The river is called Yamuna and the goddess generally Kalindi in sources related to Krishna. She is also, according to some sources, a form of Nila Devi.
In an myth related to Krishna's birth, Krishna's father Vasudeva was carrying the new-born Krishna to safety was crossing the Yamuna River, he asked Yamuna to make a way for him to cross the river, which she did by creating a passage. This was the first time that she saw Krishna whom she marries in later life. Yamuna wanted to touch the feet of the baby which she did at deeper depths of the river and as a result the river became very calm.
Krishna also spent most of youth in Vrindavan on the banks of Yamuna, playing the flute and playing with his lover Radha and the gopis on the banks. It is said that Kalindi fell in love with Him as she saw His Eternal love with Radha.
The Bhagwata Purana narrates: Once, an adult Krishna visited his cousins the five Pandava brothers with their common wife Draupadi and their mother Kunti in their capital Indraprastha, located on the banks of the Yamuna. The eldest PandavaYudhistira requests Krishna to stay with them for a couple of days. One day, Krishna and the middle Pandava Arjuna go for hunting in the forest. During their hunting, Arjuna was tired. He and Krishna went to the Yamuna and bathed and drank the clear water. There, a lovely girl was strolling along the river bank. Krishna who saw her and asked Arjuna to meet her to know who she was. When Arjuna inquired, the girl told him that she was Kalindi, the daughter of Surya, and that she was living in a house constructed by her father in the river where she has been was performing austerities with intent to have Vishnu as her husband and would remain there, until she finds him. Arjuna conveys Kalindi's message to Krishna, the avatar of Vishnu, who readily agreed to marry the beautiful damsel. Then they traveled to Indraprastha with Kalindi in the chariot and met Yudhishthira. After a stay of few days there, Krishna and Kalindi returned to his capital Dwarka with their entourage and duly married each other .
Yamuna bountifully makes her love for Krishna accessible to all her devotees and with her divine powers increases their bhaktibhava (feeling for devotional love). She removes all obstructions and impurities that keep devotees from direct access to the divine love of Krishna. At Braj, the bluish-green waters of Yamuna assume a darker hue, indicating her divine union with the dark-skinned Krishna.
According to Bhagavata Purana she had ten sons: Shruta, Kavi, Vrsa, Vira, Subahu, Bhadra, Santi, Darsa, Purnamasa and the youngest, Somaka.The Vishnu Purana mentions that she had many sons headed by Shruta.
The Bhagavata Purana also narrates: Krishna's elder brother Balarama was staying in Ambadi on Yamuna's banks for a few months. Once, he was frolicking with the gopis on the river banks and desired to play in the waters. Intoxicated with liquor and experiencing heat of the alcohol, Balarama felt to take a bath in the river. However, he refused to walk to the waters and called upon the river to come near him, but the chaste Yamuna refused despite repeated orders from Balarama. An angry Balarama dragged the river by his weapon the plough and changed its course, hurting the river goddess. Terrified, the river assumed her form as a goddess and bowed to Balarama and asked his forgiveness. A calmed Balarama ordered the river to flood the forest so he could bathe and play in her waters, and the river complied.
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