Newtonian Fluids: Vs. Non-Newtonian Fluids
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Chapter 4: Standard FlowsCM4650Polymer RheologyMichigan TechNewtonian fluids:non-Newtonian fluids:vs. How can we investigate non-Newtonian behavior?CONSTANTTORQUEMOTOR t fluid1 Faith A. Morrison, Michigan Tech U.Chapter 4: Standard Flows for RheologyCM4650Polymer RheologyMichigan Techv1 ( H ) V 0 H 0 constantHv1 ( x2 )x2shearx1x3x1 , x2elongation2 Faith A. Morrison, Michigan Tech U.1
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018On to . . . Polymer Rheology . . .We now know how to model Newtonian fluid motion, v 0 tContinuity equation v v v p g t v v Tv ( x , t ), p ( x , t ) : Cauchy momentum equationNewtonian constitutive equation3 Faith A. Morrison, Michigan Tech U.Rheological Behavior of Fluids – Non-NewtonianHow do we model the motion of Non-Newtonian fluid fluids? v 0 tContinuity equation v v v p g t f ( x, t )Cauchy Momentum EquationNon-Newtonian constitutive equation4 Faith A. Morrison, Michigan Tech U.2
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Rheological Behavior of Fluids – Non-NewtonianHow do we model the motion of Non-Newtonian fluid fluids? v 0 tContinuity equation v v v p g t f ( x, t )Cauchy Momentum EquationNon-Newtonian constitutive equationThis is themissing piece5 Faith A. Morrison, Michigan Tech U.Chapter 4: Standard Flows for RheologyChapter 4: Standard flowsChapter 5: Material FunctionsChapter 6: Experimental DataChapter 7: GNFChapter 8: GLVEChapter 9: AdvancedTo get to constitutiveequations, we mustfirst quantify hownon-Newtonian fluidsbehaveNew Constitutive Equations,6 Faith A. Morrison, Michigan Tech U.3
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018What do we observe?Rheological Behavior of Fluids – Newtonian1. Strain response toimposed shear stress x2 v1 ( x2 ) shear rate is constantd constantdtx1trrzz2. Pressure-driven flow ina tube (Poiseuille flow)3. Stress tensor in shearflow only two components arenonzero viscosity is constantQ PR 48 L 0 12 21 0 0 0 R 4 constant8 LQ P0 0 0 1237 Faith A. Morrison, Michigan Tech U.What do we observe?Rheological Behavior of Fluids – Non-Newtonian 1. Strain response to imposed shearstressx2Releasestressv1 ( x2 ) shear rate is variablex1trrzz2. Pressure-driven flow in a tube(Poiseuille flow) all 9 components are nonzero viscosity is variableNormalstressesQQ f P 2Q1Q1 P13. Stress tensor in shear flow2 P1 P 11 12 13 21 22 23 31 32 33 1238 Faith A. Morrison, Michigan Tech U.4
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Non-Newtonian Constitutive Equations We have observations that some materialsare not like Newtonian fluids. How can we be systematic about developingnew, unknown models for these materials?Need measurementsVWFor Newtonian fluids, measurementswere easy: independent of flow (use shear flow) one stress, one material constant, (viscosity)x2v1(x2) Hx1x39 Faith A. Morrison, Michigan Tech U.Non-Newtonian Constitutive EquationsNeed measurementsFor non-Newtonian fluids,measurements are not easy: Depends on the flow (shear flow is not the only choice)Four non-zero stresses even in shear,, ,,Unknown number of material constants inUnknown number of material functions in?VWx2x3v1(x2) Hx110 Faith A. Morrison, Michigan Tech U.5
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Non-Newtonian Constitutive EquationsWe know weneed to makemeasurements toknow more,Need measurementsFor non-Newtonian fluids,measurements are not easy: Depends on the flow (shear flow is not the only choice)Four non-zero stresses even in shear,, ,,Unknown number of material constants inUnknown number of material functions in?VWx2v1(x2) Hx1x311 Faith A. Morrison, Michigan Tech U.Non-Newtonian Constitutive EquationsWe know weneed to makemeasurements toknow more,Need measurementsFor non-Newtonian fluids,measurements are not easy: Depends on the flow (shear flow is not the only choice)Four non-zero stresses even in shear,, ,,Unknown number of material constants inUnknown number of material functions in?But, because we do notVWknow the functionalform of,x2we don’t know what weHneed to measurev1(xto2) knowx1 more!x312 Faith A. Morrison, Michigan Tech U.6
