Stokes flow In
Stokes flow, at very low
Reynolds number, the
convective acceleration terms in the
Navier–Stokes equations are neglected. Then the flow equations become, for an
incompressible steady flow: :\begin{align} &\nabla p = \mu\, \nabla^2 \mathbf{u} = - \mu\, \nabla \times \mathbf{ \boldsymbol{\omega} }, \\[2pt] &\nabla \cdot \mathbf{u} = 0, \end{align} where: • is the
fluid pressure (in Pa), • is the
flow velocity (in m/s), and • is the
vorticity (in s−1), defined as \boldsymbol{\omega}=\nabla\times\mathbf{u}. By using some
vector calculus identities, these equations can be shown to result in
Laplace's equations for the pressure and each of the components of the vorticity vector: : u_z = \frac{1}{r}\frac{\partial\psi}{\partial r}, \qquad u_r = -\frac{1}{r}\frac{\partial\psi}{\partial z}, with and the flow velocity components in the and direction, respectively. The azimuthal velocity component in the –direction is equal to zero, in this axisymmetric case. The volume flux, through a tube bounded by a surface of some constant value , is equal to and is constant. : \omega_\varphi = \frac{\partial u_r}{\partial z} - \frac{\partial u_z}{\partial r} = - \frac{\partial}{\partial r} \left( \frac{1}{r}\frac{\partial\psi}{\partial r} \right) - \frac{1}{r}\, \frac{\partial^2\psi}{\partial z^2}. The
Laplace operator, applied to the vorticity , becomes in this cylindrical coordinate system with axisymmetry: : \psi(r,z) = - \frac{1}{2}\, u\, r^2\, \left[ 1 - \frac{3}{2} \frac{R}{\sqrt{r^2+z^2}} + \frac{1}{2} \left( \frac{R}{\sqrt{r^2+z^2}} \right)^3\; \right]. The solution of velocity in
cylindrical coordinates and components follows as: :\begin{align} u_r(r, z) &= \frac{3R r z u}{4 \sqrt{r^2 + z^2}} \left( \left( \frac{R}{r^2+z^2} \right)^2 - \frac{1}{r^2+z^2} \right) \\[4pt] u_z(r, z) &= u + \frac{3Ru}{4 \sqrt{r^2 + z^2}} \left( \frac{2 R^2 + 3 r^2}{3 (r^2 + z^2)} -\left(\frac{r R}{r^2 + z^2}\right)^2 - 2 \right) \end{align} The solution of vorticity in cylindrical coordinates follows as: :\omega_\varphi(r, z) = - \frac{3 Ru}{2} \cdot \frac{r}{\sqrt{r^2+z^2}^3} The solution of pressure in cylindrical coordinates follows as: :p(r, z) = - \frac{3 \mu R u}{2} \cdot \frac{z}{\sqrt{r^2+z^2}^3} The solution of pressure in
spherical coordinates follows as: : p(r, \theta) = - \frac{3\mu R u}{2} \cdot \frac{\cos\theta}{r^2} The formula of pressure is also called
dipole potential analogous to the concept in electrostatics. A more general formulation, with arbitrary far-field velocity-vector \mathbf{u}_{\infty}, in
cartesian coordinates \mathbf{x}= (x, y, z)^T follows with: \begin{align} \mathbf{u}(\mathbf{x}) &= \underbrace{\underbrace{\frac{R^3}{4} \cdot \left(\frac{3 \left(\mathbf{u}_{\infty} \cdot \mathbf{x}\right)\cdot \mathbf{x}}{\|\mathbf{x}\|^5} - \frac{\mathbf{u}_{\infty}}{\|\mathbf{x}\|^3} \right)}_{\text{conservative: curl=0,}\ \nabla^2\mathbf{u}=0} + \underbrace{\mathbf{u}_{\infty}}_{\text{far-field}}}_{\text{Terms of Boundary-Condition}} \; \underbrace{- \frac{3R}{4}\cdot\left( \frac{\mathbf{u}_{\infty}}{\|\mathbf{x}\|} + \frac{\left(\mathbf{u}_{\infty} \cdot \mathbf{x}\right)\cdot \mathbf{x}}{\|\mathbf{x}\|^3} \right)}_{\text{non-conservative: curl}=\boldsymbol{\omega}(\mathbf{x}),\ \mu\nabla^2\mathbf{u}=\nabla p} \\[8pt] &= \left[ \frac{3R^3}{4} \frac{\mathbf{x\otimes\mathbf{x}}}{\|\mathbf{x}\|^5} - \frac{R^3}{4} \frac{\mathbf{I}}{\|\mathbf{x}\|^3} - \frac{3R}{4} \frac{\mathbf{x}\otimes\mathbf{x}}{\|\mathbf{x}\|^3} - \frac{3R}{4} \frac{\mathbf{I}}{\|\mathbf{x}\|} + \mathbf{I} \right]\cdot \mathbf{u}_{\infty} \end{align} :\boldsymbol{\omega}(\mathbf{x}) = - \frac{3R}{2} \cdot \frac{\mathbf{u}_{\infty}\times \mathbf{x}}{\|\mathbf{x}\|^3} :p\left(\mathbf{x}\right)= - \frac{3\mu R}{2} \cdot \frac{\mathbf{u}_{\infty} \cdot \mathbf{x}}{\|\mathbf{x}\|^3} In this formulation the
