Chapter6 - 57:020 Mechanics of Fluids and Transport...

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Unformatted text preview: 57:020 Mechanics of Fluids and Transport Processes Chapter 6 Professor Fred Stern Fall 2006 1 1 Chapter 6 Differential Analysis of Fluid Flow Fluid Element Kinematics Fluid element motion consists of translation, linear defor- mation, rotation, and angular deformation. Types of motion and deformation for a fluid element. Linear Motion and Deformation : Translation of a fluid element Linear deformation of a fluid element 57:020 Mechanics of Fluids and Transport Processes Chapter 6 Professor Fred Stern Fall 2006 2 2 Change in δ∀ : ( ) u x y z t x δ δ δ δ δ ∂ ⎛ ⎞ ∀ = ⎜ ⎟ ∂ ⎝ ⎠ the rate at which the volume δ∀ is changing per unit vo- lume due to the gradient ∂ u/ ∂ x is ( ) ( ) 1 lim t d u x t u dt t x δ δ δ δ δ → ∀ ∂ ∂ ⎡ ⎤ ∂ = = ⎢ ⎥ ∀ ∂ ⎣ ⎦ If velocity gradients ∂ v/ ∂ y and ∂ w/ ∂ z are also present, then using a similar analysis it follows that, in the general case, ( ) 1 d u v w dt x y z δ δ ∀ ∂ ∂ ∂ = + + = ∇ ⋅ ∀ ∂ ∂ ∂ V This rate of change of the volume per unit volume is called the volumetric dilatation rate. Angular Motion and Deformation For simplicity we will consider motion in the x–y plane, but the results can be readily extended to the more general case. 57:020 Mechanics of Fluids and Transport Processes Chapter 6 Professor Fred Stern Fall 2006 3 3 Angular motion and deformation of a fluid element The angular velocity of line OA , ω OA , is lim OA t t δ δα ω δ → = For small angles ( ) tan v x x t v t x x δ δ δα δα δ δ ∂ ∂ ∂ ≈ = = ∂ so that ( ) lim OA t v x t v t x δ δ ω δ → ∂ ∂ ⎡ ⎤ ∂ = = ⎢ ⎥ ∂ ⎣ ⎦ Note that if ∂ v / ∂ x is positive, ω OA will be counterclockwise. Similarly, the angular velocity of the line OB is lim OB t u t y δ δβ ω δ → ∂ = = ∂ In this instance if ∂ u / ∂ y is positive, ω OB will be clockwise. 57:020 Mechanics of Fluids and Transport Processes Chapter 6 Professor Fred Stern Fall 2006 4 4 The rotation, ω z , of the element about the z axis is defined as the average of the angular velocities ω OA and ω OB of the two mutually perpendicular lines OA and OB . Thus, if counterclockwise rotation is considered to be positive, it follows that 1 2 z v u x y ω ⎛ ⎞ ∂ ∂ = − ⎜ ⎟ ∂ ∂ ⎝ ⎠ Rotation of the field element about the other two coordinate axes can be obtained in a similar manner: 1 2 x w v y z ω ⎛ ⎞ ∂ ∂ = − ⎜ ⎟ ∂ ∂ ⎝ ⎠ 1 2 y u w z x ω ∂ ∂ ⎛ ⎞ = − ⎜ ⎟ ∂ ∂ ⎝ ⎠ The three components, ω x , ω y , and ω z can be combined to give the rotation vector, ω , in the form: 1 1 2 2 x y z curl ω ω ω = + + = = ∇ × ω i j k V V since 1 1 2 2 x y z u v w ∂ ∂ ∂ ∇ × = ∂ ∂ ∂ i j k V 1 1 1 2 2 2 w v u w v u y z z x x y ⎛ ⎞ ⎛ ⎞ ∂ ∂ ∂ ∂ ∂ ∂ ⎛ ⎞ = − + − + − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ i j k 57:020 Mechanics of Fluids and Transport Processes...
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This note was uploaded on 12/08/2011 for the course MECHANICS 57:020 taught by Professor Fredrickstern during the Fall '10 term at University of Iowa.

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Chapter6 - 57:020 Mechanics of Fluids and Transport...

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