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Non-Newtonian Constitutive EquationsWhat should we do?13 Faith A. Morrison, Michigan Tech U.Non-Newtonian Constitutive EquationsWhat should we do?1. Pick a small number of simple flows Chapter 4: Standard flowsStandardize the flowsMake them easy to calculate withMake them easy to produce in the lab14 Faith A. Morrison, Michigan Tech U.7
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Non-Newtonian Constitutive EquationsWhat should we do?1. Pick a small number of simple flows Chapter 4: Standard flowsStandardize the flowsMake them easy to calculate withMake them easy to produce in the lab2. Make calculations3. Make measurementsChapter 5: Material FunctionsChapter 6: Experimental Data15 Faith A. Morrison, Michigan Tech U.Non-Newtonian Constitutive EquationsWhat should we do?1. Pick a small number of simple flows Chapter 4: Standard flowsStandardize the flowsMake them easy to calculate withMake them easy to produce in the lab2. Make calculations3. Make measurements4. Try to deduceChapter 5: Material FunctionsChapter 6: Experimental DataChapter 7: GNFChapter 8: GLVEChapter 9: Advanced16 Faith A. Morrison, Michigan Tech U.8
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Tactic: Divide theProblem in halfModeling CalculationsExperimentsDream up modelsCalculate materialfunctions frommodel stressesStandard FlowsCompareBuild experimentalapparatuses thatallow measurementsin standard flowsDeterminematerial functionsfrom measuredstressesPass judgmenton modelsCollect models and their reportcards for future use17 Faith A. Morrison, Michigan Tech U.Calculate modelpredictions forstresses instandard flowsStandard flows – choose a velocity field (not an apparatus or aprocedure) For model predictions, calculations are straightforward For experiments, design can be optimized for accuracy andfluid varietyMaterial functions – choose a common vocabulary of stress andkinematics to report results Make it easier to compare model/experiment Record an “inventory” of fluid behavior (expertise)18 Faith A. Morrison, Michigan Tech U.9
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Newtonian fluids:non-Newtonian fluids:vs. How can we investigate non-Newtonian behavior?CONSTANTTORQUEMOTOR t fluid19 Faith A. Morrison, Michigan Tech U.Simple Shear Flowvelocity fieldv1 ( H ) V 0 H 0 constantv1 ( x2 )Hx2x1x1 (t )x1 (t t )Vx2 (t)x2 v 0 0 123x1path lines20 Faith A. Morrison, Michigan Tech U.10
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Near solid surfaces, theflow is shear flow.21 Faith A. Morrison, Michigan Tech U.Experimental Shear Geometries(z-planesection)y xHr(z-planesection)yr( -planesection)x(z-planesection)( planesection) ( -planesection) (z-planesection)( -planesection) r22 Faith A. Morrison, Michigan Tech U.11
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Standard Nomenclature for Shear Flowx2gradientdirectionneutraldirectionx3x1flow direction23 Faith A. Morrison, Michigan Tech U.Why is shear a standard flow? simple velocity field represents all sliding flows simple stress tensorx2x124 Faith A. Morrison, Michigan Tech U.12
