Chapter 3: Newtonian Fluids CM4650 Michigan Tech Navier-Stokes Equation v vv p t 2 v g 1 Chapter 3: Newtonian Fluid TWO GOALS Derive governing equations (mass and momentum balances Solve governing equations for velocity and stress fields QUICK START W First, before we get deep into derivation, let s do a Navier-Stokes problem to get you started in the mechanics of this type of problem solving. x 3 x 2 x 1 v 1 (x 2 ) H 2 1
EXAMPLE: Drag flow between infinite parallel plates Newtonian steady state incompressible fluid very wide, long uniform pressure W x 2 v 1 (x 2 ) H x 3 x 1 3 EXAMPLE: Poiseuille flow between infinite parallel plates Newtonian steady state Incompressible fluid infinitely wide, long x 2 W x 1 x 3 2H x 1 =0 v 1 (x 2 ) x 1 =L p=p o p=p L 4 2
Engineering Quantities of Interest (any flow) In more complex flows, we can use general expressions that work in all cases. volumetric flow rate average velocity Using the general formulas will help prevent errors. Here, is the outwardly pointing unit normal of ; it points in the direction through 5 The stress tensor was invented to make the calculation of fluid stress easier. Total stress tensor, Π: Π b (any flow, small surface) Force on the Π surface (using the stress convention of Understanding Rheology) S ds nˆ Here, is the outwardly pointing unit normal of ; it points in the direction through 6 3
To get the total force on the macroscopic surface, we integrate over the entire surface of interest. Π Fluid force on the surface S, and evaluated at the surface (using the stress convention of Understanding Rheology) 7 Engineering Quantities of Interest (any flow) Using the general formulas will help prevent errors (like forgetting the pressure). force on the surface, S z component of force on the surface, S (using the stress convention of Understanding Rheology) 8 4
Engineering Quantities of Interest (any flow) Total Fluid Torque on a surface, S Π is the vector from the axis of rotation to (using the stress convention of Understanding Rheology) 9 Common surface shapes: rectangular : ds dxdy circular top : ds r drd surface of cylinder : ds Rddz sphere : ds ( Rd ) r sind R 2 sindd Note: spherical coordinate system in use by fluid mechanics community uses 0 as the angle from the =axis to the point. For more areas, see Exam 1 formula handout: pages.mtu.edu/~fmorriso/cm4650/formu la_sheet_for_exam1_2018.pdf 10 5
Review: Chapter 3: Newtonian Fluid TWO GOALS Derive governing equations (mass and momentum balances Solve governing equations for velocity and stress fields QUICK START First, before we get deep into derivation, let s do a Navier-Stokes problem to get you started in the mechanics of this type of problem solving. x 3 x 2 x 1 W v 1 (x 2 ) H We got a Quick Start with Newtonian problem solving 2 Now Back to exploring the origin of the equations (so we can adapt to non Newtonian) Chapter 3: Newtonian Fluid TWO GOALS Derive governing equations (mass and momentum balances Solve governing equations for velocity and stress fields Mass Balance Consider an arbitrary control volume enclosed by a surface S rate of increase net flux of of mass in C mass into C 12 6
Mathematics Review S ds nˆ b 13 Chapter 3: Newtonian Fluid Mass Balance rate of increase of mass in net flux of mass into through surface S d dt nˆ S d v ds outwardly pointing unit normal Consider an arbitrary volume enclosed by a surface S 14 7
Chapter 3: Newtonian Fluid Mass Balance Leibnitz rule d dt d d t t S S nˆ v ds nˆ v ds vd v d 0 Gauss Divergence Theorem 15 Chapter 3: Newtonian Fluid Mass Balance Since is arbitrary, t v d 0 Continuity equation: microscopic mass balance t v 0 16 8
Chapter 3: Newtonian Fluid Mass Balance Continuity equation (general fluids) v t v t D Dt 0 v 0 For =constant (incompressible fluids): v 0 v 0 17 Chapter 3: Newtonian Fluid Momentum is conserved. Consider an arbitrary control volume enclosed by a surface S rate of increase net flux of of momentum in C momentum into C sum of forces on C resembles the rate term in the mass balance resembles the flux term in the mass balance Forces: body (gravity) molecular forces 18 9
