3 z z p B w V a z z ρ µ ρ JG NAVIER STOKES EQUATION N N N N 2 1 3 body

3 z z p b w v a z z ρ µ ρ jg navier stokes

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3 z z p B w V a z z ρ µ ρ + + = JG
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NAVIER STOKES EQUATION N N N N 2 1 ( . ) ( . ) 3 body deformation pressure convective force stresses force local due to acceleration acceleration compressibility V B p V V V V t ρ µ µ ρ ρ − ∇ + + ∇ ∇ = + JG JG JG JG JG JG ±²³²´ ±²³²´ for ρ = const . . 0 V = JG 2 DV B p V Dt ρ µ ρ −∇ + = JG JG JG General equation coordinate independent Cartesian coord. x – dir. 2 2 2 2 2 2 x p u u u u u u u B u v w x x y z t x y z ρ µ ρ + + + = + + + ±²²²³²²²´ ( ) x g if B g gravitional acceleration = JG JG 1 1 Force acting on the fluid element as a result of viscous stress distribution on the surface of element
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Note: Cylindrical coord. pg. 60 viscosity is constant ρ = const. . 0 V = JG isothermal flow. For non-isothermal flows, esp. for liquids, viscosity is often highly temp. dependent. CAUTION CONSERVATION OF ENERGY: THE ENERGY EQ. 1st law of Thermodinamics for a system t dE dQ dW = + 3 ( / ) J m work done on system heat added increase of energy of the system N 2 1 . 2 t gz E e V g r ρ + = + JG G 3 ( / ) J m x z y r G g JG E t : total energy of the system (per unit volume) e : internal enregy per unit mass : displacement of particle r G
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moving system such as flowing fluid particle need Material derivative : time rate of change, following the particle t DQ DW Dt DE D Dt t = + (J/m 3 .s) Energy eq. for a flowing fluid t De DV V gV Dt D DE Dt t ρ + = JGJG Q & W in terms of fluid properties x q x w dx x x q q dx x + x x w w dx x + • assume heat transfer Q to the element is given by Fourier ’s law x T q k x = − heat flows from positive to the neg. temp. ( decreasing temperature gradient ) (HEAT FLOW) q k T = − ∇ G x y z
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Heat flow (rate) into the element in x - dir. : Heat flow (rate) out of the element in x - dir. : The net heat transfer to the element in x - dir. : Hence, the net heat transfer to the element = x q dydz x x q q dx dydz x + W/m 2 x q dxdydz x . . x x x q q q d q d x y z + + ∀ = −∇ G ( ) . . DQ q k T Dt = −∇ = ∇ G [W/m 3 ] neglect internal heat generation Rate of work done to the element per unit area on the left face negatif because work is done on the system ( ) x xx xy xz W u v w σ τ τ = − + + surface direction derivation : force on the left face: ( ) xx xy xz i j k dydz σ τ τ + + G G G
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Rate of work done on the element by this force ( ) ( ) ( ) . xx xy xz xx xy xz i j k ui v j wk dydz u v w dydz σ τ τ σ τ τ = − + + + + = − + + G G G G G G Similarly, rate of work done by the right face stresses is x x W W dx x = − + • Net rate of work done on the element ( ) ( ) ( ) xx xy xz yx yy yz zx zy zz u v w x u v w y u v w z σ τ τ τ σ τ τ τ σ + + + + + + + + .
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