CM4650 Lectures 1-3: Intro, MathematicalReviewHow doparticles moveapart in shearflow?2/5/2018t 0VP2 00,,ll22,00 loConsider twoparticles in thesame x1-x2 plane,initially along thex2 axis.PP11 00,,l1l1,00 v1 x 2 x2x1t 0lo otP22 o0l2t ,l, l22,00 l lo o tlov1 x 2 at longtimesP11 o0ll11tt,l, l11,00 x2x125 Faith A. Morrison, Michigan Tech U.How do particlesmove apart inshear flow?Consider twoparticles in thesame x1-x2 plane,initially along the x2axis (x1 0). 0 x2 v 0 0 123Each particle has a differentvelocity depending on its x2position:v1 0 x2P1 : v1 0l1P2 : v1 0l2The initial x1 position of each particle is x1 0. After tseconds, the two particles are at the followingpositions:P1 (t ) : x1 0l1tP2 (t ) : x1 0l2t length location initial time time 26 Faith A. Morrison, Michigan Tech U.13
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018What is the separation of theparticles after time t? 0l2t 0t l2 l1 l0 0l1tl l 0t l2 l1 2220 l02 02t 2l02 l02 1 02t 2l l0 0t l l0 1 tIn shear the distance betweenpoints is directly proportional totime l0 0t2 20lnegligible ast 27 Faith A. Morrison, Michigan Tech U.Uniaxial Elongational Flowx2x3x1x1 , x2velocity field (t ) x1 2 (t ) xv 2 2 (t ) x3 (t) 0 12328 Faith A. Morrison, Michigan Tech U.14
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Uniaxial Elongational Flowx2x3x1path lines (t ) x1 2 (t ) v x2 2 (t ) x3 123x1 , x2 (t) 029 Faith A. Morrison, Michigan Tech U.Elongational flow occurs when there isstretching - die exit, flow through contractionsfluid30 Faith A. Morrison, Michigan Tech U.15
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Experimental Elongational Geometriesx1x3air-bed to support samplex1fluidx3toto tto 2 tx3x1thin, lubricatinglayer on eachplateR(t) h(t)h(to)R(to)31 Faith A. Morrison, Michigan Tech U.Sentmanat Extension Rheometer (2005) Originally developed for rubbers,good for melts Measures elongational viscosity,startup, other material functions Two counter-rotating drums Easy to load; xpansioninstruments.com/rheo-optics.htm32 Faith A. Morrison, Michigan Tech U.16
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Why is elongation a standard flow? simple velocity field represents all stretching flows simple stress tensor33 Faith A. Morrison, Michigan Tech U.How do particlesmove apart inelongational flow?x3Consider twoparticles in thesame x1-x3 plane,initially along thex3 axis.t 0loP1P2 l P1 0,0, o 2 x1 l P2 0,0, o 2 34 Faith A. Morrison, Michigan Tech U.17
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018How do particles move apart in elongational flow?Consider two particles in the same x1-x3 plane, initially along the x3 axis.00varies0022t 0x3P1ln0l lo e o l P1 0,0, o e ot 2 x1P2 l P2 0,0, o e o t 2 Particles move apart exponentially fast.35 Faith A. Morrison, Michigan Tech U.A second type of shear-free flow: Biaxial StretchingbeforebeforePafterair underpressure (t ) x1 2 (t ) x2 (t) 0v 2 (t ) x3 123after36 Faith A. Morrison, Michigan Tech U.18
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018How do uniaxial and biaxial deformations differ?Consider a uniaxialflow in which aparticle is doubled inlength in the flowdirection.aaafluid2aaa2237 Faith A. Morrison, Michigan Tech U.How do uniaxial and biaxial deformations differ?Consider a biaxialflow in which aparticle is doubled inlength in the flowdirection.aaa2a2aa/438 Faith A. Morrison, Michigan Tech U.19
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018A third type of shear-free flow:Planar Elongational Flow (t ) x1 av 0 (t) 0 (t ) x a3 123 aa2aa/239 Faith A. Morrison, Michigan Tech U.All three shear-free flows can be written together as: 1 (t )(1 b) x1 2 1 v (t )(1 b) x2 2 (t ) x3 123Elongational flow: b 0, (t) 0Biaxial stretching: b 0, (t ) 0Planar elongation: b 1, (t) 040 Faith A. Morrison, Michigan Tech U.20
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Why have we chosen these flows?ANSWER:Because these simple flows havesymmetry.And symmetry allows us to drawconclusions about the stress tensorthat is associated with these flowsfor any fluid subjected to that flow.41 Faith A. Morrison, Michigan Tech U.In general: 11 12 13 21 22 23 31 32 33 123But the stress tensor is symmetric – leaving 6 independentstress components.Can we choose a flow to use in which there are fewer than 6independent stress components?Yes we can – symmetric flows42 Faith A. Morrison, Michigan Tech U.21
