Newton's third law of motion ( Action and Reaction ! )

When a bullet is fired from a gun equal and opposite forces are exerted on the bullet and the gun during the time the bullet is passing down the barrel

Newton's second law of motion ( momentum , force , weight , free falling )

A motortruck requires a larger force to set in motion when it is heavily loaded than when it is empty

The nature of friction: Static and dynamic friction

Friction is the name given to the force which opposes the relative sliding motion of two surfaces in contact with one another. It plays a notable part in our daily lives. For example

Newton's Laws of motion:1- Newton's first law of motion

In 1687 Newton published a book written , as was the routine in those days , in Latin and given the title , Philosophiae naturalis principia mathematica Translated , this means

Derivation of Kinetic energy formula and worked examples

Where there are no opposing forces, a moving body needs noforce to keep it moving with a steady velocity ( Newton's first law of motion ). If, however, a resultant force does act on a moving body

Thursday, November 13, 2014

How Dressing Well Makes You Feel Better

How Dressing Well Makes You Feel Better

There are many reasons why you need to dress well every single day. No matter where you are heading to or what you are doing, you need to be in your best and most comfortable outfit. Dressing well can be rewarding and this can certainly change your mood and your day.

Dressing confidently is more than just wearing the latest trend in fashion. It is also about feeling better about what you are wearing and the way you look. Dressing well also makes you feel self-assured in many diverse situations. Dressing up requires extra effort but it definitely makes you feel great especially when you receive wonderful compliments.  


How Dressing Well Can Make You Feel Better

Dressing Well Gives you the Feeling of Self Improvement

Finding and wearing clothes that perfectly suit your style and taste is a means of improving yourself and your lifestyle. These things definitely make you feel better. Other people may start to look at you differently, paving way for more self respect and self pride. This may also inspire you to work better..

Dressing Well Earns You Great Impressions

When you are always dressed up properly, chances are, people around you will get great impressions of you and offer great compliments. Hearing these things will surely make you feel better and more proud of yourself. Dressing well can also mean having a great personality, so many people will start to like you. If you dress well, you will be remembered.

Dressing Well Draws the Right Kind of Attention

It feels good to get noticed, especially if you really deserve to. Dressing well is just one way of getting the right kind of attention. You can generate attention if you know how to dress in style and most importantly, you display a good attitude and character. Ideally, a well-dressed person usually stands out from the crowd. Remember that your outfit can sometimes tell people that there is something special in you.

Dressing Well Makes you Appreciate Better Things

If you aim to look good, you actually learn to appreciate better things. Dressing well enables you to appreciate the things that contribute to your self-improvement.

Dressing Well Makes you Feel More Confident

Dressing well improves self esteem and self confidence and this can certainly make you feel great. If you feel confident with the way you look, you become more motivated and determined to fulfill your works and duties. Feeling confident will also improve your performance and productivity.  


These are just few of the many ways dressing well can make you feel better. Being a fabulous fashionista actually entails numerous advantages, so do your best to always look stunning and fashionable. You might never know how many individuals are motivated and inspired by your unique and fantastic style!

The Best Styles for the Full Figure

The Best Styles for the Full Figure

The full figure is becoming more common in many women these days. Some consider having this figure as a challenge while others take this figure as an asset. There are many ways to look great with a fuller figure. There are great styles and approaches that individuals can keep in mind when choosing clothes and dressing up.

Some women with full figures find it frustrating to find good styles to suit their body shape that will make them look elegant and flattering.

However, there are some rules that can help in creating a unique style and making life easier when you choose clothes and shop. These rules will help you create the right style that will give emphasis to the beauty of your body shape.  


Some Fundamental Rules in Dressing for the Full Figure

 Wear skirts and dresses below the knees

 Choose bold prints and patterns

 Go for tailored suits

 Choose dark colors for these create a slimming effect

 Wear same tones on bottom and top

 Wear draped scarves, deep necklines and long necklaces

 Consider vertical stripes

 Wear loose fitting dresses or clothes that flatter the body


Best Style Dresses

The best dresses that look stunning and flattering to a person with a full figure are nipped at the waist such as belted or wrapped dresses. You can also look great in maxi dresses however; these have to be snug below the bust in order to make you look slimmer. There are several styles for the full figure that may look fabulous and awesome on you but this will still depend on factors other than your body shape.

There are some details that you need to look upon when shopping for dresses that conceal problem areas like your hips, waist and stomach. These are a few:

 Vertical pleats on bottom elongate the lower bottom and make the legs and hips look leaner

 Ruchings around the waist conceals a round belly

 V-necks create leaner upper bodies

 Waist cinchs define slim torsos



Essential Fashion Tips for the Full Figure

There are essential fashion tips that women with a full figure must know. These tips include but are not limited to the following:

Wearing Light Colors


Light colors are proven to maximize the size that things appear to be but this can be fixed by means of adding some touches that elongate the figure. Color blocking shortens your body figure so adding heels to compensate for this is a good option.

