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Lecture 7: Non-Newtonian Fluids • Classification of Non-Newtonian Fluids • Laminar Flow of a Non-Newtonian fluid in Circular Pipes • Recommended text-book: W.F. Hughes, J.A. Brighton, Schaum's outline of theory and problems of fluid dynamics, New York: McGraw Hill, 1999 1 Examples • Water and simple liquids; air and simple gases… are Newtonian fluids. • Fluids in food industry, gels, polymers, slurries, drilling muds, blood… are Non-Newtonian fluids. • The Non-Newtonian behaviour is frequently associated with complex internal structure: fluid has large complex molecules (like a polymer) or fluid is a heterogeneous solution (like a suspension)... 1: Coal slurries having consistency of over 80% by volume of powdered or crushed coal in water can be pumped long distances with much less power requirements for pumping than pure water. 2 Examples 2: In the fracturing treatment of oil wells, materials have been developed which when added to water make a fluid so thick that it suspends sand, glass or metal pellets. Yet the same fluid can be pumped down a well at enormous rates with less than half the friction loss of water. Such materials are used for fracturing of an oil reservoir. Fracturing is used to increase the production of the well. First, a crack is initiated in the producing zone. Then, fluid pumped down the well under high pressure greatly extends this crack. The sand, or pellets act as propping agents to hold the fracture open after the treatment. Fluid must be delivered at a rapid rate to overcome the loss by diffusion of fluid within the pores of the fractured rock. 3 Definition: Newtonian/Non-Newtonian fluids The viscous stress tensor for incompressible Newtonian fluid is ik ik v v x x i k k i ik -- rate of shear strain Newtonian fluids: linear proportionality between the shearing tensor and the shearing rate. Non-Newtonian fluids: any different relation between the shearing stress and the shearing rate. 4 Classification • Time Independent Fluids (the relation between shearing stress and rate is unique but non-linear) • Bingham plastics • Pseudoplastic fluids • Dilatant plastics • Time Dependent Fluids (the shear rate depends on the shearing time or on the previous shear rate history) • Thixotropic fluids • Reopectic fluids • Viscoelastic fluids (the shear stress is determined by the shear strain and the rate of shear strain) 5 Time Independent Fluids ik f ik τ 0 – Newtonian fluid 1 – Bingham plastics 2 – pseudoplastic fluids 3 – dilatant fluids 1 2 0 slope η 3 1. Bingham plastics y Examples: slurries, plastics, emulsions such as paints, and suspensions of finely solids in a liquid (e.g. drilling muds, which consist primarily of clays suspended in water). 6 2. Pseudoplastic (shear thinning) fluids A progressively decreasing slope of shear stress vs. shear rate. The slope can be defined as apparent viscosity: At very high rates of shear in real fluids the apparent viscosity becomes constant. a Examples: paper pulp in water, latex paint, blood, syrup, molasses, ketchup, whipped cream, nail polish The simplest empirical model is the power law due to Ostwald: k a n 1 , n 1 The power law model is popular due to its simplicity. For the cases where power law model does not give an adequate representation, it might be practical to use the actual measured properties of the fluid. 7 3. Dilatant (shear thickening) fluids The apparent viscosity increases with increasing shear rate. Can be represented by the power law model with n>1. Less common (suspensions of corn starch or sand in water). Applications: Traction control: some all-wheel drive systems use a viscous coupling unit full of dilatant fluid to provide power transfer between front and rear wheels. On high traction surfacing, the relative motion between primary and secondary drive wheels is the same, so the shear is low and little power is transferred. When the primary drive wheels start to slip, the shear increases, causing the fluid to thicken. As the fluid thickens, the torque transferred to the secondary drive wheels increases, until the maximum amount of power possible in the fully thickened state is transferred. Body armour: application of shear thickening fluids for use as body armour, allowing the wearer flexibility for a normal range of movement, yet providing rigidity to resist piercing by bullets, stabbing knife blows, and similar attacks. 