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018How does the stress tensor simplify forshear (and later, elongational) flow?P 3,1,0 123ê2e1ê1P 3,1,0 1 2 3e243 Faith A. Morrison, Michigan Tech U.What would the velocity function be for aNewtonian fluid in this coordinate system?2Hx2x1 v1 v 0 0 123V2V244 Faith A. Morrison, Michigan Tech U.22
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018What would the velocity function be for aNewtonian fluid in this coordinate system?2H v1 v 0 0 123V2x1x2V245 Faith A. Morrison, Michigan Tech U.Vectors are independent of coordinate system, but in general thecoefficients will be different when the same vector is written in twodifferent coordinate systems: v1 v1 v v2 v 2 v 3 123 v 3 123For shear flow and the two particular coordinate systems we havejust examined, however: V V x2 x2 2H 2H v 0 0 0 0 123 12346 Faith A. Morrison, Michigan Tech U.23
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018 V Vx2 x2 2H 2H v 0 0 0 0 123 123 x2x1x1x2If we plug in the same number in for 2 and ̅ 2, we will NOT beasking about the same point in space, but we WILL get the sameexact velocity vector.Since stress is calculated from the velocity field, we will get the sameexact stress components when we calculate them from eithervector representation.̅̅This is an unusualcircumstance only true forthe particular coordinatesystems chosen.47 Faith A. Morrison, Michigan Tech U.What do we learn if we formally transform v fromone coordinate system to the other?48 Faith A. Morrison, Michigan Tech U.24
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018What do we learn if we formally transformfrom one coordinate system to the other?̂̅̂̅̂̅49 Faith A. Morrison, Michigan Tech U.What do we learn if we formally transform v fromone coordinate system to the other?̂̂̅̅̅(now, substitute from previousslide and simplify)You try.50 Faith A. Morrison, Michigan Tech U.25
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Conclusion:Because of symmetry, there are only 5 nonzero components of theextra stress tensor in shear flow.SHEAR: 11 12 0 21 22 0 00 33 123 This greatly simplifies the experimentalists tasks as only four stress.components must be measured instead of 6 recall51 Faith A. Morrison, Michigan Tech U.Summary:We have found a coordinate system (the shearcoordinate system) in which there are only 5non-zero coefficients of the stress tensor. Inaddition,.This leaves only four stress components to bemeasured for this flow, expressed in thiscoordinate system.52 Faith A. Morrison, Michigan Tech U.26
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018How does the stress tensor simplify forelongational flow?x2x3x1x1 , x2There is 180o of symmetry around all threecoordinate axes.53 Faith A. Morrison, Michigan Tech U.Because of symmetry, there are only 3 nonzero components of theextra stress tensor in elongational flows.ELONGATION:0 11 0 0 22 0 00 33 123 This greatly simplifies the experimentalists tasks as only threestress components must be measured instead of 6.54 Faith A. Morrison, Michigan Tech U.27
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Standard Flows SummaryChoose velocity field:Symmetry alone implies:(no constitutive equation needed yet) (t)x2 v 0 0 123 11 12 0 21 22 0 00 33 123 1 (t )(1 b) x1 2 1 v (t )(1 b) x2 2 (t ) x3 123 0 11 0 0 22 0 00 33 123 By choosing these symmetric flows, we have reduced the number of stresscomponents that we need to measure.55 Faith A. Morrison, Michigan Tech U.Tactic: Divide theProblem in halfModeling CalculationsExperimentsDream up modelsCalculate materialfunctions frommodel stressesStandard FlowsCompareBuild experimentalapparatuses thatallow measurementsin standard flowsDeterminematerial functionsfrom measuredstressesPass judgmenton modelsCollect models and their reportcards for future use56 Faith A. Morrison, Michigan Tech U.Calculate modelpredictions forstresses instandard flows28
