When prana moves, chitta (the mental force) moves. When prana movement seizes, so does the movement of chitta - Svātmārāma

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The purpose of all pranayama practices is to create a perfectly still state of body and mind, so that inhalation and exhalation stop with the cessation of pranic movement. Before we dwell into the intrinsic details of practice, lets understand how the breath flows in our body. As we all know, breathing keeps the body and mind alive. Every breath we take simulates these five vayus, that regulate the functioning of the body and mind.
Five Vayus
Prana is upward flowing vayu from heart region to the head. It nourishes brain and controls the mental activites.
Apana is downward flowing vayu from navel region to the perinium. This is responsible for excretory functions and reproduction.
Samana is inward flowing from periphery of the body towards the pranic center (navel region). This regulates digestion.
Vyana is outward flowing from pranic center to the periphery of the body. This takes care of supplying the nourishment to all the cells of the body and hence prevails all over the body.
Udana is upward flowing energy from the throat region to head, arms and legs. This vayu is the result of right functioning of other vayus. This is responsible for growth and expression (all forms of communication).
The importance and function of these vayus are explained in detail in this five vayus section.

Pranayama is meticulously controlling these vayus to obtain mastery over the breath, hence taking control over aging of the body, mental patterns, emotions, expression and intelligence.

Raising the apana upward and bringing the prana down from the throat, yogi becomes free from old age - Svātmārāma (Pranayama - Verse 47)

In general, breath flows through left nostril (ida nadi) or right nostril (pingala nadi). Science calls it ‘alternate rhinitis.’ In yoga it is known as swara. This keeps alternating every 90 minutes in a healthy body.
Pingala Nadi Ida Nadi
When we breath through right nostril, pingala nadi is active. When the breath is flowing through the left nostril, ida nadi is predominant.
It is connected to the left brain hemisphere and it processes information logically, sequentially and functions according to time sequence.  It is connected to the right hemisphere and hence is concerned with intuition, mental creativity and orientation in space.
The body functions under dynamic masculine principle. (irrespective of actual gender, everybody oscillate between masculine and feminine principles). Mind is extroverted and body is heated. Passive feminine principle governs. This is denoted as sakthi (feminine) and shiva (for masculine) in the east. It has cooling effect on body and the mind is introverted.
During this time, sun is said to be governing the body and rajas guna (over-active temperament) is exhibited by the person. This is the right time to do physically active work like waking-up, workout, celebration, eating, etc. Moon governs the body and tamas guna (inert temperament) prevails. Apt time for subtle activities like taking a nap, desk-work, art-work, etc. Basically, physical rest and creative work.
When both nostrils operate simultaneously the energy is being transferred from one hemisphere to the other. It passes through a thin sheet of membrane between the two hemispheres called the corpus callosum. At this time the whole brain can function and perception will not be limited to one mode of processing. This happens naturally when the breath flow changes from left to right, but lasts just few seconds or lesser. These are the moments of higher intelligence.

With basic practice of pranayama, this balance can be obtained and retained for minutes. It takes a sustained and dedicated practice to get the lasting effect, that is when both ida and pingala get dormant and the prana flow through susumana nadi (middle channel). When a person can retain this state and can make the yoga (join) between jiva-atma (soul) and parama-atma (total conciousness), he/she is enlightened.

Pranayama does purify all the 72,000 nadis (energy channels), but that is not the sole purpose of pranayama. There is an efficient way to purify the nadis as preparation, so the real benefits of pranayama is obtained. It is called shat-karmas or shat-kriyas. These six methods of cleansing balances the three doshas (bodily humor), namely, vatha (wind), pitta (bile) and kapha (mucus). When these are not balanced, it'll be a huge hinderance to practice pranayama.
Shat-karmas (or) Shat-kriyas
Dauti (esophagus and stomach cleansing). Removing the excess mucus from esophagus and stomach and thus hinderances to pranayama like, coughs, asthma, diseases of the spleen, leprosy and twenty kinds of diseases caused by excess mucus are destroyed.
Basti (yogic enema) cleanses large intestine. It is not considered a common practice as it involves sucking water through rectum and releasing. Enlargement of the glands and spleen, and all diseases arising from excess mind, bile and mucus are eliminated from the body through the practice of basti. 
Neti (nasal cleansing). Cleanses the nasal passage, between two nostrils and between the nostrils and mouth. There are two ways of doing it; sutra (thread) neti and jala (water) neti. Neti cleanses the cranium and bestows clairvoyance. It also destroys all diseases which manifest above the throat.
Trataka (concentrated gazing, no blinking). Looking intently with an unwavering gaze at a small point until tears are shed is known as trataka. Commonly practiced with candle light, it can be practiced with moon, stars, idol, dot, etc. Trataka eradicates all eye diseases, fatigue, sloth, depression, insomnia, allergy, anxiety, postural problems, poor concentration, memory and closes the doorway causing these problems.
Nauli (massaging internal abdominal organs). Lean forward, protrude the abdomen and rotate (the muscles) from right to left with speed. This is called nauli by the siddhas. Nauli kindles the digestive fire, removing indigestion, sluggish, digestion, and all disorders of the doshas.
Kapalbathi (brain frontal lobe cleansing). Perform exhalation (forced) and inhalation (not forced) rapidly 108 times. This is called kapalbhati and it destroys all mucous disorders.
These six methods are discussed in detail in the internet. By the shat-karma one is freed from excesses of the doshas. Then pranayama is practiced and success is achieved without strain.

Pranayama constitutes of 3 parts:  pooraka (inhalation), rechaka (exhalation), and kumbhaka (retention). Kumbhaka is again of two types: sahita (with conscious effort) and kevala (automatic). Until kevala kumbhaka is perfected, sahita kumbhaka (retention with conscious effort) has to be practiced.