S ds nˆ b 19 rate of increase of momentum in d dt t v d v d Leibnitz rule net flux of momentum into S nˆ vv ds vvd Gauss Divergence Theorem 20 10
Forces on Body Forces (non-contact) force on dueto g g d 21 Chapter 3: Newtonian Fluid Molecular Forces (contact) this is the tough one stress f at P ds on ds the force on that surface choose a surface through P P We need an expression for the state of stress at an arbitrary point P in a flow. 22 11
Molecular Forces Think back to the molecular picture from chemistry: The specifics of these forces, connections, and interactions must be captured by the molecular forces term that we seek. 23 Molecular Forces We will concentrate on expressing the molecular forces mathematically; We leave to later the task of relating the resulting mathematical expression to experimental observations. First, choose a surface: arbitrary shape small ds nˆ f stress at P ds on ds f What is f? 24 12
Consider the forces on three mutually perpendicular surfaces through point P: x 2 P x 1 ê 1 x 3 ê 3 ê 2 b a c 25 Molecular Forces a is stress on a 1 surface at P b c a surface with unit normal ê 1 is stress on a 2 surface at P is stress on a 3 surface at P We can write these vectors in a Cartesian coordinate system: stress on a 1 surface in the 1- direction a a1e ˆ1 a2 2 a3 3 11 1 12 2 13 3 26 13
Molecular Forces a ae 1ˆ1 a2 2 a3e ˆ3 11 1 12 2 b be 1ˆ1 b2e ˆ2 b3e ˆ3 21 1 22 2 c ce 1ˆ1 c2 2 c3e ˆ3 31 1 32 2 13 3 23 3 33 3 a b c is stress on a 1 surface at P is stress on a 2 surface at P is stress on a 3 surface at P So far, this is nomenclature; next we relate these expressions to force on an arbitrary surface. Stress on a p surface in the k-direction pk 27 Molecular Forces How can we write f (the force on an arbitrary surface ds) in terms of the P pk? ds nˆ f f f 1 is force on ds in 1-direction f e f e e 1ˆ1 2ˆ2 3ˆ3 f f 2 is force on ds in 2-direction f 3 is force on ds in 3-direction There are three P pk that relate to forces in the 1-direction: 11 21,, 31 28 14
Molecular Forces How can we write f (the force on an arbitrary surface ds) in terms of the quantities P pk? f f e f e f e 1ˆ1 2ˆ2 3ˆ3 ds nˆ f f 1, the force on ds in 1-direction, can be broken into three parts associated with the three stress components:. nˆ 1 ds 11 21,, 31 first part: projection of 11 da onto the 11nˆ 1 ds surface 1 force area area 29 Molecular Forces f 1, the force on ds in 1-direction, is composed of THREE parts: stress on a 2 -surface in the 1- direction first part: second part: third part: 11 21 projection of da onto the 1 surface projection of da onto the 2 surface projection of 3 surface nˆ nˆ ds ds da onto the n e ds 31 11 21 31 ˆ 1 ˆ 2 3 the sum of these three = f 1 30 15
31 Molecular Forces f 1, the force in the 1-direction on an arbitrary surface ds is composed of THREE parts. f1 11nˆ 1 ds 21nˆ 2 ds 31nˆ 3 ds stress appropriate area Using the distributive law: f1 nˆ 11 1 21 2 31 3dS Force in the 1-direction on an arbitrary surface ds 32 16
Molecular Forces The same logic applies in the 2-direction and the 3-direction f1 nˆ f2 nˆ f nˆ 3 11 1 1 12 13 1 21 2 31 3 ds 2 32 3 ds ds 22 23 2 33 3 Assembling the force vector: f f1e ˆ1 f2 2 f3 3 ds nˆ 11 1 ds nˆ ds nˆ 21 2 31 3 1 12 1 22 2 32 3 2 13 1 23 2 33 3 3 33 Molecular Forces Assembling the force vector: f f 1 1 f 2 2 f 3 3 ds nˆ 11 1 ds nˆ ds nˆ 12 1 22 21 2 13 1 31 3 2 1 23 2 32 3 2 33 3 3 ds nˆ 11ee ˆ1 ˆ1 21 2 1 31 3 1 12ee ˆ1 ˆ2 22 2 2 32 3 2 ee ˆ ˆ 13 1 3 23 2 3 33 3 3 linear combination of dyadic products = tensor 34 17
Molecular Forces Assembling the force vector: f f ds nˆ ds nˆ 11ee ˆ1 ˆ1 21 2 1 31 3 1 12ee ˆ1 ˆ2 22 2 2 32 3 2 ee ˆ ˆ 3 ds nˆ ds nˆ p1m1 pm 3 13 1 3 p m pm p Total stress tensor (molecular stresses) m 23 2 3 33 3 3 35 rate of increase net flux of sum of of momentum in momentum into forces on t vd vv molecular forces S S d nˆ molecular forces on ds ds molecular g d forces d We use a stress sign convention that requires a negative sign here. Gauss Divergence Theorem 36 18