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Next, build and assume thisSymmetry alone implies:Choose velocity field:(no constitutive equation needed yet) (t)x2 v 0 0 123 11 12 0 21 22 0 00 33 123 1 (t )(1 b) x1 2 1 v (t )(1 b) x2 2 (t ) x3 123 Measure andpredict this0 11 0 0 22 0 00 33 123 57 Faith A. Morrison, Michigan Tech U.One final comment on measuring stresses. . .What is measured is the total stress, : p 11 21 31 12p 22 32 13 23 p 33 123For the normal stresses we are faced with thedifficulty of separating p from tii.Compressible fluids:Get p fromnRT measurements ofp T and V.VIncompressible fluids:?58 Faith A. Morrison, Michigan Tech U.29
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Density does not vary (much) with pressure for polymeric fluids.4gas densitypolymer density incompressible fluid3 g / cm3MPRT210050100150200250300Pressure (MPa)59 Faith A. Morrison, Michigan Tech U.For incompressible fluids it is not possible to separate p from tii.Luckily, this is not a problem since weonly need p Equation of motion v v v g t P gWe do notneed tii directlyto solve forvelocitiesSolution? Normal stress differences60 Faith A. Morrison, Michigan Tech U.30
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Normal Stress DifferencesFirst normal stressdifferenceN1 11 22 11 22Second normal stressdifferenceN 2 22 33 22 33In shear flow, three stressquantities are measuredIn elongational flow, two stressquantities are measured 21 , N1 , N 2 33 11 , 22 1161 Faith A. Morrison, Michigan Tech U.Normal Stress DifferencesFirst normal stressdifferenceN1 11 22 11 22Second normal stressdifferenceN 2 22 33 22 33In shear flow, three stressquantities are measuredIn elongational flow, two stressquantities are measured 21 , N1 , N 2Are shearnormalstressdifferencesreal? 33 11 , 22 1162 Faith A. Morrison, Michigan Tech U.31
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018First normal stress effects: rod climbing 11 22 0Extra tension in the 1-direction pullsazimuthally and upward (see DPL p65).Newtonian - glycerinViscoelastic - solution ofpolyacrylamide in glycerinBird, et al., Dynamics of Polymeric Fluids, vol. 1,Wiley, 1987, Figure 2.3-1 page 63. (DPL)63 Faith A. Morrison, Michigan Tech U.Second normal stress effects: inclined openchannel flow 22 33 0Extra tension in the 2-direction pulls down the freesurface where dv1 /dx2 is greatest (see DPL p65).Newtonian - glycerinViscoelastic - 1% soln ofpolyethylene oxide in waterN2 -N1 /10R. I. Tanner, Engineering Rheology,Oxford 1985, Figure 3.6 page 10464 Faith A. Morrison, Michigan Tech U.32
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Example: Can the equation of motion predict rodclimbing for typical values of N1, N2?cross-section A:z fluid 0 v v 0 r z RRrAWhat isd zz?drwww.chem.mtu.edu/ fmorriso/cm4650/rod climb.pdf65Bird et al. p64 Faith A. Morrison, Michigan Tech U.What’s next?Shear (t)x2 v 0 0 123Even with just these 2 (or 4)standard flows, we can still generatean infinite number of flows byvarying (t ) and (t ).Shear-free(elongational,extensional) 1 (t )(1 b) x1 2 1 v (t )(1 b) x2 2 (t ) x3 123Elongational flow: b 0,Biaxial stretching: b 0,Planar elongation: b 1, (t) 0 (t ) 0 (t) 066 Faith A. Morrison, Michigan Tech U.33
