Sphere velocity in liquid-fluid mechanics

[omri3012]

Homework Statement

a rigid sphere of radius R and density $$\rho$$1 mooving (linearly) inside a liquid
of densiity $$\rho$$0 and viscosity constant $$\mu$$
the liquid is in hydrostatic equilibrium under gravitational field g.
what is the velocity of the sphere after a very long time assuming Raynolds constant is
very small?

Homework Equations

N-S equations

The Attempt at a Solution

my only problem is how should i deal with the sphere density $$\rho$$1, can i ignored it?
since the density in N-S equations is the liquid density.

Thankes,
Omri

 

[mark726]

Dear Omri,

Thank you for your post. it is important to consider all relevant factors in a problem, including the density of the sphere. In this case, it is not appropriate to ignore the density of the sphere in the Navier-Stokes equations.

The Navier-Stokes equations describe the motion of a fluid, and in this case, the liquid is the fluid. Therefore, the density in the equations should correspond to the density of the liquid, which is denoted by \rho0. However, the density of the sphere, \rho1, will affect the overall motion of the system.

To solve this problem, you will need to consider the forces acting on the sphere, including gravity and the drag force from the liquid. The drag force is dependent on the velocity of the sphere, which in turn is affected by the density of the sphere. Therefore, the density of the sphere cannot be ignored in this problem.

I recommend approaching this problem by using the conservation of momentum and the equation of motion for the sphere. This will allow you to account for the density of the sphere and determine the velocity of the sphere after a very long time.

I hope this helps. Best of luck with your calculations.