Motion of fluid can be described by Navier-Stokes Equations. In Lagrangian frame, the Navier-Stokes Equation takes form:

where ρ, p, v and g are density, pressure, velocity and body force (e.g. gravity). Θ is refers to viscous term.

With SPH method, incompressible fluid modeled by slightly compressible fluid where pressure can be directly calculated as function of density

here c and ρ₀ are speed of sound and initial pressure respectively as proposed in [5].

Heat transfer in material is governed by heat energy equation:

where c_p is specific heat, T is temperature and κ is coefficient of conductivity.

Equations (1), (2), (4) are solved numerically using SPH scheme, while Equation (3) is used directly to calculate pressure field.

An open non isothermal system can interact with the environment by heat transfer described by law of cooling.

The rate of change of heat energy to the system is proportional to temperature difference between fluid and environment T − T_env and area of the contact section. Here the free surface area represents contact section with the environment [7]. The heat flow into/from the free surface can be written as:

SPH scheme is used to discretize the governing Equation are based on integral of interpolation:

The value of function f₁ as function of position r can be calculated by integration of entire space. W is weighting kernel function with smoothing parameter h. This continuous form can be approximate by summation form for numerical work as

To approximate the value of some physical quantity f_a by its all surrounding neighbours f_b at the system boundaries. The gradient of f_a is

Using formulation (7) the governing Equations (1) and (2) now take forms, for complete derivation see [7].

Π_ab denote artificial viscosity for stabilization of the scheme.

The discretization of heat Equation is done by modified SPH scheme proposed by [3,4]

while the kernel used in this simulation is based on cubic spline kernel [7].

where and β = 0.7πh². Equations (7), (8) and (9) now are ordinary differential equation which could be solved by numerical integration. Meanwhile motion of solid body is describe by Newton Equation of motion

for translational motion and

for rotational motion.

In Equation (10) v_s denoted translational velocity of solid, m is mass of solid, g is gravity force and in Equation (10) and (11) is pressure of fluid nearest to the solid, da is cross sectional area of contact force, n is normal vector to solid surface. I and denoted moment of inertia and lever arm of torque respectively.

To apply Equation (5) we need to perform detection of particle at the edge this problem can be solved by calculating particle number density of fluid-solid system as illustrated in Figure 1.

To compute particle number density, an SPH scheme available:

Birds, et al. [8] use condition N_a < 0.84N_max for particles to be classified as free surface particle.

The simulations were run at QC-cluster in Department of Physics—ITB, the melting process simulation screenshot as seen in Figure 3 shows the effect of heat transferred into ice cube by surrounding waters and ambient temperature. In this simulation, we use 400 particles, 16% is ice particles and the other is water. The different temperature was setting in 10 K. The ice start melting after 0.2 second, and the ice will move around surrounding water.

The ice cube can move freely in translation or rotation, with this movement, more interaction between ice and water will happen and heat transfer from water or ambient air more active. As the result more ice particle will be change to water. Figure 3, shows this phenomena, in (c) and (d) more ice particle change to water particle.

To confirm the model and simulation with natural melting phenomena, we made three different temperature simulation as can be read in Table 1. The results show increasing of temperature difference between ice and

water, the ice more faster melting, as can be found in our daily observation.

Study on temperature difference effect is important to see the possibility of global warming. Bintanja et al. [9] reported that the global warming on Artic and Antartic make increasing sea level significant in late 5 years. In the Arctic, observations reveal that sea-ice area, extent and volume have been declining rapidly during the past decades, with the trend in extent being −5.3% per decade since 1985 and has reached as much as −27% per decade since 1995. Our simple simulation on difference temperature gives a good agreement to this phenomenon.

Figure 4 shows sequence of an ice melting in experiments apparatus. The dark color has lower temperature, as can be seen the ice has darkest color and ambient air has lighter color.

The melting of ice is not in linear form, as can be seen in Figure 5. Vertical axis shows the percentage of ice remains compared to original size of ice, while horizontal axis is time in seconds. The melting process can be seen with decreasing of ice part.

The melting process is relative slow until approximately 220 seconds in exponential form, and faster after that. Starting from that point, the ice melting in linear form until all ice melted. The simulation and experiment data show this process clearly.