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018We seek to quantify Procedure:the behavior of non- 1. Choose a flow type (shear or a type of elongation).Newtonian fluids2. Specify (t ) or (t ) as appropriate.3. Impose the flow on a fluid of interest.4. Measure stresses.shearelongation 21 , N1 , N 2 33 11 , 22 115. Report stresses in terms of material functions.6a. Compare measuredmaterial functions withpredictions of these materialfunctions (from proposedconstitutive equations).7a. Choose the mostappropriate constitutiveequation for use in numericalmodeling. Faith A. Morrison,Michigan Tech U.6b. Compare measuredmaterial functions withthose measured on othermaterials.7a. Draw conclusions onthe likely properties of theunknown material based onthe comparison.67Done withStandard Flows.Let’s move on toMaterialFunctions68 Faith A. Morrison, Michigan Tech U.34
CM4650 Lectures 1-3: Intro, MathematicalReview2/5/2018Chapter 5: Material FunctionsSteady Shear Flow Material FunctionsCM4650Polymer RheologyMichigan TechKinematics: (t)x2 v 0 0 123 (t) 0 constantMaterial Functions: 21 0ViscosityFirst normal-stresscoefficient 1 11 22 02Second normalstress coefficient 2 22 33 02 Faith A. Morrison, Michigan Tech U.69 Faith A. Morrison, Michigan Tech U.35
Feb 05, 2018 · How can we investigate non-Newtonian behavior? . 18 Standard flows – choose a velocity field (not an apparatus or a procedure) For model predictions, calculations are straightforward For experiments, design can be optimized for accuracy and fluid variety . section) r H r .
Fluid Dynamics - Math 6750 Generalized Newtonian Fluids 1 Generalized Newtonian Fluid For a Newtonian uid, the constitutive equation for the stress is T pI 2 E, where E 1 2 (ru ruT) is the rate of strain tensor. In deriving this constitutive equation, we have used the fact that the ow is incompressible and the stress isotropic and .
Newtonian fluids are not the main scope of this course: ES206 Fluid Mechanics is a Newtonian Fluid Mechanics. Examples of Non-Newtonian fluids include: Shear thinning Shear thickening Bingham plastic Page 10 of 1
Experimental Investigation of Pressure Drop and Flow Regimes of Newtonian and Non-Newtonian Two-Phase Flow Master's Thesis In Mechanical Engineering Prepared by: Serag Alfarek . mm-ID pipe was observed by a high-speed camera. For the Newtonian model, water and air were instilled. For the non-Newtonian model, xanthan gum solution made out of .
Mixing Newtonian Fluids The ability of the modeling framework given by Eq. 4to capture the mixing of Brownian particles over more than five orders of Péclet number is clearly demonstrated in Fig. 3 A using three different Newtonian fluids (water μ 0.001 Pa·s, 20:80 wt % water:glycerol μ 0.054 Pa·s, and glycerol μ 1.2 Pa·s, Table S1).
non-Newtonian fluids. The property of these fluids is that the stress tensor is related to the rate of deformation tensor by some non-linear relationship. These fluids flow problems present some interesting challenges to researchers in engineering, applied mathematics and computer science.
Petroleum-Based Fluid Natural Gas Compressor Fluids Vacuum Pump Oils Refrigeration Compressor Fluids Cleaning and Flushing Fluids The Effective Alternative to OEM Compressor Fluids ISO 9001 . Sullube Sullair AW F 32 SRF-1/4000 LUBRIPLATE Syn Lube 68 Syn Lube 100 AC-1 Syncool or Syncool FG* SynXtreme AC-32 Syncool or Syncool FG* Synac 68 Synac
Requirements for Fully Developed Laminar Pipe Flow of Inelastic Non-Newtonian Liquids In the current study, we report the results of a detailed and systematic numerical inves-tigation of developing pipe flow of inelastic non-Newtonian fluids obeying the power-law model. We are able to demon
Batch baking is an economical way of having baked goods for the family which will last days. Owning a freezer makes batch baking an even more viable method of cooking as a variety of baked items can be frozen ahead of time and used as required. This is beneficial if you have less time to spend on meal preparation as well as helping to cater for unexpected guests and large numbers. Filling the .
