I am following this document on Stokes flow. It is stated that "if we have a falling sphere, doubling the velocity will double $\sigma_{ij} (= -p\delta_{ij} + 2\mu e_{ij})$", but I am having trouble seeing why, since in the definition of $\sigma_{ij}$ we have the term $-p\delta_{ij}$. I can see why the rest of the formula for $\sigma_{ij}$ doubles, but not this problematic pressure term. Can anyone explain?
The equations in this case are those of Stokes (Navier-Stokes with no inertia term). Here $p$ denotes pressure and $e_{ij}$ the strain rate tensor.
Two other questions I have about the more general Stokes flow: It is also stated that flow responds instantly to boundary motion, because there are no time derivatives of the velocity. Can anyone provide an illustrative example of this, as I am having a hard time visualising what responding instantly to boundary motion means? Finally, what is meant by a flow being "forced" by a boundary motion?
In component form, the Stokes equations are
$$\mu \sum_{j=1}^3\frac{\partial ^2 u_i}{\partial x_j^2} = \frac{\partial p}{\partial x_i} - F_i, \quad (i=1,2,3)$$
Using a velocity scale $U_0$ and length scale $L$, we define the dimensionless variables $\hat{u}_i$ and $\hat{x}_j$ by
$$u_i = U_0\hat{u}_i,\quad x_j = L \hat{x}_j$$
Substituting into Stokes equations we get
$$\frac{\mu U_0}{L^2}\sum_{j=1}^3\frac{\partial ^2 \hat{u}_i}{\partial \hat{x}_j^2} = \frac{1}{L}\frac{\partial p}{\partial \hat{x}_i} - F_i, \\ \implies\sum_{j=1}^3\frac{\partial ^2 \hat{u}_i}{\partial \hat{x}_j^2} =\frac{L}{\mu U_0}\frac{\partial p}{\partial \hat{x}_i} - \frac{L^2}{\mu U_0}F_i$$
With the characteristic pressure $p_c = \frac{\mu U_0}{L}$ and characteristic body force $F_c = \frac{\mu U_0}{L^2}$ we can define dimensionless variables $\hat{p} = p/p_c$ and $\hat{F}_i = F_i/F_c$ and the Stokes equations reduce to
$$\sum_{j=1}^3\frac{\partial ^2 \hat{u}_i}{\partial \hat{x}_j^2} = \frac{\partial \hat{p}}{\partial \hat{x}_i} - \hat{F}_i$$
Solving the dimensionless PDE we obtain solutions for the dimensionless velocity and pressure fields that are independent of the characteristic scales.
The constitutive equation for the components of the stress tensor is
$$\sigma_{ij} = -p\delta_{ij} + 2\mu e_{ij} = -p\delta_{ij} + \mu\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) \\ = -p_c\hat{p}\delta_{ij} + \frac{\mu U_0}{L}\left(\frac{\partial \hat{u}_i}{\partial \hat{x}_j} + \frac{\partial \hat{u}_j}{\partial \hat{x}_i}\right)\\ = \frac{\mu U_0}{L}\left(-\hat{p}\delta_{ij} + \frac{\partial \hat{u}_i}{\partial \hat{x}_j} + \frac{\partial \hat{u}_j}{\partial \hat{x}_i} \right)$$
Since the expression inside the parentheses is independent of the velocity scale $U_0$, it is clear that doubling $U_0$ will double $\sigma_{ij}$.