Question

vector calculus

(2) The Navier-Stokes equation is the fundamental equation of fluid dynamics that models the motion of water in everything from bathtubs to oceans. In one of its many forms (incompressible, viscous flow), the equation is Ae +(V.V)V)-Vp+ p(V.V at In this 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 is the constant viscosity. Write out the three component equations of this vector equation, i.e., the equations of u, v, and w.

Answer 1

First, we have expressed every term of the Navier-Stokes equation in vector form. It helps to understand the components present and it becomes easier to express the component equations of this vector equation. The complete solution is provided in the attachments.

The Navies - Stokes equation for incomprensible, viscous flow is given by fa ox + (v. v) V) = - pp +M (P.V) V -O Hue, V= <u, u, w) is three-dimensional velocity field , p is the scalar pressure, f is the constant suid density, Mis the constant viscosity. D=< 3 3 7 is the gradient vector in three-dimensional vector field.

V= ui + vj + wk > On - su i + 24 jt annet V.0 = 10 m + oo ay two / - (V.V) V = (u ou + v ceny + w au) i + ne 3 7 din the Jw n + 1 w he op = op i + ab hij + 3 pk (V. v) V = content be Di + + zn az? 24 + à ine T an? zze

i of the vector The three equation component o are as equations follows - Equation of u Equation of w: Equation of w: f en het aan toe by wyraz)