# Chapter 3 Non-Newtonian fluid

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1 Chapter 3 Non-Newtonian fluid 3-1. Introduction: The study of the deformation of flowing fluids is called rheology; the rheological behavior of various fluids is sketchen Figure 3-1. Newtonian fluids, defined as fluids for which the shear stress is linearly proportional to the shear strain rate. Newtonian fluids (stress proportional to strain rate). Many common fluids, such as air and other gases, water, kerosene, gasoline, and other oil-based liquids, are Newtonian fluids. Fluids for which the shear stress is not linearly related to the shear strain rate are called non-newtonian fluids. Examples include slurries and colloidal suspensions, polymer solutions, blood, paste, and cake batter. Some non- Newtonian fluids exhibit a "memory"-the shear stress depends not only on local strain rate, but also on its history. A fluid that returns (either fully or partially) to its original shape after the applied stress is releases called viscoelastic. n = 1 n < 1 τ B n = 1 n > 1 Figure 3-1. Rheological behavior of fluids shear Stress as a function of shear strain rate. Lecturer: Dr Jafar Ghani Majeed Page 1

2 Some non-newtonian fluids are called shear thinning fluids or pseudoplastic fluids, because the more the fluis sheared, the less viscous it becomes. A good example is paint. Paint is very viscous when poured from the can or when picked up by a paintbrush, since the shear rate is small. However, as we apply the paint to the wall, the thin layer of paint between the paintbrush and the wall is subjected to a large shear rate, ant becomes much less viscous. Plastic fluids are those in which the shear thinning effect is extreme. In some fluids a finite stress called the yield stress is required before the fluid begins to flow at all; such fluids are called Bingham plastic fluids. Certain pastes such as acne cream and toothpaste are examples of Bingham plastic fluids. If you hold the tube upside down, the paste does not flow, even though there is a nonzero stress due to gravity. However, if you squeeze the tube (greatly increasing the stress), the paste flows like a very viscous fluid. Other fluids show the opposite effect and are called shear thickening fluids or dilatant fluids; the more the fluis sheared, the more viscous it becomes. The best example is quicksand, a thick mixture of sand and water. As we all know from Hollywood movies, it is easy to move slowly through quicksand, since the viscosity is low; but if you panic and try to move quickly, the viscous resistance increases considerably and you get "stuck" (Figure 3-2). You can create your own quicksand by mixing two parts cornstarch with one part water-try it! Shear thickening fluids are usen some exercise equipment-the faster you pull, the more resistance you encounter. Figure 3-2. Figure 3-3. Lecturer: Dr Jafar Ghani Majeed Page 2

3 The viscosity takes various forms, depending on the nature of the fluid. Some of these are illustraten Figure 3-3 ann Figure 3-4. The classes of behavior illustrated are the Bingham plastic, pseudoplastic, dilatant, and structural. Both the Bingham plastic and the pseudoplastic are shear thinning, since the viscosity decreases with increasing shear rate, whereas the dilatant fluis shear thickening. The structural fluid exhibits newtonian behavior at very low and very high values of shear rate ans shear thinning at intermediate shear rates. Figure 3-4. In general the definitions of Newtonian and non-newtonian fluids as follows: Newtonian fluids: fluids which obey Newton's law of viscosity [τ w = μ (- du dy )]. Non-Newtonian fluids: fluids which do not obey Newton's law of viscosity [for Bingham plastic fluids[τ w = τ B + μ a (- du dy )]. Lecturer: Dr Jafar Ghani Majeed Page 3

4 3-2. Generalized Reynolds number for flow in pipes: For Newtonian flow in a pipe, the Reynolds number is defined by: Re = ρu μ (3.1) In case of Newtonian flow, it is necessary to use an appropriate apparent viscosity (μ a ). In flow in a pipe, where the shear stress varies with radial location, the value of μ a varies. The value of μ a evaluated at wall is given by: shear stress at wall τ w μ a = - = (3.2) shear rate at wall ( du/dy) w Another definition is based, not on the true shear rate at the wall, but on the flow characteristic. This quantity, which may be called the apparent viscosity for pipe flow, is given by: μ ap = shear stress at wall flow characteristic = τ w (3.3) / For laminar flow, μ ap has the property that it is the viscosity of a Newtonian fluid having the same flow characteristic as the non-newtonian fluid when subjected to the same value volumetric flow rate for the same pressure gradient in the same pipe. This suggests that μ ap might be a useful quantity for correlating flow rate-pressure gradient data for non-newtonian flow in pipes. This is found to be the case ant is on μ ap that a generalized Reynolds number Re is based: Re = ρu μ ap (3.4) Representing the fluid's laminar flow behavior in terms of K and n : τ w = K n (3.5) The pipe flow apparent viscosity, defined by equation 3.3, is given by: Lecturer: Dr Jafar Ghani Majeed Page 4