Choosing the Right Length


The length is an important factor when thinking about the best styles for a full figure. Calf length dresses look best on taller and fuller figures. The best dress for a full figured woman is the one that stops at the knee or even shorter. Shorter hemlines reveal more legs and make you look slimmer.

Simple Rules for the Full Figured

For women will full figures, the following rules should apply:

 Select outfits with monochromatic themes because these create slimming effects.

 Match and mix bold colors together with the darker separates.

 Select fabrics that drape over your curves. Clothes with excessive spandex cinch the body too tight. Instead of making a streamlined and smooth effect, you end up with unwanted bulges.


Regardless of their type or shape of body, it is a must for individuals to dress well according to their body types. There are plenty of ways and styles to help them look their best.  

Best Styles for the Hourglass Shape

Best Styles for the Hourglass Shape The hourglass body shape is characterized with curvy, well-defined and round waistlines with generous thighs and a fuller bust and hips. Due to the change in the concept of fashion and beauty, some women with hourglass body shapes try to conceal their curves. Others are facing difficulty in finding the best styles for this particular type of body shape or figure.

Since the hourglass body shape is considered to be the most ideal feminine shape, there is really not much that needs to be done in order to accentuate the body figure. If you are one among the lucky individuals that are blessed with this body shape, all you need to do is to dress appropriately and smartly and look for the best clothes or styles that will flaunt your shape well. Dive deeper into some more helpful information on how to appropriately dress an hourglass body shape.  


How to Dress an Hourglass Figure or Body Shape

One of the ultimate keys in dressing an hourglass figure or body type is to proportionally dress the bottom and top of your body while accentuating the beauty and shape of your waist. Wide belts over dresses and shirts are excellent ways of showing off your waist. For a more curvaceous effect, one can create this by means of adding some volume to the lower and upper body while maintaining balance. Women with hourglass figures usually have great legs so you should enjoy the freedom of playing with different skirt lengths.


Choose tops that accentuate the waist and retain the balanced look of your hourglass figure. Look for tops with nipping or banding at the waist, belted tops, wrap style tops, form-fitting tops and tailored jackets and shirts.


You are fortunate for having an hourglass body shape because you can wear any bottom as long as it is paired with the right top. You can try pencil skirts and high-waist pants, straight legged pants, boot cut, skirts showing off your great legs, skinny jeans, pants paired with balanced tops.

Coats and Jackets for Hourglass Body Shapes

Tailored and fitted jackets and coats are excellent choices for a hourglass body shape. Well fitted jackets aid in slimming down the waist and this will emphasize the your figure beautifully. With this type of figure, you are able to choose any length, however; tailored long coats work well especially with pleated waist details and belts.

Single-breasted jackets are also perfect for those with this body shape. Simplicity is still the best key when looking for the perfect coat or jacket. As much as possible, avoid over accentuated pocket or shoulder detailing.


If you are one of the many who are blessed with an hourglass body shape, you are advised to cling to shorter jackets ending at your hips. See to it that you do not conceal your shape under bulky jackets. Stick to the one that follows your body curves.

You can also try neat and small sleeves that soften your upper and fleshy arms. Your hourglass body shape will do the major talking in terms of your coat so make sure that you get the right size and fit. Do these things and you can never go wrong when it comes to your style.

When it comes to the best styles for the hourglass body shape, you also need to think about the right neckline styles, collar styles and sleeve styles. All these styles pose impact to your over-all look and image. You also need to devote time and effort in choosing fashion accessories that best suit your hourglass figure.  

The Best Styles for Pear Body Shapes

The Best Styles for Pear Body Shapes Women with pear body shapes commonly have triangular body shapes. This means that their torso is narrow, their bust is small and the hips are large. A person with a pear body shape may have a smaller waist.

In order to achieve an hour glass figure, women need to successfully create a stark variance on their hips, waist and bust. The best styles for pear body shapes can also help them make the most remarkable and fashionable transformation.

By choosing the right clothes, you can become more gorgeous and attractive. Knowing the best style that will match your body shape is crucial to reveal the fashionista within you.  


Select the Right Style for you

The following are some ways to accomplish balance and have the best style that will certainly captivate the interest and attention of many:



Wear A-line skirts that smoothen out your hips and emphasize your waist. This will highlight the mid section of your body. Instead of jeans and slacks of light colors, choose bottoms that flow straight down your hips. You can also wear straight-legged pants.



Just like bottoms, A-line dresses can also become a women’s best friend. Wear dresses with flutter or capped sleeves. A-line dresses below the knee or that fit around your knee will certainly look great. Wrap dresses are also ideal options.



