A HANDBOOK OF ELEMENTARY RHEOLOGY
CONTENTS 1 THE VISCOSITY OF LIQUIDS - SOME SIMPLE IDEAS 1 1 INTRODUCTION ………………………………………………………………………………… 1 2 RHEOLOGY, PACKAGING AND PRODUCT DELIVERY ……………………………………………… 1 3 WHERE RHEOLOGY IS IMPORTANT ……………………………………………………………… 2 WHAT IS FLOW AND DEFORMATION? 2 1 INTRODUCTION ………………………………………………………………………………… 2 2 SHEAR RATE AND SHEAR STRESS ………………………………………………………………… 2 3 THE RANGE OF SHEAR RATES …………………………………………………………………… 2 4 BASIC DIMENSIONS AND UNITS ………………………………………………………………… 2 5 EVERYDAY RHEOLOGY UNITS …………………………………………………………………… 3 HOW TO INTERPRETE GRAPHICAL DATA 3 1 INTRODUCTION ………………………………………………………………………………… 3 2 LOGARITHMIC PLOTTING ………………………………………………………………………… 3 3 DIFFERENT WAYS OF PLOTTING THE DATA ……………………………………………………… 3 4 PLOTTING START-UP …………………………………………………………………………… 3 5 LONG-DISTANCE RUNNING ……………………………………………………………………… 3 6 OTHER RUNNING REGIMES ……………………………………………………………………… 3 7 SUMMARY ……………………………………………………………………………………… 4 DRAWING RHEOLOGICAL GRAPHS 4 1 INTRODUCTION ………………………………………………………………………………… 4 2 FIGURE LEGENDS ……………………………………………………………………………… 5 THE NEWTONIAN LIQUID 5 1 INTRODUCTION ………………………………………………………………………………… 5 2 VISCOSITY ……………………………………………………………………………………… 5 3 VISCOSITY OF COMMON NEWTONIAN LIQUIDS ………………………………………………… 5 4 THE VARIATION OF NEWTONIAN VISCOSITY WITH TEMPERATURE ……………………………… 5 5 THE EFFECT OF PRESSURE ON VISCOSITY ………………………………………………………… 5 6 THE EVERYDAY PERCEPTION OF VISCOSITY ……………………………………………………… 5 7 THE LIMIT OF NEWTONIAN BEHAVIOUR ………………………………………………………… REFERENCES ………………………………………………………………………………………… 6 SOME EQUATIONS FOR THE FLOW OF NEWTONIAN LIQUIDS 6 1 INTRODUCTION ………………………………………………………………………………… 6 2 FLOW IN ROTATIONAL VISCOMETERS ……………………………………………………………… 6 2 1 NARROW-GAP CONCENTRIC CYLINDER GEOMETRY ………………………………… 6 2 2 SMALL-ANGLE CONE-AND-PLATE GEOMETRY ………………………………………… 6 3 FLOW IN STRAIGHT PIPE CIRCULAR PIPES ………………………………………………………… 6 3 1 VELOCITY PROFILE AND PRESSURE DROP IN FULLY DEVELOPED LAMINAR FLOW IN A TUBE ……………………………………………………………… 6 3 2 FLOW FROM LARGE RESERVOIRS INTO SMALL TUBES ………………………………… 6 3 3 FILLING A TUBE WITH CONSTANT SUCTION …………………………………………… 6 3 4 TUBE EMPTYING UNDER A CONSTANT PRESSURE …………………………………… 6 3 5 FLOW THROUGH A PACKED BED OF SPHERES IN A TUBE …………………………… 6 4 SPHERES FALLING IN NEWTONIAN LIQUIDS ……………………………………………………… 6 4 1 AN ISOLATED FALLING SPHERE ……………………………………………………… 6 4 2 A FALLING/RISING CLOUD OF SPHERES ……………………………………………… 6 4 3 A SPHERE ROLLING DOWN AN INCLINED TUBE ……………………………………… 6 5 OTHER IMPORTANT FLOWS ……………………………………………………………………… 6 5 1 DRAINAGE DOWN A WALL ………………………………………………………… 6 5 2 FLOW ALONG A SLOT ………………………………………………………………… 