Question

Derive the Navier Stokes equations to obtain the velocity profile and the flowrate Q for a cylindrical tank with a stirrer.

Answer 1

Navier-Stokes equations are used to represent the characteristics of turbulence and form the
basis of describing the flow phenomena. The chaotic nature of turbulent fluxes act as a direct result of
non-linear terms in the N-S equations. These equations are based on the conservation laws namely the
continuity, momentum and energy conservation laws as respectively given below .
∂ρ/∂t  + ∇·(ρu) = o (1)
∂(ρu)/∂t+ ∇·(ρuu) = −∇·P

∂(ρe)/∂t+ ∇·(eu) = −∇(u·P) − ∇·q (3)
where u, ρ, e and q are the velocity components, density, total energy per unit volume, and heat
flux, respectively.
The stress tensor, P for a Newtonian fluid is defined by:
P = p(ρ, T)I +2/3µ(∇·u)I − µ((∇u) + (∇v))^T(4)
where, p(ρ, T) is the scalar pressure, I, is a unit diagonal tensor, T is the temperature, and µ is the
dynamic viscosity coefficient.
Thus, the Navier-Stokes equation can be given by:
∂Ui/∂t+ Uj*∂Ui/∂xj= −∂/∂xi(P/ρ)+∂/∂xj(v*∂Ui/∂xj)