There are several fashion accessories that are specifically designed for individuals with pear body shapes. Every woman with a pear body shape must have particular undergarments like padded underwear, girdles and body shapers to enhance their thighs, butt and hips. The top portion of your body should be accessorized rather than putting on heavy belts on lower portions.

What to Avoid

There are some great styles for pear body shapes but there are also things that need to be avoided and these are as follows:

 Avoid wearing pants with high waists.

 Stay away from bottoms that make your hips appear wider.




Avoid clothes with flashy prints, larger pockets near the hip area and heavy patterns.

 Avoid flared or tapered pants.

 Avoid long jackets because they will just make the bottom portion look bigger.

 Stay away from extremely loose or tight tops and shirts. Button down or fitted tops are great alternatives. Choose bright and bold colors for the tops and simplify the lower half.


Guide to Shopping for the Best Dresses for a Pear Body Shape

If you have the desire to end up with the best style for a pear body shape, you can consider the following guidelines when shopping:

 Wearing a strapless dress can be an ideal option for women with pear body shapes. Shop for this dress if you want your shoulders to look broader. This dress also helps to balance childbearing hips.

 A-line dresses with defined waist play an essential role in accentuating the waist. There are benefits to selecting flared skirts. Little or too much flare can cause a problem, so shop for dresses with enough flare to graze the hips.

 Purchase a dress that perfectly fits your hips despite the fit on top. The lower portion of the body can be hard to fit, so purchase a dress that fits your butt and hips. You can take the dress to a tailor and have the upper portion or half fitted perfectly for you. This move will surely make the dress amazing and look expensive.


Your pear body shape should not really be a problem when dressing up. There are some great styles for pear body shapes that can help you look stunning. With these styles, you can now flaunt what you have with confidence and pride. Your pear body shape can be your best asset, so dress in style at all times and become a fabulous fashionista!  

Best Styles for Apple Body Shapes

Best Styles for Apple Body Shapes Women with apple body shapes are said to be fuller around the middle. When they gain weight it usually shows up in their upper body and around the belly. Individuals with this body shape are commonly larger-breasted and their upper shoulders and arms are broader. Their legs are thin and butts are flat. The primary traits of apple body shapes are:

 Narrower hips

 Larger bust

 Less Defined waists

 Broader shoulders


If you have an apple body shape, then you should be proud of your figure and be confident to flaunt it. There are certain styles for apple body shapes that will help you look your best. If you really want to show what you have, you need to dress appropriately based on your body type and shape. If you want to dress in a manner that will flatter others, consider the following styles for apple body shapes.  


Blouses and Tops for Apple Body Shapes

Wear a top with a rounded hem and long shirttails to camouflage your tummy. Wrap around a top, fitted but not too tight and create folds on the belly area. Tops having v-necks are ideal choices because these pull the eyes away from the shoulders while emphasizing or accentuating the person’s best feature.

As much as possible, avoid showing much cleavage for this might end up less flattering. Avoid tops that are low cuts, instead, wear shirts with thick straps and high necklines. A tank top or Cami with a matching jacket is a good option, especially if you are planning for a short trip or walk to the grocery store. Avoid baggy tops with full or puffy sleeves for these make the arms and shoulder look bigger.

Coats and Jackets

Belted jackets give the illusion of a smaller waist and tummy. Jackets must fall right in the waist to the middle thigh and not on the widest area of the hips for this will make the hips appear wider. Wear blazers and jackets open for ideal vertical lines to lengthen the body. Get rid of shoulder pads, pleats and gathers because your very own shoulders are perfect as they already are.

Avoid excessive fabrics on the bust around the hips and tummy. To do this, choose a simple tailored jacket that nips in your waist or a belted jacket, giving an illusion of a smaller waist. This will also trim your belly area.

Trousers and Skirts

Wear shorter skirts and dresses that draw the eyes towards your awesome thighs and hips. Shorter can mean about one to five inches above your knee. A-line skirts can give you an hourglass body figure.

Find pants and skirts with flat fronts or the ones that clasp on the side. You can also go for pants and skirts with zippers at the back or on the side. As much as possible, avoid pleats, gathers and pockets near your waist for these will just add inches to your belly.


Silks and chiffons make excellent skirts for women with apple body shapes. Women can even try bold colors and patterns with this fabric.

Avoid bottoms that are extremely tight or skirts and pants that are ultimately straight for this will make you appear bigger. Try flared legs and boot-cut pants and then A-line skirts for you to look leaner. Even with an apple body shape, the right style can make you look stunning and beautiful. You just have to keep in mind the techniques and styles of how to dress appropriately according to this particular body type. With the best styles, you will be able to accentuate your best assets and feature.  