6 5 3 FLOW THROUGH AN ANNULUS ……………………………………………………… 6 5 4 A ROD BEING PUSHED INTO A CUP FILLED WITH LIQUID …………………………… 6 5 5 FLOW BETWEEN SQUEEZED CIRCULAR PLATES ……………………………………… 6 5 6 ONE TUBE MOVING INSIDE ANOTHER WITH THE SPACE BETWEEN FILLED WITH LIQUID 6 5 7 AN ANGLED PLATE MOVING ACROSS A TRAPPED LIQUID …………………………… 6 5 8 A CYLINDRICAL INDENTER BEING PUSHED INTO A VISCOUS LIQUID ………………… 6 5 9 A DISC ROTATING IN A SEA OF LIQUID ……………………………………………… 6 6 INTERACTION BETWEEN VISCOSITY AND DENSITY IN FLOWS ……………………………………… 6 6 1 FLOW INTO THE ENTRANCE OF A TUBE ……………………………………………… 6 6 2 TURBULENT FLOW IN TUBES ………………………………………………………… 6 6 3 THE SIZE OF THE LIQUID JET COMING OUT OF A TUBE ……………………………… 6 6 4 STOKES FLOW WITH INERTIA ………………………………………………………… 6 6 5 TAYLOR VORTICES IN CONCENTRIC-CYLINDER FLOW ………………………………… 6 7 SOME FLOWS WHERE SURFACE TENSION IS IMPORTANT ………………………………………… 6 7 1 FREE COATING ……………………………………………………………………… 6 7 2 LEVELLING OF SURFACE UNDULATIONS ……………………………………………… 6 7 3 FLOW UP A CAPILLARY TUBE ……………………………………………………… 6 7 4 A LARGE BUBBLE MOVING IN A TUBE ……………………………………………… 7 VISCOMETRY 7 1 INTRODUCTION ………………………………………………………………………………… 7 2 SOME IMPORTANT THINGS TO NOTE ABOUT USING VISCOMETERS ……………………………… 7 2 1 CALIBRATION ……………………………………………………………………… 7 2 2 ARTEFACTS ………………………………………………………………………… 7 2 3 WALL EFFECTS ……………………………………………………………………… 7 2 4 EVAPORATION ……………………………………………………………………… 7 2 5 SEDIMENTATION/SEPARATION ……………………………………………………… 7 2 6 CHEMICAL DAMAGE ………………………………………………………………… 7 2 7 MECHANICAL DAMAGE ……………………………………………………………… 7 2 8 SAMPLING ………………………………………………………………………… 7 2 9 HIGH-SPEED TESTING ……………………………………………………………… 7 2 10 SUSPENDED PARTICLES …………………………………………………………… 7 3 VISCOMETER DESIGN 7 3 1 GENERAL CONSIDERATIONS ………………………………………………………… 7 3 2 COMMERCIAL CONTROLLED-STRESS VISCOMETERS …………………………………… 7 3 3 EVOLUTION OF THE SPECIFICATIONS OF CONTROLLED STRESS VISCOMETERS ………… 7 4 NON-SIMPLE VISCOMETER GEOMETRIES ……………………………………………………… 7 4 1 WIDE-GAP CONCENTRIC CYLINDERS ………………………………………………… 7 4 2 THE ROTATING PARALLEL-PLATE VISCOMETER ……………………………………… 7 3 6 TUBE/PIPE VISCOMETER ………………………………………………………… 8 YOUR RHEOLOGY LABORATORY 8 1 WHAT’S IN A GOOD RHEOLOGY LABORATORY? …………………………………………………… 8 2 WHICH PROBLEM? ……………………………………………………………………………… 8 3 WHICH RHEOMETER? …………………………………………………………………………… 8 3 1 STANDARD LABORATORY VISCOMETERS ……………………………………………… 8 3 2 RHEOMETERS ……………………………………………………………………… 8 3 3 ELONGATIONAL VISCOMETERS ……………………………………………………… 9 SHEAR-THINNING LIQUIDS 9 1 QUALITATIVE FEATURES OF FLOW CURVES …………………………………………………………… 9 2 MATHEMATICAL DESCRIPTIONS OF FLOW CURVES …………………………………………………… 9 2 1 THE CROSS