Problem 4 NAVIER-STOKES EQUATION The Navier-Stokes equation is the fundamental equation of fluid dynamics. In one of its many forms (incompressible and viscous flow) the equation is p +(V.V)V) = -Vp+μ(V.V)V. In the av Ət notation, V =< u, v, w> is the three-dimensional velocity field, p is the (scalar) pressure, p is the constant density of the fluid, and u is the constant viscosity. (i) Take the dot product of V and the nabla V operator, then apply the result to a scalar function af af of f to show that (V.V)f=u) +v +w ər ду (ii) Assume fry²2³ and V =< 1, 2,1>. Find (VV)f at (1,1,1). = (iii) Write out the 1st component equation of the Navier-Stokes vector equation.
Problem 4 NAVIER-STOKES EQUATION The Navier-Stokes equation is the fundamental equation of fluid dynamics. In one of its many forms (incompressible and viscous flow) the equation is p +(V.V)V) = -Vp+μ(V.V)V. In the av Ət notation, V =< u, v, w> is the three-dimensional velocity field, p is the (scalar) pressure, p is the constant density of the fluid, and u is the constant viscosity. (i) Take the dot product of V and the nabla V operator, then apply the result to a scalar function af af of f to show that (V.V)f=u) +v +w ər ду (ii) Assume fry²2³ and V =< 1, 2,1>. Find (VV)f at (1,1,1). = (iii) Write out the 1st component equation of the Navier-Stokes vector equation.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Problem 4 NAVIER-STOKES EQUATION
The Navier-Stokes equation is the fundamental equation of fluid dynamics. In one of its many forms
Ꮩ .
(incompressible and viscous flow) the equation is p +(V. V)V) = -√₂ + μ(V · V)V. In the
Ət
notation, V =< u, v, w > is the three-dimensional velocity field, p is the (scalar) pressure, p is the
constant density of the fluid, and u is the constant viscosity.
(i) Take the dot product of V and the nabla V operator, then apply the result to a scalar function
af af af
f to show that (V.V)f=u) +v +w
əx ду əz
(ii) Assume f = ry²2 2³ and V=< 1, 2,1>. Find (V-V)f at (1,1,1).
(iii) Write out the 1st component equation of the Navier-Stokes vector equation.
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