Best Styles for Straight Body Shapes

Best Styles for Straight Body Shapes

When you are dressing up, it is important to make sure that you are wearing the right dress that complements your body shape and brings out the best in you. Humans come in different sizes and shapes, so finding the right clothes that will flatter their specific body type can become a daunting task for most of them.

If you have a straight body shape, chances are that you at times have a hard time choosing clothes and fashion accessories that will perfectly suit you. The key to rectify this is to know your proportions and use fashion in accentuating your best physical features. The following are some helpful tips on how to dress your straight body shape:  


How to Dress your Straight Body Shape

One of the ultimate keys to dressing your straight body shape is to dress the bottom and top of your body proportionally while enhancing the waist. You can also wear form-fitting clothes like tube-type dresses and create a waist by means of adding a dark and wide belt. However, you can also create more attractive and curvaceous effects by adding volume proportionally to the lower or upper body and matching and mixing suggested separates.

Instant Fashion Tips for your Straight Body Shapes

There are numerous fashion tips that specifically apply to individuals with straight body shapes. So, if you have this kind of body shape, dressing up should not really be a challenge. Your objective in dressing should be creating an illusion of an hourglass shape or figure. No matter what your body shape or type is, balancing the body will always create a pleasing effect.

Creating Curves with Appropriate Colors

Wear color of identical or similar brightness on your lower and upper body while putting in darker color on the waist. Examples are wearing dresses that are light colored with dark and wide belts and wearing shirts that are light colored around the shoulders or bust and dark colored below the bust. Pair these with light colored skirts or pants.

Suggested Dresses for Girls with Straight Body Shapes

If you are a girl with a straight body shape, you can select dresses with fitted waists, full skirts and dresses with embellishments and cuts that add volume to your shoulders and bust and draw attention at the same time.

Additional Fashion Tips

Go for attention-seeking tops with necklines, cuts and embellishments that add volume to your shoulders and busts while emphasizing your waistline. When choosing a jacket,


pick the one with structured shoulders, nipped-in waist and a flared bottom. Jackets which come to the hip-bone are the most ideal and flattering for your straight body type.

Accessories and Bottoms for Straight Body Shapes

For a straight body shape, ladies are advised to use fashion accessories like padded bras, wide belts and body shapers trimming your midline. Choose full skirts with amazing embellishments and details and the ones that generate attention. Choose pants that have flap-pockets and jeans that are whiskering in the hips. Slightly flared, boot cut and trouser cuts are the most flattering for this body shape.

Having a straight body shape may require you to have a specific look or fashion style that may not be practical for other people. But remember, you have all the liberty to dress in style according to your body shape. You can take cues from shapes and cuts and create your own unique style. You can experiment, but of course, always be guided with fashion tips that are meant for individuals like you with a straight body shape. Shop for fashion accessories and garments that will make you look your best with your straight body shape.  

Determine Your Body Type

Determine Your Body Type Different individuals have different features, physical attributes and body types. You will notice that some people are naturally skinny, fat or naturally muscular or lean.

People have different body types, which play a big role in how they look. The following information will help you learn about the different body types, ways to determine your body type and the advantages of understanding what your body type is.

All individuals are not created equal. One displays features that the other one does not possess. You have to understand that each person is unique in his/her own ways. People have their own facial features, body shapes and body types.

It is important to determine your own body type because this defines what clothes to wear and how to style yourself. There are several aspects of life where determining your body type is necessary.  


The Three Different Body Types

1. Ectomorphs


Ectomorphs have a naturally skinny type of body. These are the women and men who are relatively thin or skinny for their whole lives without really putting much effort towards their workouts or diets. This body type is distinguished by absence of excessive muscle or fat tissue. Ectomorphs have narrow clavicles and hips, thin builds, small joints, long limbs and stringy belly muscles.

2. Endomorphs


This is the fat body type. Endomorphs are characterized by the prevalence of their body fat. These are women and men who will likely be bigger and thicker most of their lives. Despite workouts and diets, Endomorphs seem to gain excessive weight while thinking about foods and eating them often. Endomorphs are blocky and they have thicker rib cage, thicker and wider joints, wider hips and shorter limbs.

3. Mesomorphs


This body type is considered as the genetic elite. Mesomorphs are characterized by well-developed musculature. Mesomorphs are typically body builders, fitness models and athletes. Endomorphs and ectomorphs both have their negative and positive sides but mesomorphs get the best of both worlds. Mesomorphs have narrow waists, wide clavicles, round and long muscular bellies and thinner joints.

How to Determine Your Body Type

Determining your body type is the very first step in learning how to dress and look your best. There are ways to help you determine your body type and positive features and they are as follows:

 Examine your own body

 Simply study the proportions and shape of your body



Stand in front of the mirror while wearing as little clothes as possible for this helps in getting an accurate picture of your real curves.

