Rheology

Strain and Strain (Shear) Rate a dimensionless quantity representing the relative deformation of a material Normal Strain Shear Strain

Shear Stress is the intensity of force per unit area, acting tang

Simple Shear Flow

Fluid Viscosity Newtonian fluids Non-Newtonian fluids viscosity is constant (Newtonian viscosity, ) Non-Newtonian fluids shear-dependent viscosity (apparent viscosity, )

Viscosity: Introduction The viscosity is measure of the “fluidity” of the fluid which is not captured simply by density or specific weight. A fluid can not resist a shear and under shear begins to flow. The shearing stress and shearing strain can be related with a relationship of the following form for common fluids such as water, air, oil, and gasoline:

Viscosity: Introduction is the absolute viscosity or dynamics viscosity of the fluid, u is the velocity of the fluid and y is the vertical coordinate as shown in the schematic below:

Viscosity Viscosity is a property of fluids that indicates resistance to flow. When a force is applied to a volume of material then a displacement (deformation) occurs. If two plates (area, A), separated by fluid distance apart, are moved (at velocity V by a force, F) relative to each other, y = y-max y = 0 = Coefficient Viscosity ( Pa s) Newton's law states that the shear stress (the force divided by area parallel to the force, F/A) is proportional to the shear strain rate . The proportionality constant is known as the (dynamic) viscosity

In cgs unit , the unit of viscosity is expressed as poise The unit of viscosity in the SI system of units is pascal-second (Pa s) Shear stress In cgs unit , the unit of viscosity is expressed as poise 1 poise = 0.1 Pa s Shear rate 1 cP = 1 m Pa s

Newtonian fluid  =  =

Newtonian From and  =  = P =   = 8v/D

Non-Newtonian Fluids

Flow Characteristic of Non-Newtonian Fluid Fluids in which shear stress is not directly proportional to deformation rate are non-Newtonian flow: toothpaste and Lucite paint

(Casson Plastic) (Bingham Plastic)

Viscosity changes with shear rate Viscosity changes with shear rate. Apparent viscosity (a or ) is always defined by the relationship between shear stress and shear rate.

Model Fitting - Shear Stress vs. Shear Rate Summary of Viscosity Models  h = g · Newtonian Pseudoplastic Dilatant Bingham Casson Herschel-Bulkley  g · n = K ( n < 1 )  g · n = K ( n > 1 )   h g · n = + y  1 2 1 2 h 1 2 g · 1 2 =  + c   g · = + K n y or  = shear stress, º = shear rate, a or  = apparent viscosity m or K or K'= consistency index, n or n'= flow behavior index

Herschel-Bulkley model (Herschel and Bulkley , 1926) Values of coefficients in Herschel-Bulkley fluid model Fluid m n 0 Typical examples Herschel-Bulkley >0 0<n< Minced fish paste, raisin paste Newtonian 1 Water,fruit juice, honey, milk, vegetable oil Shear-thinning (pseudoplastic) 0<n<1 Applesauce, banana puree, orange juice concentrate Shear-thickening 1<n< Some types of honey, 40 % raw corn starch solution Bingham Plastic Toothpaste, tomato paste

Non-Newtonian Fluid Behaviour The flow curve (shear stress vs. shear rate) is either non-linear, or does pass through the origin, or both. Three classes can be distinguished. (1) time-independent fluids (2) time-dependent fluids (3) visco-elastic fluids

Time-Independent Fluid Behaviour Fluids for which the rate of shear at any point is determined only by the value of the shear stress at that point at that instant; these fluids are variously known as “time independent”, “purely viscous”, “inelastic”, or “Generalised Newtonian Fluids” (GNF). 1. Shear thinning or pseudoplastic fluids 2. Viscoplastic Fluid Behaviour 3. Shear-thickening or Dilatant Fluid Behaviour

Shear thinning or pseudoplastic fluids Viscosity decrease with shear stress. Over a limited range of shear-rate (or stress) log (t) vs. log (g) is approximately a straight line of negative slope. Hence  yx = m(gyx)n (*) where m = fluid consistency coefficient n = flow behaviour index Eq. (*) is applicable with 0<n<1 Re-arrange Eq. (*) to obtain an expression for apparent viscosity ma (= yx/gyx)

Pseudoplastics Flow of pseudoplastics is consistent with the random coil model of polymer solutions and melts. At low stress, flow occurs by random coils moving past each other without coil deformation. At moderate stress, the coils are deformed and slip past each other more easily. At high stress, the coils are distorted as much as possible and offer low resistance to flow. Entanglements between chains and the reptation model also are consistent with the observed viscosity changes.

Why Shear Thinning occurs Unsheared Sheared Aggregates break up Anisotropic Particles align with the Flow Streamlines In the same form as the cube used to define the other terms of stress and strain, we see a change in the molecules or particles/aggregates as shear is applied the result is a lower energy dissipation, so the viscosity drops when the sample is back at rest, then Brownian motion randomises the particles over time to rebuild the viscosity. if the rebuild process is slow enough then the sample is said to be thixotropic [time dependant] Random coil Polymers elongate and break Courtesy: TA Instruments

Shear Thinning Behavior Shear thinning behavior is often a result of: Orientation of non-spherical particles in the direction of flow. An example of this phenomenon is the pumping of fiber slurries. Orientation of polymer chains in the direction of flow and breaking of polymer chains during flow. An example is polymer melt extrusion Deformation of spherical droplets to elliptical droplets in an emulsion. An industrial application where this phenomenon can occur is in the production of low fat margarine. Breaking of particle aggregates in suspensions. An example would be stirring paint. Courtesy: TA Instruments