non-conservative term represents a kind of so-called
Stokeslet. The Stokeslet is the
Green's function of the Stokes-Flow-Equations. The conservative term is equal to the
dipole gradient field. The formula of vorticity is analogous to the
Biot–Savart law in
electromagnetism. Alternatively, in a more compact way, one can formulate the velocity field as follows: :\mathbf{u}(\mathbf{x}) = \left[ \mathbf{I} + \mathrm{H} \left(\frac{R^3}{4}\frac{1}{\|\mathbf{x}\|}\right) - \mathrm{S}\left(\frac{3R}{4} \|\mathbf{x}\|\right) \right] \cdot \mathbf{u}_{\infty} , \quad \|\mathbf{x}\| \ge R, where \mathrm{H} = \nabla\otimes\nabla is the Hessian matrix differential operator and \mathrm{S} = \mathbf{I} \nabla^2 - \mathrm{H} is a differential operator composed as the difference of the Laplacian and the Hessian. In this way it becomes explicitly clear, that the solution is composed from derivatives of a
Coulomb potential (1/\|\mathbf{x}\|) and a
Biharmonic potential (\|\mathbf{x}\|). The differential operator \mathrm{S} applied to the vector norm \|\mathbf{x}\| generates the Stokeslet. The following formula describes the
viscous stress tensor for the special case of Stokes flow. It is needed in the calculation of the force acting on the particle. In
Cartesian coordinates the vector-gradient \nabla \mathbf{u} is identical to the
Jacobian matrix. The matrix represents the
identity matrix. :\boldsymbol{\sigma} = - p \cdot \mathbf{I} + \mu \cdot \left((\nabla \mathbf{u}) + (\nabla \mathbf{u})^T \right) The force acting on the sphere can be calculated via the integral of the stress tensor over the surface of the sphere, where represents the radial unit-vector of
spherical-coordinates: :\begin{align} \mathbf{F} &= \iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \;\boldsymbol{\sigma}\cdot \text{d}\mathbf{S} \\[4pt] &= \int_{0}^{\pi}\int_{0}^{2\pi} \boldsymbol{\sigma}\cdot \mathbf{e_r}\cdot R^2 \sin\theta \text{d}\varphi\text{d}\theta \\[4pt] &= \int_{0}^{\pi}\int_{0}^{2\pi} \frac{3\mu \cdot \mathbf{u}_{\infty}}{2 R}\cdot R^2 \sin\theta \text{d}\varphi\text{d}\theta \\[4pt] &= 6\pi\mu R \cdot \mathbf{u}_{\infty} \end{align}
Rotational flow around a sphere :\begin{align} \mathbf{u}(\mathbf{x}) &= - \;R^3 \cdot \frac{ \boldsymbol{\omega}_{R} \times \mathbf{x}}{\|\mathbf{x}\|^3} \\[8pt] \boldsymbol{\omega}(\mathbf{x}) &= \frac{R^3 \cdot \boldsymbol{\omega}_{R}}{\|\mathbf{x}\|^3} - \frac{3 R^3 \cdot (\boldsymbol{\omega}_{R} \cdot \mathbf{x})\cdot \mathbf{x}}{\|\mathbf{x}\|^5} \\[8pt] p(\mathbf{x}) &= 0 \\[8pt] \boldsymbol{\sigma} &= - p \cdot \mathbf{I} + \mu \cdot \left( (\nabla \mathbf{u}) + (\nabla \mathbf{u})^T \right) \\[8pt] \mathbf{T} &= \iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \mathbf{x} \times \left( \boldsymbol{\sigma} \cdot \text{d}\boldsymbol{S} \right) \\ &= \int_{0}^{\pi} \int_{0}^{2\pi} (R \cdot \mathbf{e_r}) \times \left( \boldsymbol{\sigma} \cdot \mathbf{e_r} \cdot R^2 \sin\theta \text{d}\varphi \text{d}\theta \right) \\ &= 8\pi\mu R^3 \cdot \boldsymbol{\omega}_{R} \end{align} ==Other types of Stokes flow==