 Focus on the torso’s shape and pay close attention to the contours extending from the thinnest part of your waist to your hip line and rib cage.

 Determine the parts of your body from largest to narrowest. Note your waist, bust, thighs, hips and shoulders. Determine the connecting contours that are straightest and curviest.

 To determine your body type, you also need to get your hip, waist and bust measurements.

 Compare all your measurements to other body types.


Determining your body type is highly essential because it helps define better dressing solutions. Aside from this, your body type also dictates the right garments that will perfectly complement your body. If you want to be a fabulous fashionista, dress well according to your body type.

Advantages of Determining your Body Type

Determining your body type is a means of reading your own body to gain ideas and insights on its uniqueness. This also means gaining complete awareness to pinpoint the things that will work best for you. Your body type can be something that motivates you or defines your authentic self. Understanding your best functions based on your body type is like being guided by your own body manual. Moreover, the physical benefits of determining your body type will allow you to discover more holistic advantages that will certainly improve the quality of your life.

In fashion, determining your body type is highly advantageous. When your body type is clearly determined, it will be easier for you to choose a fashion accessory or garment and dress according to your body type.

Style Basics

Style BasicsIndividuals, regardless of their age and gender have this strong passion and inclination to fashion these days. Fashion is defined as a popular trend pertaining to dressing behavior and style. This also covers accessories and ornaments to make one look fabulous and fashionable. Fashion is also a means of doing or expressing something through style and modes of dressing. This is a prevailing style that will forever be linked to an individuals’ way of life.

One can fully understand and appreciate everything about fashion if he or she has a clear insight and understanding about fashion style basics. Fashion covers a broad scope and it would be a helpful advantage as a fashionista to be aware of the fundamentals of fashion, style and designs. The following information can help widen your knowledge and familiarity about fashion and designs.  

The Basics

The basics of fashion consist of building a suitable and well-balanced wardrobe. This also includes the do’s and don’ts in fashion, the mistakes that you need to avoid, especially in terms of dressing and styling yourself and certain fashion rules to follow. The basics also cover the fundamentals of styles and designs.

Although all designs integrate the elements or components of fashion design, the principles of design as well as its fundamentals or guiding rules whether governed or not by specific design is aesthetically pleasing and successful. The basics of fashion designs are as follows:



There are several fashion techniques that fashion designers use in order to put an emphasis on a fashion accessory or garment feature. This technique might include adding cut offs, embellishments and so on. Using colors and their placement is also a way of creating emphasis. Learning about emphasis can help an individual choose the perfect fashion accessory or garment to wear.



In fashion design and style, balance can be obtained with the use of certain features such as hemlines, seams and necklines. Symmetrical and asymmetrical balance applications can be both effective depending on the features and preference of the person.



In fashion design, proportion is very important. The size of several components of fashion garments should correlate so they look great together. Proportion is one of the most vital principles of fashion.





When the concept of unity in fashion is applied appropriately, the over-all look of a fashion accessory or garment is expected to be great. It looks as though all design and style elements contribute to create a harmonious whole. When unity is effectively applied, every element will enhance the other rather than distract the style or design.

Elements of Fashion

The four fundamental elements of fashion design are imperative in the design of any fashion accessory or garment. Recognizing several elements could be important in coming up with a pleasing garment can sometimes help the individual along. The elements of fashion are as follows:



This is a very important element in fashion design and there are actually plenty of color schemes that work great. There is a color wheel showing the compatible and appropriate color combinations which can be useful when individuals are trying to figure out certain colors that will go well with their wardrobe or accessories. In the fashion world, neutral colors still play a huge role for these can be worn with almost anything while different rules apply to dark and light colors. To make colors work, individuals should know the most flattering and stunning colors that will look great on them.



In other design types, form and shape are considered elements. In fashion design, form and shape amount to a silhouette of a fashion accessory or garment.



Lines play a crucial role in the field of fashion design. Creation of garments simply means that there is a seam line holding a group of fabric sections together.


Lines can be curved or straight. Lines in particular patterns and seam lines running vertically can suit petite women more than the lines running horizontally. For short women, asymmetrical lines are perfect but these will still depend on their placement.



This is another important element of fashion design. Other materials or fabrics like findings and trims for fashion accessories and clothing can have a series of textures. All of these can affect the look of the fashion accessory or garment. The texture and weave of fabrics pose an impact on how this drapes which therefore affects how the garment looks when being worn.

All these are important ideas and information that one should know in order to become a fabulous fashionista.