MODEL ………………………………………………………………… 9 2 2 THE BINGHAM EQUATION …………………………………………………………… 9 2 3 THE POWER-LAW EQUATION ………………………………………………………… 9 2 4 THE SISKO MODEL …………………………………………………………………… 9 3 FITTING MODELS TO DATA ………………………………………………………………………… 10 EQUATIONS FOR THE FLOW OF NON-NEWTONIAN LIQUIDS 10 1 INTRODUCTION ………………………………………………………………………………… 10 2 FLOW IN A STRAIGHT CIRCULAR PIPE …………………………………………………………… 10 2 1 POWER-LAW LIQUIDS …………………………………………………………… 10 2 2 SISKO MODEL …………………………………………………………………… 10 2 3 CASSON MODEL ………………………………………………………………… 10 2 4 BINGHAM FLUID ………………………………………………………………… 10 3 DRAINAGE DOWN A WALL ……………………………………………………………………… 10 4 A SPHERE MOVING UNDER GRAVITY IN A POWER-LAW LIQUID ………………………………… 10 5 FLOW IN A SLIT (A VERY WIDE SLOT) …………………………………………………………… 10 6 RADIAL FLOW BETWEEN PARALLEL DISCS ……………………………………………………… 10 7 FLOW OUT OF A TANK THROUGH A LONG EXIT PIPE …………………………………………… 10 8 FLOW OF A POWER-LAW LIQUID BETWEEN SQUEEZING PLATES ……………………………… 10 9 FLOW THROUGH A CONVERGING NOZZLE ……………………………………………………… 10 10 LEVELLING OF SURFACE UNDULATIONS UNDER THE ACTION OF SURFACE TENSION ……………… 10 11 A DISC ROTATING IN A POWER-LAW LIQUID ………………………………………………… 10 12 MORE FORMULAS FOR LIQUIDS WITH AN APPARENT YIELD STRESS ……………………………… 10 12 1 FLOW IN A MIXER ……………………………………………………………… 10 12 2 WIDE-GAP CONCENTRIC CYLINDERS ……………………………………………… 10 12 3 CAPILLARY RISE FOR A YIELD STRESS LIQUID ……………………………………… 11 VERY SHEAR-THINNING OR ‘YIELD - STRESS’ FLUIDS 11 1 INTRODUCTION ………………………………………………………………………………… 11 2 THE HISTORY OF THE ‘YIELD STRESS’ …………………………………………………………… 11 3 VALUES OF YIELD STRESS ……………………………………………………………………… 11 4 THE ARRIVAL OF COMMERCIAL CONTROLLED-STRESS RHEOMETERS ……………………………… 11 5 THE VANE GEOMETRY ………………………………………………………………………… 11 6 SOME FLOW EQUATIONS WITH YIELD STRESSES ………………………………………………… 12 THE FLOW OF ‘SOLIDS’ 12 1 INTRODUCTION ………………………………………………………………………………… 12 2 NON-LINEAR ‘VISCOSITY’ OF SOLIDS …………………………………………………………… 12 3 CONCLUSION …………………………………………………………………………………… 13 LINEAR VISCOELASTICITY AND TIME EFFECTS 13 1 INTRODUCTION ………………………………………………………………………………… 13 2 MECHANICAL ANALOGUES OF VISCOELASTIC BEHAVIOUR ……………………………………… 13 3 MEASURING LINEAR VISCOELASTICITY ………………………………………………………… 13 3 1 GENERAL ………………………………………………………………………… 13 3 2 CREEP TESTS ……………………………………………………………………… 13 3 3 THE BEHAVIOUR OF MODEL MATERIALS IN CREEP TESTS …………………………… 13 3 4 OSCILLATORY TESTS – OR MECHANICAL VIBRATIONAL SPECTROSCOPY ……………… 13 3 5 OSCILLATORY TESTS WITH A MAXWELL MODEL …………………………………… 13 3 6 DOUBLE OR MULTIPLE MAXWELL MODELS IN OSCILLATION ………………………… 13 3 7 OSCILLATION WITH A VOIGT MODEL ……………………………………………… 13 3 8 THE