Navier-Stokes Equations

The Navier-Stokes equations are differential equations of motion that will allow you to incorporate the viscous effects of a fluid.  Recall that viscosity is the fluids willingness to flow.  In other words it will give you the ability to also consider the fluids frictional forces.  Finally, to use the Navier-Stokes equations you will need establish a relationship between the stresses on the fluid element as well as the elements velocity.

Stress and Deformation

First, before the the Navier-Stokes equations can be defined we will need to linearly relate stress to the rates of deformation for an incompressible Newtonian fluid.  First, let’s take a look at the normal stresses expressed on the Cartesian coordinate system.

(Eq 1) $σ_{xx}=-p+2μ\frac{∂u}{∂x}$

(Eq 2) $σ_{yy}=-p+2μ\frac{∂ν}{∂y}$

(Eq 3) $σ_{zz}=-p+2μ\frac{∂w}{∂z}$

In addition, to the normal stresses, the shearing stresses also need to be defined.

(Eq 4) $τ_{xy}=τ_{yx}=μ\left(\frac{∂u}{∂y}+\frac{∂ν}{∂x}\right)$

(Eq 5) $τ_{zy}=τ_{yz}=μ\left(\frac{∂ν}{∂z}+\frac{∂w}{∂y}\right)$

(Eq 6) $τ_{zx}=τ_{xz}=μ\left(\frac{∂w}{∂x}+\frac{∂u}{∂z}\right)$

Furthermore, in equations 1-3 $-p$ is the pressure of the average of the three normal stresses.  As a result, $-p=(1/3)(σ_{xx}+σ_{yy}+σ_{zz})$.  Hence, the pressure for viscous fluids has to be defined  as an average of the normal stresses due to the fact that the normal stresses are not necessarily the same in different directions. In addition, another thing that needs to be considered is the difference between an elastic solid and a Newtonian fluid.  For instance, for an elastic solid the stress is linearly related to deformation or strain.  However,  for a Newtonian fluid stress is linearly related to the rate of deformation or rate of strain.

Finally, sometimes it is easier to express the stresses on the fluid element in polar coordinates.  As a result, the following equations are the polar coordinates for normal stress.

(Eq 7) $σ_{rr}=-p+2μ\frac{∂ν_r}{∂r}$

(Eq 8) $σ_{θθ}=-p+2μ\left(\frac{1}{r}\frac{∂ν_θ}{∂θ}+\frac{ν_r}{r}\right)$

(Eq 9) $σ_zz=-p+2μ\frac{∂ν_z}{∂z}$

In addition, the polar coordinates for the shearing stresses are as follows.

(Eq 10) $τ_{rθ}=τ_{θr}=μ\left[r\frac{∂}{∂r}\left(\frac{ν_θ}{r}\right)+\frac{1}{r}\frac{∂ν_r}{∂θ}\right]$

(Eq 11) $τ_{θz}=τ_{zθ}=μ\left(\frac{∂ν_θ}{∂z}+\frac{1}{r}\frac{∂ν_z}{∂θ}\right)$

(Eq 12) $τ_{zr}=τ_{rz}=μ\left(\frac{∂ν_r}{∂z}+\frac{∂ν_z}{∂r}\right)$

Navier-Stokes Equations

By applying the stresses defined in the equations above to the differential equation of motion, the Navier-Stokes equation can be derived.  As a result, the Navier-Stokes equations in the x, y, and z directions can be seen below.

  • x-direction

(Eq 13) $ρ\left(\frac{∂u}{∂t}+u\frac{∂u}{∂x}+ν\frac{∂u}{∂y}+w\frac{∂u}{∂z}\right)$$=-\frac{∂p}{∂x}+ρg_x+μ\left(\frac{∂^2u}{∂x^2}+\frac{∂^2u}{∂y^2}+\frac{∂^2u}{∂z^2}\right)$

  • y-direction

(Eq 14) $ρ\left(\frac{∂ν}{∂t}+u\frac{∂ν}{∂x}+ν\frac{∂ν}{∂y}+w\frac{∂ν}{∂z}\right)$$=-\frac{∂p}{∂y}+ρg_y+μ\left(\frac{∂^2ν}{∂x^2}+\frac{∂^2ν}{∂y^2}+\frac{∂^2ν}{∂z^2}\right)$

  • z-direction

(Eq 15) $ρ\left(\frac{∂w}{∂t}+u\frac{∂w}{∂x}+ν\frac{∂w}{∂y}+w\frac{∂w}{∂z}\right)$$=-\frac{∂p}{∂z}+ρg_z+μ\left(\frac{∂^2w}{∂x^2}+\frac{∂^2w}{∂y^2}+\frac{∂^2w}{∂z^2}\right)$

In the above equations $u$, $ν$, and $w$ represent the velocity components in the x, y, and z directions.  In addition, the acceleration terms of the the Navier-Stokes equations are on the left side of the equation.  On the other hand, the force terms are on the right side.  Furthermore, when the Navier-Stokes equations are combined with conservation of of mass, this will give you the ability to completely mathematically describe the flow of an incompressible Newtonian fluid.  However,  the Navier-Stokes equations are complex equations.  As a result, there are only a few instances where you will be able to have an exact mathematical solution.  Even so these equations are the governing differential equations of motion when you are working with an incompressible Newtonian fluid.

Finally, the Navier-Stokes equations can also be written in cylindrical polar coordinates.

  • r-direction

(Eq 16) $ρ\left(\frac{∂ν_r}{∂t}+ν_r\frac{∂ν_r}{∂r}+\frac{ν_θ}{r}\frac{∂ν_r}{∂θ}-\frac{ν_θ^2}{r}+ν_z\frac{∂ν_r}{∂z}\right)$$=-\frac{∂p}{∂r}+ρg_r+μ\left[\frac{1}{r}\frac{∂}{∂r}\left(r\frac{∂ν_r}{∂r}\right)-\frac{ν_r}{r^2}+\frac{1}{r^2}\frac{∂^2ν_r}{∂θ^2}-\frac{2}{r^2}\frac{∂ν_θ}{∂θ}+\frac{∂^2ν_r}{∂z^2}\right]$

  • $θ$-direction

(Eq 17) $ρ\left(\frac{∂ν_θ}{∂t}+ν_r\frac{∂ν_θ}{∂r}+\frac{ν_θ}{r}\frac{∂ν_θ}{∂θ}+\frac{ν_rν_θ}{r}+ν_z\frac{∂ν_θ}{∂z}\right)$$=-\frac{1}{r}\frac{∂p}{∂θ}+ρg_θ+μ\left[\frac{1}{r}\frac{∂}{∂r}\left(r\frac{∂ν_θ}{∂r}\right)-\frac{ν_θ}{r^2}+\frac{1}{r^2}\frac{∂^2ν_θ}{∂θ^2}+\frac{2}{r^2}\frac{∂ν_r}{∂θ}+\frac{∂^2ν_θ}{∂z^2}\right]$

  • $z$-direction

(Eq 18) $ρ\left(\frac{∂ν_z}{∂t}+ν_r\frac{∂ν_z}{∂r}+\frac{ν_θ}{r}\frac{∂ν_z}{∂θ}+ν_z\frac{∂ν_z}{∂z}\right)$$=-\frac{∂p}{∂z}+ρg_z+μ\left[\frac{1}{r}\frac{∂}{∂r}\left(r\frac{∂ν_z}{∂r}\right)+\frac{1}{r^2}\frac{∂^2ν_z}{∂θ^2}+\frac{∂^2ν_z}{∂z^2}\right]$


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