# lecture19 - 2.57 Nano-to-Macro Transport Processes Fall...

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2.57 Fall 2004 – Lecture 19 1 2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 19 In last lecture, we talked about the Newton’s shear stress law. Assuming that the number density of particles is n, the number density of particles having velocity v is f o (v x ,v y ,v z )=nP(v z ,v y ,v z ) ( ) T 2 / v v u v m 2 / 3 B B 2 z 2 y 2 x e T π 2 m n κ + + κ = From + = o o o f m f f f v r F v τ ( F is zero here because this is no external fields here and gravity is neglected), the distribution function is ± ± () ± ± oo o o o o oy ff f f f u τ f τ vf τ v yy xy z vi v j vk i j k z u ⎛⎞ ∂∂ =− + + + + ⎜⎟ ⎝⎠ ±± . Note: We can also prove xyz v fdv dv dv x u ∞∞∞ −∞ −∞ −∞ = ∫∫∫ . The shear stress along the x-direction, on a plane perpendicular y-axis can be calculated by considering the momentum exchange across the plane, [] y u μ dv dv dv f v τ v y u dv dv fdv v v τ z y x o x 2 y z y x x y xy = ∫∫ = = u m m The dynamic viscosity can be 2 o yx x y z f τ vm v d v d v d v u µ x y U o u v y v x

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2.57 Fall 2004 – Lecture 19 2 2 '2 2 B B zB 3/2 mv /(2 T) mv /(2 T) mv /(2 T) 22 ' z v τ ed v v ved v 2 π y x x yy x BB m mn TT κ κκ ∞∞ −∞ −∞ −∞ ⎛⎞ = ⎜⎟ ⎝⎠ ∫∫ .
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lecture19 - 2.57 Nano-to-Macro Transport Processes Fall...

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