Viscous Dissipation in spherical polar coordinates

Viscous Dissipation termWhen the dot product of each term in the Navier-Stokes equation is taken with thevelocity vector u , the result can be cast in the form of an equation for the time rate ofchange of the kinetic energy of the fluid per unit volume. An important term that appearsin the result for this quantity is the rate at which the work done against viscous forces isirreversibly converted into internal energy. This is known as “viscous dissipation.” Theviscous dissipation per unit volume is written as τ : ∇u= µ Φvwhere Φvfor aNewtonian fluid is given below in different coordinate systems.Rectangular Cartesian Coordinates ( xyz , , )⎡⎛∂ux⎞ ⎛ ⎞⎤Φv= 2 ⎢⎜⎟ + ⎜ ⎟ + ⎜ ⎟ ⎥ + ⎜ +⎢⎝∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂z ⎠ ⎥ ⎝ ∂y∂x⎣⎦2 2 22⎛∂uy⎞ ∂u⎛ uz∂u∂x y2 2⎛∂uy ∂u⎞z ⎛∂uz∂ux⎞ 2+ ⎜ + ⎟ + ⎜ + ⎟ − ∇•⎝ ∂z ∂y ⎠ ⎝ ∂x ∂z⎠ 3( u)2⎞⎟⎠Cylindrical Polar Coordinates ( r, θ , z)⎡⎛∂ur⎞ u ⎛ u ⎞Φv= 2 ⎢⎜ ⎟ + ⎜ + ⎟ + ⎜ ⎟⎢⎣⎝ ∂r ⎠ ⎝r ∂θr ⎠ ⎝ ∂z⎠2 22⎛1∂uθr ⎞ ∂z22 2⎡ ∂ ⎛uθ⎞ 1∂ur⎤ ⎡1∂uz∂uθ⎤+ ⎢r ⎜ ⎟+ + +∂ r r r θ⎥⎝ ⎠ ∂ ⎣⎢ r ∂θz ⎦⎥⎣⎦∂⎡∂ur∂uz⎤ 2+⎢+ − ∇•⎣ ∂z∂r⎥⎦ 3( u)2⎤⎥⎥⎦1

Spherical Polar Coordinates ( r, θφ , )⎡22⎛∂ur ⎞ ⎛1∂uur 1 uθ ⎞ ⎛ ∂φ uruθ⎞Φv= 2⎢⎜ ⎟ + ⎜ + ⎟ + ⎜ + + cotθ⎟⎢⎝ ∂r ⎠ ⎝r ∂θ r ⎠ ⎝rsinθ ∂φr r⎣⎠⎡ 1 ∂uur∂ ⎛ φ ⎞⎤2+ ⎢ + r ⎜ ⎟⎥− ∇•⎣rsinθ∂φ∂r⎝ r ⎠⎦32⎡ ∂ ⎛u1rsin uθ ⎞ ∂u⎤ ⎡ θ ∂ ⎛ φ ⎞ 1 ∂u⎤θ+ ⎢r ⎜ ⎟+ r r r θ⎥ + ⎢ ⎜ ⎟+⎥⎣ ∂ ⎝ ⎠ ∂ ⎦ ⎣ r ∂θ ⎝sinθ ⎠ r sinθ∂φ⎦2( u)222⎤⎥⎥⎦2