Derivation of Viscous Force

Question

For viscous drag, the formula for the force is,

$$ \mathbf F=\pm\eta A\nabla\mathbf u $$ where $\eta$ is the viscosity coefficient, $A$ the area and $\nabla\mathbf u$ the velocity gradient. How is this formula derived?

Answer 1

Personally, i think understanding the fundamentals of that equation is beyond that of a 11th grader but i will give it a go.

I'm going to start with something which seems completely unrelated; a warm house losing heat to its colder surroundings. Assuming no wind is blowing outside, the difference between the temperature inside and outside drives the heat flow $j$, this is know as Newton's law of cooling and is quantified as: $$j \propto \frac{\Delta T}{\Delta x}$$

Obviously, the larger the temperature difference ($\Delta T$ becomes larger), the more heat will be lost ($j$ will become larger) and vice versa. On the other hand, if the wall thickness is increased ($\Delta x$ becomes larger), we insulate the house more and less heat will be lost ($j$ will become smaller). The proportionality constant $k$ is known as the thermal conductivity and describes how well a material (such as a wall) conducts heat.

The above equation is an example of heat diffusion which is a process in which molecules exchange heat by colliding with each other. Molecules have larger kinetic energies at higher temperatures and when they collide with molecules at smaller kinetic energies, some of the kinetic energy is transferred. This transfer goes from high to low kinetic energies (and temperatures) as described by the above equation and fundamentally is known as Fourier's law of diffusion.

Now heat is not the only quantity which can diffuse, e.g. molecules of different types exhibit diffusion also from places of high concentration to places of low concentrations. Imagine releasing a colored volume of air in the atmosphere; disregarding winds again, the color will quickly fade due to mixing with transparent air. This is again explained by the motion of molecules as colored molecules on average tend to move away from the colored patch and so their transport is analogous to the above equation: $$j\propto \frac{\Delta c}{\Delta x}$$ this is known as Fick's law of diffusion. The proportionality constant here is known as the diffusion constant and describes the rate at which a molecule moves a certain distance.

Now finally, besides heat and mass, the final quantity which can diffuse is momentum and is the basis of the equation you are looking for. Unfortunately, diffusion of momentum is difficult to explain which is why i draw the parallels with heat and mass transfer. Diffusion of momentum occurs like heat and mass from places of high momentum to low momentum according to Newton's law of viscosity: $$j_{xy}\propto \frac{\Delta v_x}{\Delta y}$$ The proportionality constant is known as the viscosity and describes how well momentum is transferred from molecule to molecule.

Often the flux $j_{xy}$ is written as $\tau_{xy}$ and is given the name 'shear-stress' because it is the amount of stress (force per area) two layers of fluid exert on each other when they are sheared. The quantity $\Delta v_x/\Delta y$ is known as the 'shear rate' and describes how a fluid parcel deforms in the $y$-direction as a shear is applied in the $x$-direction. The amount of deformation is directly related to how well momentum is transferred.

Since the shear-stress is effectively a friction force per surface area you could find the friction force by multiplying the shear-stress with the surface area. This provides you with the equation for the magnitude of the force between liquid layers you have in your textbook: $$F=\tau A=\mu A \frac{\Delta v_x}{\Delta y}$$

Answer 2

The original equation you wrote is incorrect. The 3D version of Newton's law of viscosity expresses the stress tensor in terms of the velocity gradient tensor as follows:

$$\vec{\sigma}=-p\vec{I}+\eta (\vec{\triangledown u}+(\vec{\triangledown u})^T)$$ where p is the pressure, $\vec{I}$ is the isotropic identity tensor, and the superscript T indicates the transpose of a tensor. Note that the equation includes the transpose of the velocity gradient tensor in addition to the velocity gradient tensor itself. Inclusion of the transpose of the velocity gradient tensor is necessary to remove any local rigid body rotation of the fluid elements, which does not constitute a deformation of the fluid, and thus, does not contribute to the stress in the fluid.