Compressible Flows: Normal Shock Waves , Oblique Shock Waves and Expansion Waves

9.1 Normal Shock Waves

Shock waves are large-amplitude waves that travel in a gas. They emanate from the wings of a supersonic aircraft, from a large explosion, from a jet engine, and ahead of the projectile in a gun barrel. They can be oblique waves or normal waves. First, we will consider the normal shock wave, as shown in Fig. 9.7. In this figure it is stationary so that a steady flow exists. If V1 were superimposed to the left, the shock

imageFigure 9.7 A stationary shock wave.

would be traveling in stagnant air with velocity V and the induced velocity behindthe shock wave would be (V1V2). The shock wave is very thin, on the order of 10−4 mm, and in that short distance large pressure changes occur causing enormous energy dissipation. The continuity equation with A1= A2 is

imagewhere the areas have divided out since A1 = A2. Assuming that the three quantities r1, p1, and V1 before the shock wave are known, the above three equations allow us to solve for three unknowns r2, p2, and V2 since, for a given gas, k is known.

Rather than solve the above three equations simultaneously, we write them in terms of the Mach numbers M and M , and put them in more convenient forms.

First, the momentum equation (9.37), using Eq. (9.35) and V 2 = M2 pk /ρ, can be written as


In like manner, the energy equation (9.36), with p = ρRT and V 2 = M2 kRT , can be written as

imageIf the pressure and temperature ratios from Eqs (9.38) and (9.39) are substituted into Eq. (9.40) the downstream Mach number is related to the upstream Mach num- ber by (the algebra to show this is complicated)

imageFor air, the preceding equations simplify to

imageSeveral normal shock-flow relations for air have been presented in Table D.2. The use of that table allows one to avoid using Eq. (9.44). In addition, the ratio p02 /p01 of the stagnation point pressures in front of and behind the shock wave are listed.

imageFigure 9.8 Flοw with shοck waves in a nοzzle.

Return to the converging-diverging nozzle and focus attention on the flow below curve C of Fig. 9.6. If the receiver pressure decreases to p /p = a in Fig. 9.8, a normal r 0 shock wave would be positioned somewhere inside the nozzle as shown. If the receiver pressure decreased still further, there would be some ratio p /p = b that would posi- r 0 tion the shock wave at the exit plane of the nozzle. Pressure ratios c and d would result in oblique shock-wave patterns similar to those shown. Pressure ratio e is associated with isentropic flow throughout, and pressure ratio f would provide an exit pressure greater than the receiver pressure resulting in a billowing out, as shown, of the exiting flow, as seen on the rockets that propel satellites into space.


A normal shock wave travels at 600 m/s through stagnant 20°C air. Estimate the velocity induced behind the shock wave. (a) Use the equations and (b) use the normal shock-flow table D.2. Refer to Fig. 9.7.


Superimpose a velocity of 600 m/s so that the shock wave is stationary and V1 = 600 m/s, as displayed in Fig. 9.7. The upstream Mach number is


(a) Using the equations, the downstream Mach number and temperature are, respectively,

imageEXAMPLE 9.4

Air flows from a reservoir maintained at 20°C and 200 kPa absolute through a converging-diverging nozzle with a throat diameter of 6 cm and an exit diameter of 12 cm to a receiver. What receiver pressure is needed to locate a shock wave at a position where the diameter is 10 cm? Refer to Fig. 9.8.


Let’s use the isentropic-flow table D.1 and the normal shock-flow table D.2. At the throat for this supersonic flow M = 1. The Mach number just before the shock wave is interpolated from Table D.1 where A1 /A* = 10 /6 = 2.778 to be


since the stagnation pressure does not change in the isentropic flow before the shock wave so that p01 = 200 kPa. From just after the shock wave to the exit, isentropic flow again exists so that from Table D.1 at M = 0.5078

imageWe have introduced an imaginary throat between the shock wave and the exit of the nozzle. The exit area A is introduced by


Using Table D.1 at this area ratio (make sure the subsonic part of the table is used), we find


using p0 e = p02 for the isentropic flow after the shock wave. The exit pressure is equal to the receiver pressure for this isentropic subsonic flow.

9.2  Oblique Shock Waves

Oblique shock waves form on the leading edge of a supersonic sharp-edged airfoil or in a corner, as shown in Fig. 9.9. A steady, uniform plane flow exists before and after the shock wave. The oblique shock waves also form on axisymmetric projectiles.

imageFigure 9.9 Οblique shοck waves: (a) flοw οver a wedge and (b) flοw in a cοrner.

The oblique shock wave turns the flow so that V2 is parallel to the plane surface.