5 μ ap = τ w / = K n 1 (3.6) When this equation for μ ap is substitutento equation 3.4, the generalized Reynolds number takes the from: Re = ρu2 n n 8 n 1 K (3.7) Use of this generalized Reynolds number was suggested by Metzner and Reed (1955). For Newtonian behavior, K = μ and n = 1 so that the generalized Reynolds number reduces to the normal Reynolds number Turbulent flow of inelastic non-newtonian fluids in pipes: Turbulent flow of Newtonian fluids is describen terms of the Fanning friction factor, which is correlated against the Reynolds number with the relative roughness of the pipe wall as a parameter. The same approach is adopted for non-newtonian flow but the generalized Reynolds number is used, The fanning friction factor is defined by: f = τ w 1 2 ρu2 (3.8) For laminar flow of a non-newtonian fluid, the wall shear stress can be expressen terms of K and n as: τ w = K n (3.9) On substituting for τ w equation 3.8, the Fanning friction factor for laminar non-newtonian flow becomes: Lecturer: Dr Jafar Ghani Majeed Page 5

6 f = 16 Re (3.10) This is of the same form as equation 2.13 for Newtonian flow ans one reason for using this form of generalized Reynolds number. Equation 3.10 provides another way calculating the pressure gradient for a given flow rate for laminar non-newtonian flow Laminar-turbulent transition: A stability analysis made by Ryan and Johnson (1959) suggests that the transition from laminar to turbulent flow for inelastic non-newtonian fluids occurs at a critical value of the generalized Reynolds number that depends on the value n. The results of this analysis are shown in Figure 3-5. This relationship has been for shear thinning and for Bingham plastic fluids and Figure 3-5 Variation of critical value of the Reynolds number with n. has been found to be accurate. Over the range of shear thinning behavior encounteren practice, 0.2 n < 1.0, the critical value of Re is in the range 2100 Re Friction factors for turbulent flow in smooth pipes : Experimental results for the Fanning friction factor for turbulent flow of shear thinning fluids in smooth pipes have correlated by Dodge and Metzner (1959) as a generalized form of the von Ka rma n equation: Lecturer: Dr Jafar Ghani Majeed Page 6

7 1 f 1/2 = 4.0 (n ) 0.75 log [f(1 n /2) Re ] 4.0 (n ) 1.2 (3.11) This correlation is shown in Figure 3-6. The broken lines represent extrapolation of equation 3.11 for values of n and Re beyond those of the measurements made by Dodge and Metzner. More recent studies tend to confirm the findings of Dodge and Metzner but do not significantly extend the range of applicability. Figure 3-6 Friction factor chart for purely viscous non-newtonian fluids. Having determined the value of the friction factor f for specified flow rate and hence Re, the pressure gradient can be calculaten the normal way using equation P f = 4f L ρu 2 2 = 2fLρu2 (3.12) Lecturer: Dr Jafar Ghani Majeed Page 7

8 3-4. Power law fluids: The methods presenten mentioned Sections are general and do not require the assumption of any particular flow model. While the flow of power law fluids and Bingham plastics can be treated by those methods, some results specific to these materials will be consideren this section. It was mentioned earlier that for laminar flow of a non-newtonian fluid, the wall shear stress can be expressen terms of K and n as: τ w = K n (3.13) The relationship between n and n, and K and K will now be demonstrated. For the conditions at the pipe wall, denoted by the subscript w, the equation of power law fluid can be written as: τ w = K ( γ w) n = K ( du dy ) w n (3.14) The minus sign has been placenside the parentheses recognizing the fact that the shear rate γ (equal to du/dy) is negative. γ w is the true shear rate at wall ans related to the flow characteristic (/ ) by the Rabinowitsch- Mooney equation: γ w= 3n + 1 4n (3.15) Therefore the behavior of a power law fluid, evaluated at the wall conditions, is given by: τ w = K n 3n + 1 4n n (3.16) Equation 3.15 shows that a plot of ln τ w against ln (/ ) has a constant gradient of n. Consequently, for a power law fluid: n = n (3.17) Lecturer: Dr Jafar Ghani Majeed Page 8

9 Comparing equation 3.16 with equation 3.13 (both with n = n), shows that: K = K 3n+ 1 4n (3.18) Nomenclature di f K K L n n Re Re u inside diameter of the pipe, m fanning friction factor, dimensionless consistency coefficient, Pa s n consistency coefficient for pipe flow, Pa s n length of pipe, m power law index, dimensionless flow characteristic, dimensionless Reynolds number, dimensionless generalized Reynolds number, dimensionless volumetric average velocity, m/s γ shear rate = du/dy, s -1 μ μ a μ ap dynamic viscosity, Pa s apparent viscosity at wall, Pa s apparent viscosity for pipe flow, Pa s ρ fluid density, kg /m 3 τ τ w τ B shear stress, Pa shear stress at wall, Pa yield stress, Pa Lecturer: Dr Jafar Ghani Majeed Page 9