OSCILLATORY RESPONSE OF REAL SYSTEMS …………………………………… 13 3 8 1 VISCOUS/TERMINAL REGION – SIMPLE MAXWELL BEHAVIOUR ……… 13 3 8 2 POLYMER SYSTEMS ………………………………………………… 13 3 8 3 DISPERSIONS ……………………………………………………… 13 4 WHAT CAN WE GET FROM OSCILLATORY DATA? ………………………………………………… 13 5 THE RELATIONSHIP BETWEEN OSCILLATORY AND STEADY-STATE VISCOELASTIC PARAMETERS … 13 5 1 GENERAL CONSIDERATIONS ……………………………………………………… 13 5 2 CREEP VERSUS OSCILLATORY TESTING …………………………………………… 13 5 3 A SIMPLE LINK BETWEEN CREEP AND OSCILLATORY PARAMETERS ………………… 13 6 NON-LINEAR RESPONSE IN OSCILLATORY TESTING ……………………………………………… 13 7 USING OSCILLATORY DATA TO MONITOR CURING, AND REBUILDING AFTER SHEAR ……………… 13 8 STRESS-RELAXATION TESTING …………………………………………………………………… 13 9 START-UP EXPERIMENTS ……………………………………………………………………… 14 NON-LINEAR VISCOELASTICITY 14 1 EVERYDAY ELASTIC LIQUIDS …………………………………………………………………… 14 2 SOME VISIBLE VISCOELASTIC MANIFESTATIONS ………………………………………………… 14 3 THE PROPER DESCRIPTION OF VISCOELASTIC FORCES …………………………………………… 14 4 THE MEASUREMENT OF VISCOELASTIC FORCES ………………………………………………… 14 5 EXAMPLES OF FIRST NORMAL-STRESS DIFFERENCES ……………………………………………… 14 6 SOME VISCOELASTIC FORMULAS ……………………………………………………………… 14 6 1 GENERAL ………………………………………………………………………… 14 6 2 EXTRUDATE FLOW ………………………………………………………………… 14 6 3 FLOW IN A SLIGHTLY INCLINED TROUGH …………………………………………… 14 6 4 PRESSURE IN HOLES AND SLITS …………………………………………………… 14 6 5 INERTIA IN A CONE-AND-PLATE GEOMETRY ………………………………………… 14 7 POSTSCRIPT – FLOW OF A VISCOELASTIC LIQUID THROUGH A CONTRACTION ……………………… 15 THE FLOW OF SUSPENSIONS 15 1 INTRODUCTION ………………………………………………………………………………… 15 2 VISCOSITY OF DISPERSIONS AND EMULSIONS ………………………………………………… 15 2 1 THE CONTINUOUS PHASE ………………………………………………………… 15 2 2 THE EFFECT OF VERY LOW CONCENTRATION OF SUSPENDED PARTICLES ……………… 15 2 2 1 THE EFFECT OF THE CONTINUOUS PHASE ……………………………… 15 2 2 2 THE EFFECT OF THE DISPERSE PHASE ………………………………… 15 2 2 3 THE EFFECT OF MEDIUM-TO-HIGH CONCENTRATIONS OF PARTICLES …… 15 3 PARTICLE-SIZE EFFECTS IN CONCENTRATED DISPERSIONS ……………………………………… 15 4 PARTICLE-SHAPE EFFECTS IN CONCENTRATED DISPERSIONS ……………………………………… 15 5 PARTICLE DEFORMABILITY ……………………………………………………………………… 15 6 PARTICLE INTERACTIONS – AN OVERVIEW ……………………………………………………… 15 7 THE VISCOSITY OF FLOCCULATED SYSTEMS ……………………………………………………… 15 8 THIXOTROPY …………………………………………………………………………………… 15 9 SHEAR THICKENING …………………………………………………………………………… 15 10 APPARENT WALL SLIP ………………………………………………………………………… 15 11 VERY HIGH CONCENTRATION PASTES ………………………………………………………… 15 12 COLLOIDAL CONTROL OF SUSPENSION VISCOSITY ……………………………………………… 15 13 THE STABILITY OF SUSPENSIONS ……………………………………………………………… 16 POLYMER