Another variable, the angle through which the flow turns, is introduced but the additional tangential momentum equation allows a solution. Consider the control volume of Fig. 9.10 surrounding the oblique shock wave. The velocity vector V1 is assumed to be in the x-direction and the oblique shock wave turns the flow through the wedge angle or deflection angle q so that V2 is parallel to the wall. The oblique shock wave makes an angle of b with V1. The components of the velocity vectors are shown normal and tangential to the oblique shock. The tangential components of the velocity vectors do not cause fluid to flow into or out of the control volume, so continuity providesimageThe pressure forces act normal to the control volume and produce no net force tan- gential to the oblique shock. This allows the tangential momentum equation to take the form

imageFigure 9.10 Oblique shock-wave control volume.

imagesince the tangential velocity terms cancel.

Observe that the tangential velocity components do not enter the three Eqs. (9.45), (9.48), and (9.49). They are the same three equations used to solve the normal shock- wave problem. So, the components V1n and V2n can be replaced with V1 and V2, respectively, of the normal shock-wave problem and a solution obtained. Table D.2 may also

To often simplify a solution, we relate the oblique shock angle b to the deflection angle q. This is done by using Eq. (9.45) to obtain


With this relationship the oblique shock angle b can be found for a given incoming Mach number and wedge angle q. A plot of Eq. (9.52) is useful to avoid a trial-and- error solution. It is included as Fig. 9.11. Three observations can be made by studying the figure.


Figure 9.11 Οblique shοck wave angle b related tο wedge angle q and Mach

For given Mach number M and wedge angle q there are two possible oblique shock angles b. The larger one is the “strong” oblique shock wave and the smaller one is the “weak” oblique shock wave.

• For a given wedge angle q there is a minimum Mach number for which there is only one oblique shock angle b.

• If the Mach number is less than the minimum for a particular q, but greater than one, the shock wave is detached as shown in Fig. 9.12. Also, for a given M1 there is a sufficiently large q that will result in a detached

shock wave.

The required pressure rise determines if a weak shock or a strong shock exists. The pressure rise is determined by flow conditions.

For a detached shock wave around a blunt body or a wedge, a normal shock wave exists on the stagnation streamline; the normal shock is followed by a strong oblique shock, then a weak oblique shock, and finally a Mach wave, as shown in Fig. 9.12. The shock wave is always detached on a blunt object. image_thumb[4]

Figure 9.12 Detached shock waves around (a) a plane, blunt object and (b) a wedge.



Air at 30°C flows around a wedge with an included angle of 60° (see Fig. 9.9a). An oblique shock emanates from the wedge at an angle of 50°. Determine the approach velocity of the air. Also find M and T .


image_thumb[5]From Table D.2 interpolation provides M2n = 0.5176 so that


9.3 Expansion Waves

Supersonic flow exits a nozzle (the pressure ratio f in Fig. 9.8), and billows out into a large exhaust plume. Also, supersonic flow does not separate from the wall of a nozzle that expands quite rapidly, as shown in Fig. 9.8. How is this accomplished? Consider the possibility that a single finite wave, such as an oblique shock, is able to turn the flow around the convex corner, as shown in Fig. 9.13a. From the tangential momentum equation, the tangential component of velocity must remain the same on both sides of the finite wave. For this to be true, V2 > V1. As before, this increase in velocity as the fluid flows through a finite wave requires an increase in entropy, a violation of the second law of thermodynamics, making a finite wave an impossibility.

A second possibility is to allow an infinite fan of Mach waves, called an expansion fan, emanating from the corner, as shown in Fig. 9.13b. This is an ideal isentropic process so the second law is not violated; such a process may be approached in a real application. Let’s consider the single infinitesimal Mach wave displayed in Fig. 9.14, apply our fundamental laws, and then integrate around the corner. Since the tangential velocity components are equal, the velocity triangles yield

image_thumb[7]Figure 9.13 Supersonic flow around a convex corner.



The solution to this relationship is presented for air in Table D.3 to avoid a trial-and- error solution for M given the angle q. If the pressure or temperature is desired, the isentropic flow table can be used. The Mach waves that allow the gas to turn the corner are sometimes referred to as expansion waves.

Observe from Table D.3 that the expansion fan that turns the gas through the angle q results in M = 1 before the fan to a supersonic flow after the fan. The gas

speeds up as it turns the corner and it does not separate. A slower moving subsonic flow would separate from the corner and would slow down. If M = ∞ is substituted into Eq. (9.60), θ = 130.5°, which is the maximum angle through which the flow could possibly turn. This shows that turning angles greater than 90° are possible, a rather surprising result.


Air at 150 kPa and 140°C flows at M = 2 and turns a convex corner of 30°. Estimate

the Mach number, pressure, temperature, and velocity after the corner.


