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MTH- 486: Fluid Mechanics Instructor: Dr. Fahad Munir Abbasi Assistant Professor Department of Mathematics Comsats Institute of Information Technology Islamabad, Pakistan Layout of lecture # 5 Summary of the previous lecture Newtonian and non-Newtonian fluids Types of non-Newtonian fluids Flow between two plates: Solved examples Summary Previously we studiedβ¦ Velocity of the fluid at a point: local and particle rate of change Viscosity of a fluid Real and Ideal fluids MTH-486, Fluid Mechanics Lec. # 05 Newtonian fluids: Fluids in which the shear stress is directly proportional to the rate of deformation are called the βNewtonian fluidsβ. For such fluids, the Newtonβs law of viscosity holds, i.e. π π πππππ ππππππ β π π Οππ π π =ΞΌ π π (1) Where the constant of proportionality βΞΌβ is known as the viscosity of the fluid. MTH-486, Fluid Mechanics Lec. # 05 Non-Newtonian fluids: Fluid in which the shear stress is not directly proportional to the rate of deformation are called the βnon-Newtonian fluidβ. Examples: Toothpaste, blood, ketchups, paint, drilling mud, biological fluids, etc. For such fluids, the βPower law modelβ holds, i.e.: π π πππππ ππππππ β π π Οππ π π π =π π π , π β π. π (2) MTH-486, Fluid Mechanics Lec. # 05 Where βnβ is flow behavior index, βkβ the consistency index. Note that Eq. (2) reduces to Newtonβs law of viscosity for n = 1 and k = ΞΌ. Also from Eq. (2) Οππ π π =Ξ· , π π (3) Where π π Ξ·=π π π Is the apparent viscosity. πβπ (4) MTH-486, Fluid Mechanics Lec. # 05 The rod climbing or the Weissenberg effect When a rod is rotated in a beaker containing a polymer (non-Newtonian) solution. The solution moves in the opposite direction of the motion of the rod and climbs up the rod. This effect is known as the βrod climbing effectβ or the βWeissenberg effectβ. For a beginner, this is a simple test to differentiate between the Newtonian and non-Newtonian fluid. MTH-486, Fluid Mechanics Lec. # 05 Types of non-Newtonian fluids: The non-Newtonian fluids are divided into three broad groups: Non-Newtonian Fluids Time independent fluids Time dependent fluids Viscoelastic fluids MTH-486, Fluid Mechanics Lec. # 05 . Time independent fluids Pseudoplastic or shear thinning fluids (n < 1) Dilatant or shear thickening fluids (n >1) Ideal or Bingham plastic fluids MTH-486, Fluid Mechanics Lec. # 05 . Time dependent fluids Thixotropic fluids Rheopectic fluids MTH-486, Fluid Mechanics Lec. # 05 Time independent non-Newtonian fluids: 1. Pseudoplastic (or shear thinning) fluids: Fluids in which the apparent viscosity decreases with increasing deformation rate, i.e. n < 1. Examples: Polymer solution such as rubber, Colloidal suspensions and paper pulp In water, blood, milk etc. 2. Dilatant (or shear thickening) fluids: Fluids in which the apparent viscosity increases with increasing deformation rate, i.e. n > 1. Examples: suspensions of starch and of sand, butter, printing ink, suger in water etc. MTH-486, Fluid Mechanics Lec. # 05 3. Ideal or Bingham plastic fluids: Fluids that behave as solids until a minimum yield strength, Οπ is exceeded and subsequently exhibits a linear relation between stress and rate of deformation. Mathematically Οππ π π = Οπ + ΞΌπ π π Examples: Drilling muds, toothpaste and clay suspensions, jellies etc. MTH-486, Fluid Mechanics Lec. # 05 Time dependent non-Newtonian fluids: 1. Thixotropic fluids: Fluids that show a decrease in Ξ· with time under a constant applied shear stress. Examples: Lipstick, some paints and enamels, etc. 2. Rheopectic fluids: Fluids that show an increase in Ξ· with time under a constant applied shear stress. Examples: gypsum suspension in water and bentonite solution, etc. MTH-486, Fluid Mechanics Lec. # 05 Viscoelastic non-Newtonian fluids: Some fluid after deformation partially return to their original shape when the applied stress is released. Such fluids are named as βviscoelastic fluids.β Viscoelastic fluids have two major types: linear viscoelastic fluids e.g. the Maxwell and Jefferyβs fluids, and non-linear viscoelastic fluids e.g. Walterβs A and B fluid, Oldroyed A and B etc. MTH-486, Fluid Mechanics Lec. # 05 Graphical representation: Fig. 1: Flow curves for time independent fluids. MTH-486, Fluid Mechanics Lec. # 05 Graphical representation: Fig. 2: Flow curves for time dependent fluids MTH-486, Fluid Mechanics Lec. # 05 Graphical representation: Fig. 3: Flow curves for time dependent fluids MTH-486, Fluid Mechanics Lec. # 05 Flow between two plates: An infinite plate is moved over a second plate on a layer of liquid. For small gap width, h = 0.3mm, we assume a linear velocity distribution in the liquid, U = 0.3 m/sec. The liquid viscosity is 0.65 X π²π βπ ππ π.πππ i) and its specific gravity is 0.88. Find: The kinematic viscosity of the fluid. ii) The shear stress on lower plate. iii) Indicate the direction of shear stress. MTH-486, Fluid Mechanics Lec. # 05 Flow between two plates: MTH-486, Fluid Mechanics Lec. # 05 density MTH-486, Fluid Mechanics Lec. # 05 Now, i) Ξ½ = ΞΌ Ο = π.ππ πΏ ππβπ π.ππ πΏ πππ ii) Οππ = Οπππππ = =ββ β π π ΞΌ π π = πΌ ΞΌ π ππ πππ = π²π π. ππ π.ππππ iii) Since the Οππ is positive so the direction of shear stress is along positive x-axis. MTH-486, Fluid Mechanics Lec. # 05 Example # 02: Suppose that the fluid being sheared between to plates is SAE 30 oil (ΞΌ = 0.29 π²π ) π.πππ at 20°C. Compute the shear stress in the oil if V=3 m/s and h = 2 cm. Solution: π²π π π π π½ π. ππ π. πππ πΏ π. π πππ Ο=ΞΌ =ΞΌ = π π π π. ππ π π²π π΅ = π. ππ = ππ π = πππ·π π π. πππ π Summary Newtonian and non-Newtonian fluids Types of non-Newtonian fluids Solved examples Todayβs quote: Knowledge is powerβ¦
