Hypothetical Battle : Arjuna vs Achilles

Ever wondered who would win - Arjuna or 

Achilles, if ever faced off each other ? and 

how?

Note: People who get offended easily are well-advised to steer away from this post.

Arjun's vast army, with shitloads of infantry, elephants, and horse riders on one side. Arjun, of course, would've already sucked up to Lord Krishna, and asked him to drive his chariot. The latter would've agreed if only because Arjun's wife is his sister, and he doesn't want Arjun beating her up in anger, like Carlo Rizzi.
Meanwhile, the great Achilles, demi-god... the son of Thetis and Peleus. Wiith hundreds of ships. His uncontrollable rage. His absolute anger.

Achilles

Arjuna with Krishna as charioteer

Situation: I imagine that the fight would have been caused because Arjuna, who's known to covet women earnestly, picked up a girl Achilles set his sights on.

Achilles challenges him for a duel at some common ground, and Arjuna accepts it. As the large armies face off each other at the battlefield, Arjun suddenly develops cold feet.

The war is fairly archaic with chariots, horses, and armour. Both armies essentially cancel each other out. And of course, in the midst is Achilles smashing his way through heads like he were Popeye on a spinach high. He's enraged, and as we know, when Achilles is enraged; boy that's bad news. After going on one particular thousand man-killing-spree, he boasts, "Haha, my names is Achilles. I-kills people. Get it?"

On the other side, Krishna is rather bored with the proceedings as Arjun calmly slots arrow after arrow into people's heads. He uses various astras after murmuring their respective shlokas, and causes wide-scale destruction. On one occasion, he has several disarmed people in front of him. He says, "Kill me, uh-January. Kill me, uh-February. Go on." The scared army people follow on with the rest of the months, and when one person says, "Kill me, uh-June", beheads him with an arrow, and says, "That's me! You asked for it, baby!" Krishna stares at him in annoyance, but lets it go, as Arjun throws him a wry smile.

Eventually, Arjun and Achilles face off. They trash-talk each other for a while.
"I, who beat Hector, will emerge the victor," says Achilles out loud, cleverly rhyming Hector and victor.
Arjun, who doesn't know who Hector is, isn't sure what to make of it. Annoyed that his trash talking went waste, Achilles who'd done his research about Arjun, says, "You Karna-t beat me!" in a clear reference to Arjun's greatest rival.
Arjun stands there red in anger. Krishna, as usual, comes to his rescue and whispers something in his ear.
Arjun gathers his composure, and says, "But unlike you, at least I will Troy."
Krishna chuckles. Achilles lets out a loud roar.

This trash-talking gets them so worked up that they fight for several days, with Krishna falling asleep after day two of their arrows cancelling each other.
Arjun, however, becomes weaker and weaker with each passing day, while Achilles' fury keeps him up and about.
"Look," says Arjun, wearily, waking Krishna up. "I have a history of sharing women. If you don't mind, you can have this chick. Let's stop fighting."
This riles up Achilles more than Arjun expected, and makes him stronger than ever.
Just as Achilles is about to beat Arjun fair and square, just as he's about to win the well-deserved war, just as Arjun realises he's no match for a demi-god who's furious... Krishna, as usual, tilts the advantage in favour of Arjun by transforming himself in disguise as the woman Arjun and Achilles were fighting over. Just as Achilles pauses to admire her, Arjun shoots an arrow straight at Achilles' heel, who promptly falls down and dies.

"How did you know this?" asks Arjun.
Krishna who's transformed back to his real body now, lest Arjun get carried away, says, "I know everything."
Arjun, who knows he may need Krishna's help to drive his chariot again in some other war, sucks up to him majorly again by saying, "I salute you Krishna. You rock."
Krishna smiles and takes off.