Stokes Flow

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STOKES EQUATIONS For this type of flow, the inertial forces are assumed to be negligible and the Navier-Stokes Equations simplify to give the Stokes equations: :$\backslash boldsymbol\{$ abla} \cdot \mathbb{P} + \boldsymbol{f} = 0 where $\backslash mathbb\{P\}$ is the Comoving Stress Tensor , and $\backslash boldsymbol\{f\}$ an applied body force. There is also an equation for Conservation Of Mass . In the common case of an incompressible Newtonian Fluid , the Stokes equations are: :$\backslash boldsymbol\{$ abla}p = \mu abla^2 \boldsymbol{u} + \boldsymbol{f} :$\backslash boldsymbol\{$ abla}\cdot\boldsymbol{u}=0 Properties The Stokes Equations represent a considerable simplification of the full Navier-Stokes Equations , especially in the incompressible Newtonian case. In this case, the equations are: ; Instantaneity :A Stokes flow has no dependence on time other than through time-dependent Boundary Condition s. This means that, given the boundary conditions of a Stokes flow, the flow can be found without knowledge of the flow at any other time. ; Time-reversibility :An immediate consequence of instantaneity, time-reversibility means then a time-reversed Stokes flow solves the same equations as the original Stokes flow. This property can sometimes be used (in conjunction with linearity and symmetry in the boundary conditions) to derive results about a flow without solving it fully. While these properties are true for incompressible Newtonian Stokes flows, the non-linear and sometimes time-dependent nature of Non-Newtonian Fluid s means that they do not hold in the more general case. Methods of solution By Streamfunction It can be shown that in 2-D, the streamfunction for an incompressible Newtonian Stokes flow satisfies the Biharmonic Equation $$ abla^4 \psi = 0. In the 3-D axisymmetric case, the streamfunction $\backslash Psi$ solves the equation $E^2\; \backslash Psi\; =\; 0$, where $E\; =\; \{\backslash partial^2\; \backslash over\; \backslash partial\; r^2\}\; +\; \{\backslash sin\{\; heta\}\; \backslash over\; r^2\}\; \{\backslash partial\; \backslash over\; \backslash partial\; heta\}\; \{\; 1\; \backslash over\; \backslash sin\{\; heta\}\}\; \{\backslash partial\; \backslash over\; \backslash partial\; heta\}.$ By Papkovich-Neuber Solution The Papkovich-Neuber Solution represents the velocity and pressure fields of an incompressible Newtonian Stokes flow in terms of two Harmonic potentials. By Boundary Element Method Certain problems, such as the evolution of the shape of a bubble in a Stokes flow, are conducive to numerical solution by the Boundary Element method. This technique can be applied in both 2- and 3-dimensional flows. By Green's Function

  :<math>p(\boldsymbol{x}) \int rac{\boldsymbol{f}(\boldsymbol{y})\cdot(\boldsymbol{x}-\boldsymbol{y})}{4 \pi \boldsymbol{x}-\boldsymbol{y}^3} \, \mathrm{d}^3\!y</math>