rate of increase net flux of sum of of momentum in momentum into forces on t vd vv molecular forces S S d nˆ molecular forces on ds ds molecular g d forces d UR/Bird choice: positive compression (pressure is positive) Gauss Divergence Theorem 37 F on surface ~ ds nˆ nˆ S S ds yx ~ yx UR/Bird choice: fluid at lesser y exerts force on fluid at greater y (IFM/ choice: (opposite) 38 19
Final Assembly: rate of increase net flux of of momentum in momentum t vd vvd g d v t sum of into forces on vv g d 0 d Because is arbitrary, we may conclude: v t vv g 0 Microscopic momentum balance 39 Microscopic momentum v vv t g balance 0 After some rearrangement: v vv g t Dv g Dt Equation of Motion Now, what to do with? 40 20
Now, what to do with? Pressure is part of it. Pressure definition: An isotropic force/area of molecular origin. Pressure is the same on any surface drawn through a point and acts normally to the chosen surface. pressure p I p p ee ˆ1 ˆ1 p 2 2 p 3 3 0 0 0 p 0 0 0 p 123 Test: what is the force on a surface with unit normal? 41 back to our question, Now, what to do with? Pressure is part of it. There are other, nonisotropic stresses Extra Molecular Stresses definition: The extra stresses are the molecular stresses that are not isotropic p I (other sign convention: Extra stress tensor, i.e. everything complicated in molecular deformation Now, what to do with? This becomes the central question of rheological study 42 21
Stress sign convention affects any expressions with or ~,, ~ p I UR/Bird choice: fluid at lesser y exerts force on fluid at greater y ~ ~ p I (IFM/ choice: (opposite) 43 Constitutive equations for Stress are tensor equations relate the velocity field to the stresses generated by molecular forces are based on observations (empirical) or are based on molecular models (theoretical) are typically found by trial-and-error are justified by how well they work for a system of interest are observed to be symmetric f ( v, material properties) Observation: the stress tensor is symmetric 44 22
Microscopic momentum balance v vv g t Equation of Motion In terms of the extra stress tensor: v vv t p g Equation of Motion Cauchy Momentum Equation Components in three coordinate systems (our sign convention): http://www.chem.mtu.edu/~fmorriso/cm310/navier2007.pdf 45 Newtonian Constitutive equation v v T for incompressible fluids (see text for compressible fluids) is empirical may be justified for some systems with molecular modeling calculations Note: (IFM choice: (opposite) 46 23
How is the Newtonian Constitutive equation related to Newton s Law of iscosity? v 1 T 21 v v x2 incompressible fluids incompressible fluids rectilinear flow (straight lines) no variation in x 3 -direction 47 Back to the momentum balance... v vv t p g Equation of Motion v v T We can incorporate the Newtonian constitutive equation into the momentum balance to obtain a momentum-balance equation that is specific to incompressible, Newtonian fluids 48 24
Navier-Stokes Equation v vv t 2 p v g incompressible fluids Newtonian fluids Note: The Navier-Stokes is unaffected by the stress sign convention because neither nor appear. 49 Navier-Stokes Equation v vv t 2 p Newtonian Problem Solving v g 50 25
EXAMPLE: Drag flow between infinite parallel plates Newtonian steady state incompressible fluid very wide, long uniform pressure W from QUICK START x 2 v 1 (x 2 ) H x 3 x 1 51 EXAMPLE: Poiseuille flow between infinite parallel plates from QUICK START Newtonian steady state Incompressible fluid infinitely wide, long x 2 W x 1 x 3 2H x 1 =0 v 1 (x 2 ) x 1 =L p=p o p=p L 52 26
EXAMPLE: Poiseuille flow in a tube Newtonian Steady state incompressible fluid long tube A z r cross-section A: z r v z (r) L fluid R 53 EXAMPLE: Torsional flow between parallel plates Newtonian Steady state incompressible fluid cross-sectional view: z H R r 54 27
Done with Newtonian Fluids. Let s move on to Standard Flows 55 Chapter 4: Standard Flows Newtonian fluids: vs. non-newtonian fluids: CM4650 Michigan Tech How can we investigate non-newtonian behavior? CONSTANT TORQUE MOTOR t fluid 56 28