# Is drag coefficient in fluid dynamics a constant in every direction?

I am referring the book 'Stochastic Processes in Cell Biology'. In this book the author says that Fokker Plank equation in isotropic diffusion: $$\frac{\partial p(x,t)}{\partial t}=-\frac{F}{\gamma}\frac{\partial p(x,t)}{\partial x}+D\frac{{\partial}^2 p(x,t)}{{\partial x}^2}$$ $\frac{F}{\gamma}=v$ where F is force, $\gamma$ is drag coefficient and $v$ is terminal velocity. For three dimensions the same book refer to Fokker Plank equation as: $$\frac{\partial p(\textbf{X},t)}{\partial t}=-\nabla\cdot [\frac{\textbf{F}}{\gamma}p(\textbf{X},t)]+D \nabla^2 p(\textbf{X},t), \label{FPE3}$$ where $\nabla^2$ is Laplacian operator. Now my question is, can I write $\frac{\textbf{F}}{\gamma}$ as equivalent terminal velocity in three dimensions which can be vectorized as:

$$\frac{\textbf{F}}{\gamma}=\textbf{v}=(u \textbf{i}+v \textbf{j}+w \textbf{k}),$$ Basically is drag coefficient a constant which on dividing force can give a three dimensional velocity?

• If the drag coefficient varied with direction, the diffusion coefficient would too. You would have to rewrite the FK equation with tensors. – Bert Barrois Feb 22 '18 at 20:10

If the force is the gradient of a potential energy function, $F=-\nabla V$, then the system should approach thermodynamic equilibrium, $p\propto \exp (-\beta V)$, which must be a time-independent solution to the F-P equation. This requires that ${{\gamma }^{-1}}=\beta D=D/kT$, in your notation.
With anisotropic and/or position-dependent diffusion, the equation should actually read: $$\dot{p}={{\partial }_{i}}[{{D}_{ij}}(\beta p{{\partial }_{j}}V+{{\partial }_{j}}p)]={{\partial }_{i}}[{{D}_{ij}}(-\beta p{{F}_{j}}+{{\partial }_{j}}p)]$$
• If I understand it right, you mean to say that if Diffusion is isotropic, i.e., \begin{bmatrix} D & 0 & 0 \\ 0 & D & 0 \\ 0 & 0 & D \end{bmatrix} Then I can write $$\frac{\textbf{F}}{\gamma}=\textbf{v}=(u \textbf{i}+v \textbf{j}+w \textbf{k}),$$ where $u,v,w$ are the velocities in three dimensions, derived by dividing a Force vector $\textbf{F}$ by constant drag coefficient $\gamma$. – Userhanu Feb 23 '18 at 6:00