How is melting time affected by flow rate and temperature of surroundings? Suppose you have a solid sphere of m, where m is an element with freezing point of 0 degrees Celsius.
In one scenario, you place your sphere in a (“static”) 25 degree Celsius environment and measure time, t, until melting. The sphere is fixed and cannot be displaced.
In the other, you place your sphere in environment with temperature, T, and with constant flow rate, v. Again, you measure time, t, until melting.
What is the equation that would relate the two scenarios? In other words, at what temperature and flow rate would time required for melting in the second scenario equal time required in the first?
 A: The answer to this is very subtle, and is the core subject of interest in convective heat transfer. In either case, you’ll find that most engineers would model either scenario using Newton’s law of cooling:
$$Q = hA(T-T_{\infty})$$
where $Q$ is the heat transfer rate, $A$ is the surface area of the object in contact with its surroundings, $T$ is the temperature of the object and $T_{\infty}$ is the (approximate) temperature of the surroundings. $h$ is a sort of catchall term called the “heat transfer coefficient”, which is affected by all sorts of things—in particular, by flow in the surroundings of the embedded object. Most engineers find this coefficient through empirical studies.
That being said, flow in general increases the amount of heat transfer, and so an object embedded in surroundings at a different temperature & a uniform flow will heat up/cool down to the surrounding temperature faster than without the flow.
In the case without flow, temperature gradients will actually cause flow themselves by changing the density of the fluid near the object with a different temperature, so there will still be some minor convective heat transfer—this is usually called natural convection.
A: For the first case the differential equation for evolution of temperature of the sphere
$$ 
 m * C_p * \frac{dT_m}{dt} =  h_{nat} (T_{amb} - T_s) \\
$$
$$
\begin{array}
 \text{where} \\
       m  & \text{mass of of the sphere}       \\
       C_p  & \text{Specific heat of the solid} \\
       T_m  &  \text{Mean temperature of the sphere}  \\ 
       T_s  &  \text{Surface temperature of the sphere} \\
       T_{amb} &  \text{Ambient temperature} \\
       h_{nat} & \text{Heat transfer coeff. (natural convection)} \\
\end{array}
$$
The above combined with internal transient conduction equation for the sphere
with thermal conductivity (k)
$$   \frac{\partial T}{\partial t} = k \nabla ^2T $$
should provide necessary equations to determine the temporal and spatial variation of the sphere over time. I have omitted other gory details of boundary and initial conditions here. Under certain conditions, one can omit the above equation and assume that sphere temperature is uniform. (high thermal conductivity and and small heat flux at the surface of sphere)
Now it is possible to evaluate the second case, simply replacing the $h_{nat}$ with appropriate forced convection heat transfer coefficient. In general for air forced convection heat transfer coefficient is proportional to $v^{0.8}$
A: In the static case, you need to give a better definition of the problem.  How big is the container that the ice sphere resides in?  Are the walls of the container insulated, or can they exchange heat with the environment?  If heat exchange occurs with the environment, what are the container walls made of, what is their thermal conductivity, is the container in shade, etc.?  Does the melted water "puddle up" around the bottom of the sphere, or is it drained in some way?  Is the ice sphere surrounded by air, water, or something else?  What is the initial temperature of the material surrounding the ice sphere?
For the dynamic case, what is flowing around the sphere, what is its temperature, and how fast is velocity "v"?  At very low velocities, you will have laminar flow, whereas at somewhat higher velocities, you will have turbulent flow.  Turbulence is one of the huge unsolved problems in physics, and no equations currently exist for this phenomenon.  Due to this, practical heat transfer problems are very dependent on the geometry of the situation, flow rates, etc., which means that a lot of empirical equations have been developed for very specific applications.  Your problem will almost certainly require the collection of a lot of data for your specific geometry and details, such that you can develop an empirical equation for this one case.