2. Viscoplastic Fluid Behaviour Viscoplastic fluids behave as if they have a yield stress (0). Until  0 is exceeded they do not appear to flow. A Bingham plastic fluid has a constant plastic viscosity for Often the two model parameters 0B and mB are treated as curve fitting constants, even when there is no true yield stress.

or Dilatant Fluid Behaviour 3. Shear-thickening or Dilatant Fluid Behaviour yx = m(yx)n (*) where m = fluid consistency coefficient n = flow behaviour index Eq. (*) is applicable with n>1. Viscosity increases with shear stress. Dilatant: shear thickening fluids that contain suspended solids. Solids can become close packed under shear.

Time-dependent Fluid Behaviour More complex fluids for which the relation between shear stress and shear rate depends, in addition, on the duration of shearing and their kinematic history; they are called “time-dependent fluids”. The response time of the material may be longer than response time of the measurement system, so the viscosity will change with time. Apparent viscosity depends not only on the rate of shear but on the “time for which fluid has been subject to shearing”.

Thixotropic : Material structure breaks down as shearing action continues : e.g. gelatin, cream, shortening, salad dressing. Rheopectic : Structure build up as shearing continues (not common in food : e.g. highly concentrated starch solution over long periods of time Thixotropic Rheopectic Shear stress Shear rate

Rheological curves of Time - Independent and Time – Dependent Liquids _ _ + + A B C D E F G Non - newtonian Rheological curves of Time - Independent and Time – Dependent Liquids

Visco-elastic Fluid Behaviour Substances exhibiting characteristics of both ideal fluids and elastic solids and showing partial elastic recovery, after deformation; these are characterised as “visco-elastic” fluids. A true visco-elastic fluid gives time dependent behaviour.

Newtonian Pseoudoplastic Dilatant Common flow behaviours Shear stress Shear stress Shear stress Shear rate Shear rate Shear rate Viscosity Viscosity Viscosity Shear rate Shear rate Shear rate Common flow behaviours

Examples Newtonian flow occurs for simple fluids, such as water, petrol, and vegetable oil. The Non-Newtonian flow behaviour of many microstructured products can offer real advantages. For example, paint should be easy to spread, so it should have a low apparent viscosity at the high shear caused by the paintbrush. At the same time, the paint should stick to the wall after its brushed on, so it should have a high apparent viscosity after it is applied. Many cleaning fluids and furniture waxes should have similar properties.

Examples The causes of Non-Newtonian flow depend on the colloid chemistry of the particular product. In the case of water-based latex paint, the shear-thinning is the result of the breakage of hydrogen bonds between the surfactants used to stabilise the latex. For many cleaners, the shear thinning behaviour results from disruptions of liquid crystals formed within the products. It is the forces produced by these chemistries that are responsible for the unusual and attractive properties of these microstructured products.

Newtonian Foods Examples: Water Milk Vegetable oils Fruit juices Shear stress Shear rate Examples: Water Milk Vegetable oils Fruit juices Sugar and salt solutions

Pseudoplastic (Shear thinning) Foods Shear stress Shear rate Examples: Applesauce Banana puree Orange juice concentrate Oyster sauce CMC solution

Dilatant (Shear thickening) Foods Shear stress Shear rate Examples: Liquid Chocolate 40% Corn starch solution

Bingham Plastic Foods Examples: Tooth paste Tomato paste Shear stress Shear rate Examples: Tooth paste Tomato paste

Measurement of Viscosity Viscosity of a liquid can be measurement Capillary Tube Viscosity Rotational Viscometer

Viscosity: Measurements A Capillary Tube Viscosimeter is one method of measuring the viscosity of the fluid. Viscosity Varies from Fluid to Fluid and is dependent on temperature, thus temperature is measured as well. Units of Viscosity are N·s/m2 or lb·s/ft2

Capillary Tube Viscometer

Newtonian  =  =

Capillary tube viscometer Reference sample with known density and time will be used (at specified temperature). Time and density of sample will be evaluated and then the viscosity can be determined.

Falling sphere method This method can be used for assumption that falling is in Stoke region. When the sphere is fell through the fluid, distance and time of falling will be measured (velocity).

Falling sphere method

Example In a capilary flow experiment using buret, 30 ml of water was emptied in 10 s. A 40% sugar solution (specific gravity 1.179) having similar volume took 52.8 s for emptying. What was the viscosity of sugar solution?

Viscometers In order to get meaningful (universal) values for the viscosity, we need to use geometries that give the viscosity as a scalar invariant of the shear stress or shear rate. Generalized Newtonian models are good for these steady flows: tubular, axial annular, tangential annular, helical annular, parallel plates, rotating disks and cone-and-plate flows. Capillary, Couette and cone-and-plate viscometers are common.

Rotational Viscometer

Non-newtonian fluid from  = 2r2L  =  = from Integrate from r = Ro Ri and  = 0i

Non-newtonian fluid  or a º obtain Linear : y = y-intercept + slope (x)

K and n n = 0.25 K = 59.02 (shear thinning)

Apparent viscosity (Pa.s) Example The shear rate for a power law fluid without yield stress when using a single spindle viscometer is given by (4N/n). Determine the flow behavior index and consistency coefficient based on the following data: Apparent viscosity (Pa.s) RPM 5.0 6 1.5 60