10 Examples-Chapter 3 Example 3-1 A general time-independent non-newtonian liquid of density 961 kg/m 3 flows steadily with an average velocity of 2.0 m/s through a tube m long with an inside diameter of m. for these conditions, the pipe flow consistency coefficient K has a value of 1.48 Pa s 0.3 and n a value of 0.3. Calculate the values of the apparent viscosity for pipe flowμ ap, the generalized Reynolds number Re and the pressure drop across the tube, neglecting end effects. Solution: Apparent viscosity for pipe flow is given by: μ ap = K n 1 and the generalized Reynolds number by: The flow characteristic is given by: and Re = ρu μ ap 8 (2.0 m/s) = = 210 s m n 1 = (210 s -1 ) ( ) = s 0.7 Hence μ ap = (1.48 Pa s 0.3 )( s 0.7 ) = Pa s Lecturer: Dr Jafar Ghani Majeed Page 10

11 and Re = (961 kg/m3 ) 2.0 m.s m ( Pa s) = 4178 From Figure 3-6, the Fanning friction factor f has a value Therefore the pressure drop is given by: P f = 4f L ρu 2 2 = 2fLρu2 P f = 2 (0.0047)(3.048 m)(961kg/m3 )(2.0 m/s) 2 (0.0762m) = 1445 Pa Example 3-2 Calculate the frictional pressure gradient P f /L for a time independent non- Newtonian fluin steady state flow in a cylindrical tube if: the liquid density ρ = 1000 kg/m 3 the inside diameter of the tube the mean velocity = 0.08 m u = 1 m/s the point pipe consistency coefficient K = 2 Pa s 0.5 and the flow behavior index n = 0.5 Solution: Apparent viscosity for pipe flow is given by: μ ap = K n 1 and the generalized Reynolds number by: Lecturer: Dr Jafar Ghani Majeed Page 11

12 The flow characteristic is given by: and Re = ρu μ ap 8 (1.0 m/s) = = 100 s m n 1 = (100 s -1 ) ( ) = 0.1 s 0.5 Hence μ ap = (2 Pa s 0.5 )(0.1 s 0.5 ) = 0.2 Pa s and Re = (1000 kg/m3 ) 1.0 m.s 0.08 m (0.2 Pa s) = 400 From Figure 3-6, at Re = 400, the Fanning friction factor f = 16 Re : then f = = Therefore the frictional pressure gradient P f /L is given by: P f /L = 2fρu2 P f /L = 2 (0.04)(1000 kg/m3 )(1.0 m/s) 2 (0.08 m) = 1000 Pa/m Lecturer: Dr Jafar Ghani Majeed Page 12

13 Example 3-3 The rheological properties of a particular suspension can be approximate reasonably well by either a "power law" or a "Bingham plastic" model and the shear rate range of 10 to 50 s -1. If the consistency coefficient, K is 10 N s n /m 2 and the flow behavior index, n is 0.2 in the power law model. What will the approximate values of the yield stress and of the plastic viscosity in the Bingham plastic model? When τ w = K ( du dy )n τ w = 10 ( du dy )0.2 du dy = 10 s-1, τ w = 10 * (10) 0.2 = N/m 2 du dy = 50 s-1, τ w = 10 * (50) 0.2 = N/m 2 Using Bingham plastic model: Thus: μ a = 0.15 N.s/m 2 and : τ B =14.35 N/m 2 τ w = τ B + μ a ( du dy ) = τ B + μ a (10), = τ B + μ a (50), Thus, Bingham plastic equation is: τ w = ( du dy ) Lecturer: Dr Jafar Ghani Majeed Page 13

14 Example 3-4 A Newtonian liquid of viscosity, μ is 0.1 N.s/m 2 is flowing through a pipe. As a result of a process change a small quantity of polymer is added to the liquid and this causes the liquid to exhibit non-newtonian characteristics; its rheology is described adequately by the "power law" model and the flow index, n is The apparent viscosity (μ a ) of the modified fluis equal to the viscosity (μ) of the original liquid at a shear rate, du/dy is 1000 s -1. Determine the rheological equation? Solution: τ w = K ( du dy )n τ w = K du dy n 1 du dy. (1) Since μ a = τ w du dy. (2) Substitute equation (1) in equation (2), get equation (3): μ a = K du dy n 1. (3) μ a = K = K K = 10 Pa s n Rheological equation is: τ w = 10 ( du dy )0.33 Lecturer: Dr Jafar Ghani Majeed Page 14

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