8 Time Dependent Fluids Thixotropic fluids (the shear stress decreases with time as the fluid is sheared) As the fluid is sheared from the state of rest, it breaks down (on molecular scale), but then the structural reformation will increase with time. An equilibrium situation is eventually reached where the breakdown rate is equal to build-up rate. If allowed to rest, the fluid builds up slowly and eventually regains its original consistency. Examples: many gels or colloids http://www.youtube.com/watch?v=oJaLV_r_kNA http://www.youtube.com/watch?v=zn2uonRGKYk Rheopectic fluids (the shear stress increases with time as the fluid is sheared). Molecular structure is formed by shear and behaviour is opposite to that of thixotropy. Example: beating and thickening of egg whites, inks. http://www.youtube.com/watch?v=cuzn8wh8Fys 9 Viscoelastic Fluids A viscoelastic material exhibits both elastic and viscous properties. The simplest type is one which is Newtonian in viscosity and obeys Hooke’s law for the elastic part: λ is a rigidity modulus. Simplest and popular model -- Maxwell liquids: Under steady flow, . If the motion is stopped the stress relaxes as exp t Movies: •http://www.youtube.com/watch?v=B1C4qNyrUjU •http://www.youtube.com/watch?v=zwz3R0IG9Xc •http://www.youtube.com/watch?v=KcNWLIpv8gc •http://www.youtube.com/watch?v=wmUx-1o3Lzs •http://www.youtube.com/watch?v=nX6GxoiCneY •http://www.youtube.com/watch?v=HQ8FP0sa_hk Examples: polymers, metals at temperature close to their melting point 10 Laminar flow of Bingham plastics R rp We consider steady plane parallel flow. The governing equations are reduced to z z projection By denoting p z A : 0 p z 1 r r r rz (*) and integrating (*) we obtain rz Ar 2 c1 r c1 = 0 as the stress tensor must be bounded at r = 0, i.e. Ar τrz rz rp R r 2 τy 11 For Bingham plastics, the rate of stress tensor is related to the shearing stress as y : 0 , y : rz y , or v r (**) at r r : p rz z 0 1 2 v r z v rA 0 z r y r p 2 for τ τ y for τ τ y y A 12 Integration of (**) gives 1 1 A R r v rA dr 2 4 r 2 z y R Setting r r p 2 2 y R r , r r R p gives y A 2 R v 1 , A r 2 p r r 0 p p The volumetric flow rate R A 4 2 1 2 Q 1 8 3 RA 3 RA 4 4 y y At τy we will have the formulae earlier obtained for the Poiseuille flow. 13 Eugene Cook Bingham, born 8 December 1878, died 6 November 1945. Bingham made many contributions to rheology. 14 Lecture 8: Flow fields with negligible inertia forces • Flow in slowly-varying channel • Lubrication theory 15 Flow in slowly-varying channels v For Poiseuille flow: t 0 Steady flow For steady flow in a slowly varying channel v v 0 v 0 Plane-parallel flow v v 0 but small v v We can always make the ratio 1 v by choosing a sufficiently slow rate of variation of the crosssection. 16 Consider a steady flow along a circular tube with R(x), with dR dx x dp and dx A x In such a flow, v r ~ v z . And u ~ V , 2 R Hence, V v v ~ R 2 v v RV ~ 1 , i.e. inertia force is negligible. v Thus, the flow profile and the volumetric flow flux are v z x ,r A 4 R 2 r 2 R A 4 Q 8 This approximation is useful in many different circumstances e.g. the flow of a fluid squeezed out radially by pressing close together two plane disks. 17 Lubrication theory It is a matter of common experience that two solid bodies can slide over one another easily when there is a thin layer of fluid between them and that under certain conditions a high positive pressure is set up in the fluid layer. This is used as a means of substituting fluid-solid friction for the much larger friction between two solid bodies in contact. In some case the fluid layer is used to support a useful load, and is then called a lubrication bearing. Reynolds’ theory, 1886 RU 1 α h1 h U h2 which is usually satisfied under practical conditions of 18 lubrication Flow profile: A u 2 y h y U Poiseuille flow Volumetric flow flux: h Q u dy 0 Ah h y h Couette flow 3 12 1 Uh 2 From here, the pressure gradient is p 2Q U A 6 2 3 x h h (1) Q must be independent of x. In addition, h h 1 x Integration of (1) gives 6 p p0 U 1 1 1 1 Q 2 2 h1 h h1 h 19 Suppose the sliding block is completely immersed in the fluid, so p=p0 when h=h2, which enables us to determine Q, Q U h 1h 2 h1 h 2 and p p0 p p0 0 if 6 U h 1 h h h 2 h 2 h1 p-p0 x p max LU h 2 1 h 2 h 1 h 2 A lubrication layer will be able to support a load normal to the layer only when the layer is so arranged that the relative motion of the two surfaces tends to drag fluid from the wider to the narrow end. pmax can be high if h1 is small (L is a layer length) 20 The total normal force exerted on either of the two boundaries by the fluid layer is L 6 U h 1 h1 h 2 p p 0 dx 2 ln h 2 h h 2 1 2 0 The total tangential force exerted by the fluid on the lower plate L u 2 U h 1 h 2 h1 y dx 3 h h 2 ln h 1 2 2 0 y 0 The tangential force on the upper boundary is L u 2 U h 1 h 2 h1 ln dx 3 y h h h y h 1 2 2 0 21 tangential force on the block normal force on the block h h f 1~ 1 h2 L This ratio is independent of the viscosity, and can be made indefinitely small by reduction of h1 with h1/h2 held constant. α is regarded as a given quantity, although in any case in which the sliding block is free to move, α may be a variable, but consideration of this is beyond of our scope. 22