I am struggling with the following problem:
A rigid sphere of radius $a$ is placed in a stream of fluid whose velocity in the undisturbed state is $V$. Determine the velocity of fluid at any point of the disturbed stream.
I have just started to learn fluid dynamics and I know the concepts of streamline, pathlines, equation of continuity etc. Is it possible to solve the above using these concepts?
Please help.
For steady, incompressible flow of a Newtonian fluid the velocity and pressure fields satisfy the Navier-Stokes equations:
$$\nabla \cdot \mathbf{u} = 0 \\ \rho \mathbf{u} \cdot \nabla \mathbf{u}= - \nabla p+ \mu\nabla^2\mathbf{u},$$
where $\rho$ is the density and $\mu$ is the viscosity.
We have boundary conditions: $\mathbf{u} \rightarrow \mathbf{V}$ far from the sphere and $\mathbf{u} = \mathbf{0}$ at the surface of the sphere. Using the radius of the sphere $a$ as the length scale and the magnitude of the far-field velocity $V$ as a velocity scale, the equations can be non-dimensionalized as
$$\nabla \cdot \mathbf{u} = 0 \\ Re\,\mathbf{u} \cdot \nabla \mathbf{u}= - \nabla p+ \nabla^2\mathbf{u},$$
where $Re = \rho V A / \mu$ is the Reynolds number.
You can obtain analytical solutions only in the limiting cases where $Re \rightarrow \infty$ (inviscid flow) and $Re = 0$ (creeping flow).
In general, for $0 < Re < \infty$, the partial differential equations must be solved numerically or approximately in terms of a perturbation expansion for small $Re$. Furthermore, as the Reynolds number increases -- unsteady, flow is observed empirically and the time derivative must be included in the equations. The wake exhibits a pattern of shed vortices and , ultimately, as the Reynolds number becomes larger, turbulent flow is observed. High-speed turbulent flow is difficult to simulate numerically as motion occurs on a multiplicity of length scales. In solving with a finite-difference method, for example, the number of mesh points becomes impractically large in order to resolve motion on the finest length scales.
In the case of creeping flow (zero Reynolds number) with $\mathbf{V} = V\mathbf{e}_z$, we can assume that the velocity field is axisymmetric. In terms of spherical coordinates $(r,\theta,\phi)$ (with origin at the center of the sphere), the azimuthal angular component of velocity $u_\phi = 0$ and the radial and polar angular components $u_r$ and $u_\theta$ depend only on $r$ and $\theta.$ Exact solution of the governing equations is facilitated by introducing a stream-function and eliminating the pressure term. This leads to the formula for the drag force known as Stoke's Law:
$$F_d = 6 \pi \mu a V\,\,\,\,(Re=0).$$