newtonian Fluids Only) is on the back: There are no `s on this side 9 Problem Solving Procedure solving for velocity and stress fields 1. sketch system 2. 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 Navier Stokes equation) 5. solve the differential equations for velocity and pressure (if applicable) ) 6. apply boundary conditions 7. calculate any engineering values of interest (flow rate, average velocity, force on wall) 10 5

6 EXAMPLE I: Flow of a Newtonian fluid down an inclined plane evisited g x g sin v 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 evisited (see hand notes) 12 6

7 As with balance we performed with a control volume 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 velocity in the x or y directions (laminar flow) 2. well developed flow 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 A r cross-section A: r L v (r) EXAMPLE II: Pressure driven flow of a Newtonian fluid in a tube: (Poiseuille flow) steady state constant well developed long tube pressure at top pressure at bottom fluid g 14 7

8 15 Navier Stokes:

9 See hand notes 17 List of Common Integrals CommonIntegrals.pdf 18 9

10 What is the force on the walls in this flow? Total wetted area force area cross-section A: r v (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 2 / Momentum Flux f ê y A surface whose unit normal is in the y-direction stress on a y-surface in the y-direction (See discussion of sign convention of stress; this is the tension-positive convention, ) in the -direction y flux of -momentum 20 10

11 What is the shear stress in this flow? Stress on an rsurface in the direction cross-section A: r L v (r) fluid 21 Force on the walls: See hand notes 22 11

12 Engineering Quantities of Interest (tube flow) average velocity volumetric flow rate component of force on the wall Must work these out for each problem in the coordinate system in use; see inside back cover of book

13 Engineering Quantities of Interest (any flow) volumetric flow rate average velocity component of force on the wall For more complex flows, we use the Gibbs notation versions (will discuss soon). 25 Volumetric Flow ate: volumetric flow rate Let s try See hand notes 26 13

14 4 1 8 Hagen Poiseuille Equation** 27 v av v v,max r rdrd Lg P 4L o Lg P 8L o P L P L r 2 rdrd 28 14

15 v v av p p 0 p p 0 L L Velocity maximum is twice the average (for incline it was 2.5 the average) r 29 EXAMPLE II: Pressure driven flow of a Newtonian fluid in a tube: Poiseuille flow v /<v> Bullet shaped; flow down an incline was a parabola cross section, but a sheet in 3d

16 Example III: Pressure-driven flow of a Newtonian fluid in a rectangular duct: Poiseuille flow A x y What is the steady state velocity profile for laminar flow of an incompressible Newtonian fluid flowing down a long duct of rectangular cross section? The duct height is 2 and the width is 2. The pressure at an upstream position is and a distance downstream the pressure is. What is the force on the walls? cross-section A: See hand notes x v (x,y) H (Example 7.11, p549) 31 Can this modeling method work for complex flows? Answer: yes. (with some qualifiers) 32 16