**3. Verification**

First, the accuracy of the Ladd method for processing moving boundaries is evaluated through the settlement process of circular particles in an infinitely long pipe. Then, the accuracy of the solute distribution model established in this paper is verified by calculating the solute field conservation of the single dendrite in the moving state. The LBM model, the dendrite growth of CA model, and the rationality of the single dendrite movement model have been confirmed in relevant researches [16,17], so this article will not be described here.