RHEOLOGY 16 1 INTRODUCTION ………………………………………………………………………………… 16 2 SOME OF THE DIFFERENT KINDS OF POLYMER CHAINS ………………………………………… 16 2 1 LONG RODS ……………………………………………………………………… 16 2 2 COILS/STRINGS …………………………………………………………………… 16 3 POLYMER SOLUTIONS ………………………………………………………………………… 16 4 HIGHLY BRANCHED POLYMERS ………………………………………………………………… 16 5 ‘LIVING POLYMERS’ OR ROD- OR WORM-LIKE MICELLES ……………………………………… 16 6 ASSOCIATIVE POLYMERS ……………………………………………………………………… 16 7 A POLYMER SOLUTION AS A STANDARD NON-NEWTONIAN LIQUID …………………………… 17 EXTENSIONAL FLOW AND EXTENSIONAL VISCOSITY 17 1 INTRODUCTION ………………………………………………………………………………… 17 2 WHERE DO WE FIND EXTENSIONAL FLOWS? …………………………………………………… 17 3 WHAT’S SO SPECIAL ABOUT EXTENSIONAL VISCOSITIES? ………………………………………… 17 4 THE TROUTON RATIO ………………………………………………………………………… 17 5 SOME EXAMPLES OF EXTENSIONAL VISCOSITY CURVES ………………………………………… 17 6 THE EFFECT OF TIME OR EXTENSION ON EXTENSIONAL VISCOSITY ……………………………… 17 7 OTHER KINDS OF EXTENSIONAL FLOWS ………………………………………………………… 17 8 VISCOMETERS TO MEASURE EXTENSIONAL VISCOSITY ………………………………………… 17 9 COMMERCIALLY AVAILABLE EXTENSIONAL VISCOMETERS ……………………………………… 17 10 SOME INTERESTING APPLICATIONS …………………………………………………………… 18 RHEOLOGY OF SURFACTANT SYSTEMS 18 1 INTRODUCTION ………………………………………………………………………………… 18 2 SURFACTANT PHASES …………………………………………………………………………… 18 3 RHEOLOGY OF SURFACTANT SYSTEMS …………………………………………………………… 18 4 NECESSARY RHEOLOGICAL PROPERTIES ………………………………………………………… 19 RHEOLOGY OF FOOD PRODUCTS 19 1 INTRODUCTION ………………………………………………………………………………… 19 2 LIQUID-LIKE FOOD RHEOLOGY ………………………………………………………………… 19 3 IN-VIVO FLOW FIELDS ………………………………………………………………………… 19 4 RHEOLOGY AND FOOD PROCESSING …………………………………………………………… 20 YOUR RHEOLOGY LIBRARY 20 1 INTRODUCTION ………………………………………………………………………………… 20 2 RHEOLOGY BOOKS IN PRINT ………………………………………………………………… 20 3 GENERAL RHEOLOGY ………………………………………………………………………… 20 4 THEORETICAL RHEOLOGY ……………………………………………………………………… 20 5 POLYMERIC SYSTEMS ………………………………………………………………………… 20 6 BLOOD ………………………………………………………………………………………… 20 7 FOOD ………………………………………………………………………………………… 20 8 OTHER SPECIFIC SYSTEMS …………………………………………………………………… 20 9 DRAG REDUCTION …………………………………………………………………………… 20 10 GEOLOGICAL SUBJECTS ……………………………………………………………………… 20 11 PROCEEDINGS OF MEETINGS (NOT MENTIONED ELSEWHERE) ………………………………… 21 RHEOLOGY AND COMPUTING 21 1 INTRODUCTION ………………………………………………………………………………… 21 2 A BRIEF DESCRIPTION OF HOW CFD WORKS ………………………………………………… 21 3 THE RHEOLOGICAL DESCRIPTION OF A NON-NEWTONIAN LIQUID AND CFD CODES …………… 21 4 WHEN SHOULD CFD BE USED? ……………………………………………………………… 21 5 SOME TYPICAL RESULTS OF CFD CALCULATIONS ………………………………………………… 21 6 A WORD OF CAUTION