Table D.3 assumes the air is initially at M = 1. So, assume the flow originates from M = 1 and turns a corner to M1 = 2 and then a second corner to M2, as shown. From Table D.3, an angle of 26.4° is required to accelerate the flow from M = 1 to M = 2. Add another 30° to 26.4° and at q = 56.4° we find that

M2 = 3.37


Using the isentropic flow table D.1, the entries from the reservoir to state 1 and also to state 2 can be used to find



Quiz No. 1

1. Two rocks are slammed together by a friend on one side of a lake. A listening device picks up the wave generated 0.45 s later. The distance across the lake is nearest

(A) 450 m

(B) 550 m

(C) 650 m

(D) 750 m

2. The Mach number for a projectile flying at 10 000 m at 200 m/s is (A) 0.62

(B) 0.67

(C) 0.69

(D) 0.74

3. A supersonic aircraft passes 200 m overhead on a day when the temperature is 26°C. Estimate how far the aircraft is from you when you hear its sound if its Mach number is 1.68.

(A) 245 m

(B) 275 m

(C) 315 m

(D) 335 m

4. A converging nozzle with exit area of 10 cm2 is attached to a reservoir maintained at 250 kPa absolute and 20°C. If the receiver pressure is maintained at 150 kPa absolute, the mass flux is nearest

(A) 0.584 kg/s

(B) 0.502 kg/s

(C) 0.428 kg/s

(D) 0.386 kg/s

5. Air flows through a converging-diverging nozzle attached from a reservoir maintained at 400 kPa absolute and 20°C to a receiver. If the throat and exit diameters are 10 and 24 cm, respectively, the receiver pressure that will just result in supersonic flow throughout is nearest

(A) 6.8 kPa

(B) 9.2 kPa

(C) 16.4 kPa

(D) 28.2 kPa

6. The temperature, pressure, and velocity before a normal shock wave in air are 18°C, 100 kPa absolute, and 600 m/s, respectively. The velocity after the shock wave is nearest

(A) 212 m/s

(B) 249 m/s

(C) 262 m/s

(D) 285 m/s

7. A large explosion occurs on the earth’s surface producing a shock wave that travels radially outward. At a particular location the Mach number of the wave is 2.0. Determine the induced velocity behind the shock wave if T1 = 15°C.

(A) 502 m/s

(B) 425 m/s

(C) 368 m/s

(D) 255 m/s

8. Air flows from a reservoir through a nozzle into a receiver. The reservoir is maintained at 400 kPa absolute and 20°C. The nozzle has a 10-cm-diameter throat and a 20-cm-diameter exit. The receiver pressure needed to locate a shock wave at the exit is nearest

(A) 150 kPa

(B) 140 kPa

(C) 130 kPa

(D) 120 kPa


(A) 164 kPa

(B) 181 kPa

(C) 192 kPa

(D) 204 kPa

10. An airflow with a Mach number of 2.4 turns a convex corner of 40°. If the temperature and pressure are 5°C and 60 kPa absolute, respectively, the Mach number after the corner is nearest

(A) 5

(B) 3

(C) 2

(D) 1

Quiz No. 2

1. An underwater animal generates a signal that travels through water until it hits an object and then echoes back to the animal 0.46 s later. How far is the animal from the object?

2. A bolt of lightning lights up the sky and 1.5 s later you hear the thunder.

How far did the lightning strike from your position?

3. A supersonic aircraft passes 200 m overhead on a day when the temperature is 26°C. Estimate how far the aircraft is from you when you hear its sound if its Mach number is 3.49.

4. A small-amplitude wave travels through the 15°C atmosphere creating a pressure rise of 5 Pa. Estimate the temperature rise across the wave and the induced velocity behind the wave.

5. A converging nozzle with exit area of 10 cm2 is attached to a reservoir maintained at 250 kPa absolute and 20°C. Calculate the mass flux if the receiver pressure is maintained at 100 kPa absolute.

6. Air flows from a converging-diverging nozzle from a reservoir maintained at 400 kPa absolute and 20°C through a 12-cm-diameter throat. At what diameter in the diverging section will M = 2?

7. The temperature and pressure before a normal shock wave in air are 20°C and 400 kPa absolute, respectively. The Mach number after the shock wave is 0.5. Calculate the pressure and velocity after the shock.

8. Air flows from a reservoir maintained at 400 kPa absolute and 20°C out a nozzle with a 10-cm-diameter throat and a 20-cm-diameter exit into a

receiver. Estimate the receiver pressure needed to locate a shock wave at a diameter of 16 cm.

9. A supersonic airflow changes direction 20° due to a sudden corner (see Fig. 9.9b). If T1 = 40°C, p1 = 60 kPa absolute, and V1 = 900 m/s, calculate p and V assuming a strong shock.

10. An airflow with M = 3.6 is desired by turning a 20°C-supersonic flow with a Mach number of 1.8 around a convex corner. If the upstream pressure is 40 kPa absolute, what angle should the corner possess? What is the velocity after the corner?