In Bharatnatyam, like in any other classical dance form, there is a close relationship between rasa and bhava. In simple words, bhava is the sentiment and rasa is the emotion or love, which it develops into.
“Anubhavyate anen vagang krotobhinaya iti”: This shloka by Bharata simply defines the bhava as the constant state of mind or a sentiment. Rasa is the dominant emotion or feeling.
Eight rasas and forty nine bhavas are described by Bharat Muni in the Natya Shastra.

The eight sthayi bhava give rise to the eight corresponding rasas.

1. The sthayi bhava of Rati (passion of love) gives rise to the Shringara Rasa (erotic).
2. The sthayi bhava of Haasa (mirth) gives rise to the Hasya Rasa (laughter).
3. The sthayi bhava of Shoka (sorrow) gives rise to the Karuna Rasa (compassion).
4. The sthayi bhava of Krodha (anger) gives rise to the Raudra Rasa (fury).
5. The sthayi bhava of Utsaha (heroism) gives rise to the Veer Rasa (heroic).
6. The sthayi bhava of Bhaya (fear) gives rise to the Bhayanaka Rasa (fearful).
7. The sthayi bhava of Jugupsa (disgust) gives rise to the Vibhatsa Rasa (revulsion).
8. The sthayi bhava of Vismaya (astonishment) gives rise to the Adbhuta Rasa (wonder).

Bharata says “Vibhaavaanubhaav vyabhichaari samyogat ras nishpatti”, meaning, out of the combination (samayoga), of the determinants (vibhava), the consequents (anubhava) and the transitory mental status (vyabhichari), the birth of emotion (rasa) takes place.
Bharata describes it beautifully by giving an example of the stimulus – response situations in life. There can be stimulus from the environment, i.e. Vibhava. These cause a certain amount of excitation in a person, which is called Anubhav. For example, a woman upon coming to a garden may give a provocative glance or a smile. This excitement may also include any of the sattvika emotions like blushing, sweating, fainting etc. All this together emerges as Rasa, or, the relishable aesthetically experienced feeling or emotion.
Thus, we see that although there is only one kind of rasa, it can have different bhavas embedded in it. For example, in the dominant emotion or Rasa, say, Shringara Rasa, or love, there can be many bhavas or sentiments. On one hand it can be shown through bhavas of playing with your companion, the soothing effect of seasons, enjoyment of garlands and ornaments, enjoying and seeing the beloved, making love with your beloved etc. On the other hand, it could be shown in the form of separation, with bhavas of pain, agony, eagerness and anxiety. But in both the cases, the dominant emotion remains the same, i.e. Shringara Rasa.
Therefore, there can be no rasa without bhava and no bhava without rasa. Hence, in histrionics, they attain perfection only when they complement each other.