# Help Deriving the Navier Stoke's Eq

• member 428835
In summary, the Navier-Stokes equations describe the motion of fluids and can be derived using the equations of motion and assumptions about the nature of the fluid. The viscous force is related to the rate of deformation tensor, which represents the rate at which material points in the fluid separate from each other. The R tensor, representing vorticity, is not included in the viscous force. The equations can also be applied to different fluids by considering their specific properties.
member 428835
hey pf!

i was hoping one of you could help me with deriving the navier stokes equations. i know the equations state something like this $$\sum \vec{F}=m\vec{a}$$ and for fluids we have something like this for the forces: $$\underbrace{-\iint Pd\vec{s}}_{\text{pressure}}+\underbrace{\vec{F}}_{\text{viscous forces}}-\underbrace{\iint \rho\vec{V}\vec{V}\cdot d\vec{s}}_{\text{momentum leaving flux}}$$ and for the acceleration terms we have $$\underbrace{\frac{\partial}{\partial t}\iiint \rho\vec{V} dv}_{\text{time rate of change of momentum}}$$

now ultimately we can use the divergence theorem, some vector/tensor identities, and arrive at the navier stokes equation: $$\rho \frac {D \vec{V}}{Dt} = - \nabla P + \mu \nabla^2 \vec{V}$$

my question is, can someone please explain to me how to arrive from my $\vec{F}$ viscous force to the $\mu \nabla^2 \vec{V}$

i'm good with the rest, but I'm just not sure how to start from an integral calculus perspective.

thanks!

joshmccraney said:
hey pf!

i was hoping one of you could help me with deriving the navier stokes equations. i know the equations state something like this $$\sum \vec{F}=m\vec{a}$$ and for fluids we have something like this for the forces: $$\underbrace{-\iint Pd\vec{s}}_{\text{pressure}}+\underbrace{\vec{F}}_{\text{viscous forces}}-\underbrace{\iint \rho\vec{V}\vec{V}\cdot d\vec{s}}_{\text{momentum leaving flux}}$$ and for the acceleration terms we have $$\underbrace{\frac{\partial}{\partial t}\iiint \rho\vec{V} dv}_{\text{time rate of change of momentum}}$$

now ultimately we can use the divergence theorem, some vector/tensor identities, and arrive at the navier stokes equation: $$\rho \frac {D \vec{V}}{Dt} = - \nabla P + \mu \nabla^2 \vec{V}$$

my question is, can someone please explain to me how to arrive from my $\vec{F}$ viscous force to the $\mu \nabla^2 \vec{V}$

i'm good with the rest, but I'm just not sure how to start from an integral calculus perspective.

thanks!
The viscous force per unit volume is $∇\centerdot \vec{σ}_v$, where $\vec{σ}_v$ is the viscous portion of the stress tensor, and is related to the kinematics of the deformation by
$\vec{σ}_v=2μ\vec{E}$
where $\vec{E}$ is the rate of deformation tensor:

$$\vec{E}=\frac{(∇\vec{v})+(∇\vec{v})^T}{2}$$

In this equation, $\vec{v}$ is the velocity vector.

Chestermiller said:
The viscous force per unit volume is $∇\centerdot \vec{σ}_v$, where $\vec{σ}_v$ is the viscous portion of the stress tensor, and is related to the kinematics of the deformation by
$\vec{σ}_v=2μ\vec{E}$
where $\vec{E}$ is the rate of deformation tensor:

$$\vec{E}=\frac{(∇\vec{v})+(∇\vec{v})^T}{2}$$

can you explain where $\vec{E}$ comes from and how it relates to the rate of deformation (this is not intuitive to me)? looking at wikipedia it appears we have two matrices; namely the $\vec{E}$ as you have listed and another, they name $\vec{R}$ defined as $\vec{R}=\frac{(∇\vec{v})-(∇\vec{v})^T}{2}$. I'm just not too sure what is going on or where these two are coming from ($\vec{E}$ and $\vec{E}$ that is)

thanks!

joshmccraney said:
can you explain where $\vec{E}$ comes from and how it relates to the rate of deformation (this is not intuitive to me)? looking at wikipedia it appears we have two matrices; namely the $\vec{E}$ as you have listed and another, they name $\vec{R}$ defined as $\vec{R}=\frac{(∇\vec{v})-(∇\vec{v})^T}{2}$. I'm just not too sure what is going on or where these two are coming from ($\vec{E}$ and $\vec{E}$ that is)

thanks!
The rate of deformation tensor relates to the rate at which the various material points in the fluid separate from one another as the fluid deforms. It's kind of a rate of stretching. The R tensor is the vorticity tensor, which relates to how rapidly fluid elements are rotating in the flow. This rotation does not contribute to the rate of separation of the material points. The relationship between the stress tensor and the rate of deformation tensor (in terms of the viscosity) is the 3D generalization of Newton's law of viscosity, and reduces to Newton's law of viscosity for the case of shear between parallel plates.

I have a similar question, but here I'm just interested in the derivation of the Stokes equation (Material Equation, etc):
Where a=dV/dt and V is a vector of d/dt(u,v,w) (i,j,k are respective here). But my fundamental question is why u, v, and w would be functions of (x,y,z,t)? I understand t and x being functions of, say u, because the u is defined as the velocity in the i direction.

Any help is appreciated.

j_phillips said:
I have a similar question, but here I'm just interested in the derivation of the Stokes equation (Material Equation, etc):
Where a=dV/dt and V is a vector of d/dt(u,v,w) (i,j,k are respective here). But my fundamental question is why u, v, and w would be functions of (x,y,z,t)? I understand t and x being functions of, say u, because the u is defined as the velocity in the i direction.

Any help is appreciated.

The x component of velocity u can also be a function of y and z. For example, suppose you have a "shear flow" where $u=γy$, where y is the distance from a horizontal wall, and γ is the "shear rate."

That was easy! Thanks much.

## 1. What is the Navier-Stokes equation?

The Navier-Stokes equation is a mathematical model that describes the motion of fluid substances, such as liquids and gases. It is used to predict the flow of fluids in various real-world scenarios, from air flow over an airplane wing to the circulation of blood in the human body.

## 2. What are the variables in the Navier-Stokes equation?

The Navier-Stokes equation includes several variables, including fluid velocity, pressure, and density. These variables are used to describe the flow of a fluid in a particular space and time.

## 3. How is the Navier-Stokes equation derived?

The Navier-Stokes equation is derived using a combination of conservation laws and assumptions about the behavior of fluids. It involves complex mathematical calculations and is often simplified for specific applications.

## 4. What are some real-world applications of the Navier-Stokes equation?

The Navier-Stokes equation is used in a wide range of fields, including aerospace engineering, meteorology, and oceanography. It is used to understand and predict fluid flow in various systems, from weather patterns to the flow of water in pipes.

## 5. What are the limitations of the Navier-Stokes equation?

The Navier-Stokes equation is a simplified model of fluid flow and has its limitations. It does not account for certain factors such as turbulence, compressibility, and viscosity. It is also a non-linear equation, making it difficult to solve for certain scenarios.

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