General guidelines for inhale:retain:exhale is 1:4:2. i.e., if you inhale for 2 seconds, retain for 8 seconds and exhale slowly for 4 seconds.

Suryabheda (right nostril inhalation) means piercing the body with solar energy. In empty stomach, sitting comfortably in siddhasana, the yogi should become fixed in his posture and slowly breathe the air in through the right nostril. Retain until you feel the breath diffuse to the roots of the hair and tips of the nails. Then slowly exhale through the left nostril. Repeat it 10 times.

Suryabheda is excellent for purifying the cranium, destroying imbalances of the wind dosha and eliminating worms. In general, when we inhale only through left nostril, it is exhaled without retention. The retention in practiced only when we inhale through the right or both nostrils.

Bhastrika (bellows breathing). Sit in padmasana to ensure sturdy posture, neck and abdomen in alignment, inhale and exhale in like pumping a ball or a tire. First start with 20 pairs of inhalation and exhalation. Then take a deep breath with right-nostril and retain breath for a count of 40 and slowly exhale through left nostril. Gradually, in weeks, increase it to 50:50:100 (inhale:exhale and retain). Repeat it 5 times. imbalances of vatha (wind), pitta (bile) and kapha (mucus) are annihilated and the digestive fire is increased.

Padmasana is emphasized here because the body is to be firmly locked so that physical movement is restricted and the spine remains straight. Nervous impulses are then able to travel directly up through the central nervous system. Bhastrika stimulates the circulation of cerebral fluid and increases the compression and decompression upon the brain, creating a rhythmic massage. The rhythmic pumping of the diaphragm and lungs stimulates the heart and blood circulation. Accelerated blood circulation and rate of gas exchange in each cell produces heat and washes out waste gases. In advanced sadhakas (practioners), bhastrika unties the three granthis (pranic knots), namely, rudra granthi, vishnu granthi and bramha granthi. These 3 knots will be discussed in detail in kundalini chapter.

Ujjayi (victorious deep breathing). Sit in siddhasana or lie down in savasana. Closing the mouth, inhale with control and concentration through both ida (left) and pingala (right), so that the breath is felt from the throat to the heart and produces a sonorous sound. Do kumbhaka (retain) as before and exhale through ida (left). Continue for as long as you can. 10 mins will be apt. This removes phlegm from the throat and stimulates the (digestive) fire.

Seetkari (hissing breath) and Sheetali (cooling breath):
Seetkari: Sit in Siddhasana, get in chin-mudra (tip of index finger and tip of thumb touching) and Inhale through the mouth, clenched teeth, making a hissing sound and exhale through the nose. repeat it 20 times.
Sheetali: Sit in Siddhasana, get in chin-mudra (tip of index finger and tip of thumb touching) and Inhale through the mouth, tongue rolled on both sides to make a passage in the middle, making a hissing sound and exhale through the nose. repeat it 20 times.

The benefits of sheetali and seetkari are basically the same. They cool the whole body and the lower energy centers, particularly those connected to the reproductive and excretory organs, is cooled down significantly. Eliminates indolence (sloth) and the need and desire to eat, drink and sleep. These are useful techniques for alleviating psychosomatic disease such as high blood pressure.

These two practices are unique because inhalation is done through the mouth. In every other yogic practice and in breathing in general, we are told to always breathe through the nose. When the breath is taken through the nose, the nose heats up and cleans the incoming air. This infringement is acceptable providing you do not practice in a dirty, polluted atmosphere or in excessively cold weather.

Bhramari (humming bee breath). In empty stomach, inhale slowly and deeply through the nose, listening to the sound of the breath. Close the ears with the index and middle fingers by pressing the middle outer part of the ear ligament into the ear hole. Keep the ears closed and exhale, making a deep soft humming sound. Concentrate on the sound, keeping it low pitched. When exhalation is complete, lower the hands to the knees and breathe in slowly. repeat it 20 times.

Bhramari awakens psychic sensitivity and awareness of subtle vibrations, therefore, it is better to practice in the early hours of the morning or late at night. The sound produced in bhramari is very soothing and thus the practice relieves mental tension and anxiety and reduces anger.

Moorchha literally means "to-faint". Sit in padmasana, hands on knees, inhale slowly and deeply through both nostrils, get into jalandara bandha (with erect spine, bend the head forward so as to chin touching collar bone) and retain breath for even longer than it is comfortable. Then relax and wait till breath is normal. Repeat it 5 times, pushing the limit of retention every time.

This pranayama clears the mind of unnecessary thoughts and reduces awareness of the senses and external world. Hence, it is an excellent preparation for meditation and enhances dharana (concentration) practices. It helps reduce anxiety and mental tension and induces relaxation and inner awareness.

Plavini means "to float". When the inner part of the abdomen is completely fitted with air, one can float like a lotus leaf on water. Sit in padmasana or siddhasana and start inhaling slowly. After inhalation the air has to be swallowed as you swallow food, retained inside and not belched out. Do it 3-5 times without expelling the air. While retaining the air inside the stomach there should be absolutely no physical movement or the air will escape. Try to retain the air in the stomach for at least thirty to ninety minutes and burp it out voluntarily.
Plavini should be practiced after asana and all other pranayama techniques, or it can be practiced during the day if you are fasting. This will keep the stomach full and prevent hunger pangs and the desire to eat. Yogis practice plavini before going into samadhi for days together so that the stomach remains full during their natural fast.

Not all these pranayamas to be practiced at the same time. Learn them all and with right practice, the need will arise automatically to perform different methods of these pranayama in different situations.

Next: Yoga: 5. Bandhas (Locks) and Mudras (Gestures)