Cows produce milk in their mammary glands, which are located in the udder. The udder on a cow is found between the hind legs.
Milk is produced by the cow--and other female mammals, including humans--through a complex interaction of food consumption, and using the nutrients from the food, once her body's needs have been met, to filter into the mammary glands via the bloodstream. Reproductive hormones including oxytocin are produced which encourages a cow to produce milk. Cows, like all female mammals, primarily only produce milk immediately after they've given birth.
Milk is produced via milk alveoli, which are basically modified sweat glands, located as part of the mammary system in the udder. These glands take nutrients from the blood, like water, vitamins and minerals, and immunoglobins, and also synthesize various compounds (such as casein and lactose)--as well as create a transport system for other compounds (proteins, long- and short-chain fatty acids) to come through the cell wall into the alveoli--and basically combine them all to make milk.
The milk is then transported through various serious of chambers and canals to the teats. From the teat canal, the milk is released from the suckling action of the calf, the skilled fingers of the hand-milker or the gentle vacuum pressure of the milk machine suction cups.
The Bovine Practicum Q&A "blog" is an informative, just-for-fun section where I find a variety of questions that are often (and not-so-often) asked by inquisitive people like you, and answer them to the best of my ability. Much of the questions have come from a wide variety of reaches from the Internet.
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