MrsDrPoe: Braves Trivia

Good evening all! Since today's main post was about Atlanta's home team, I thought you might enjoy a bit of Braves trivia. Here's a timed quiz that will definitely test your team knowledge...be sure to share with us how you do!

This quiz and others can be found on at Sporcle- a site truly full of "mentally stimulating diversions." Mr. Poe and I love Sporcle quizzes- especially the daily word ladders. Definitely check out the rest of the site when you get a chance; you may just find your new favorite hobby!

MrsDrPoe: So We'll Root, Root, Root for the Bravos...

This Foto Friday I have something for you sports fans out there! We recently went to (my first) Braves game at Turner Field with our friends, the Granthams:


We parked at the CNN building:


And then walked through downtown Atlanta, past the capitol building:


And across the sea of swirly twirly gumdrops (just kidding) to the stadium. I've been to two other baseball stadiums in my lifetime (Minute Maid Park and Busch Stadium), and I don't remember the view from either of them being as lovely as the one from Turner Field...but I'm probably just partial to my new hometown:


That night the Bravos beat the Arizona Diamondbacks, 4-2 with 3 homers; although, I must confess that I yacked more than I watched the game:


The whole experience was quite relaxing, with the exception of parts of the walk back through downtown to the cars. Mr. Poe and I had a ton of fun, and we were so happy that we could go!

MrsDrPoe: True Friendship

Welcome to Theology Thursday here at the blog! I invite you all to open your Bibles with me today as well examine what true friendship looks like according to the word of God.

At any given time in our lives, each of us has had one or more persons that could be called a friend; these people are definitely blessings from God and a great encouragement through life's trials. Although we form these friendships in various places and through various circumstances, each of these relationships shares that we have something in common with and enjoy the company of our friend. We must ask ourselves, however, do our friends exhibit the qualities that true friends should? And do we exhibit these qualities toward our friends? To determine what a true friend should be, we should examine some Biblical examples of friendship.

If we look in 1 Kings 12:1-15, we see the children of Israel requesting of King Rehoboam to reduce the heavy burdens that his father (King Solomon) had placed on them. Before making a decision, Rehoboam sought counsel from the elders who served his father; they told him to grant the people's request. Rehoboam didn't like this answer; however, so he further sought the counsel of his friends that he had grown up with. These friends advised him to not only deny the people's request but to further increase their burden. Rehoboam approved of this answer and, as a result, later lost the majority of his kingdom to Jeroboam (as was prophesied).

If we go back a little further in the Old Testament to 2 Sam 13, we see one of King David's sons, Amnon, who lusted after his half-sister, Tamar. Just like Rehoboam, Amnon had a friend named Jonadab who told him exactly what he wanted to hear. Jonadab helped Amnon devise a plan that allowed Amnon to physically take advantage of Tamar, humbling her, disgracing the royal family, and eventually leading to the murder of Amnon.

Were the friends of Rehoboam and Amnon good friends? The world would say, "Of course! They helped give them what they wanted. They helped make them happy (at least for a little while)." Solomon would disagree in Proverbs 1, however. In this passage, the wise old king tells his son to avoid the companionship of those who would entice and encourage sinful behavior, contrasting their calls for friendship with the call of wisdom to be holy and seek the Lord.

We must conclude then that true friendship should be anchored by more than a common workplace/sports team/hobby. To be a good friend, one must keep another's spiritual well-being in mind. If we have a friend who is lusting after a woman/man not his/her wife/husband, the 'friendly' thing to do is to help them overcome these sinful thoughts through prayer, Bible study, counseling, etc. The 'unfriendly' thing to do is outlined in the 2 Samuel passage we just examined. Similarly, if we are the ones lusting, a true friend would help us overcome this sin NOT encourage us to continue in it.

Look with me at 2 Samuel 12. In this passage, the prophet, Nathan, comes before King David to tell him a parable about a rich man who steals a lamb from a poor man who loved the lamb as his family; outraged at the injustice, David vows to kill the rich man. Nathan then tells David that he is the rich man and that the parable is a parallel to what he had done to Uriah and Bathsheba. Even though he was speaking to the king, Nathan had the courage to point out his sin as God had sent him to do. Unlike the friends of Rehoboam and Amnon, Nathan was a true friend to David.

So what's the best way to be a good friend? If our friend is not a Christian, we can invite him/her to Bible study, introducing him/her to Jesus and allowing His teaching throughout the New Testament to point out the sin in their lives. If our friend is a Christian, Matt 18, 1 Cor 5, and 2 Thess 3 tell us that sin in the church must be addressed; and if it is not remedied, withdrawal is required.

It's really hard to be a true friend. I've never known anyone who particularly enjoys confrontations, and telling someone what he/she wants to hear is a lot less confrontational that telling that same person that what he/she is doing or wants to do is wrong. No one desires the social stigma associated with one who's labeled as "nosy," "rude," or "intolerant." And of course it's always difficult to determine the best way to approach a sinful situation- we don't want to hurt or embarrass our friends. But we must remember that our friends' souls are more important than an awkward moment or temporary strain on/loss of a friendship.

MrsDrPoe: Sliders

It's another What's for Dinner Wednesday, and tonight I'll actually be sharing a main course recipe for two- old school sliders. Technically, we prepare regular sized hamburgers this way too, but mini-burgers are cuter. The breadcrumbs and Worcestershire make for a super moist and delicious burger, no matter the size!

Ingredients: 1/2 pound lean ground beef, 2 Tablespoons Worcestershire sauce, 1/4 cup Italian breadcrumbs, small buns or rolls (not pictured), your favorite burger toppings (not pictured).


In a small bowl, combine the beef, Worcestershire sauce, and breadcrumbs with your hands. After the mixture is thoroughly combined, divide in fourths, forming a small bun-shaped patty out of each fourth. Place the patties on a hot grill (or electric grill or grill pan if you're cooking inside):


Cook the burgers to your desired doneness (15-20 minutes till well-done on a grill pan, depending on patty shape and pan temp), flipping once about half-way through the cooking time. Place on the buns (we love potato rolls) and top with your favorite flavors (we like mustard, ketchup, dill pickles, bacon, lettuce, tomato, and cheese; the Lukes love BBQ sauce, ranch, bacon, cheese, lettuce, and tomato). Bon Appetit!!

MrsDrPoe: Gone with the Wind

For the next reading review, I'll be examining some fiction for a change. I present to you "Gone with the Wind" by Margaret Mitchell.


I had read this ginormous book (over 1000 pages!) once before in high school and had seen the movie at least once, so I knew all about Scarlett, Rhett, Ashley, and Melanie, and the major events in the story. While it was nice to fill in the blanks that were lacking in my memory bank, I was a little disturbed by the amount of profane language used by the characters, especially considering the negative publicity associated with the single word present in the 1939 making of the movie of the same title. All the characters except for Melanie and Ellen (Scarlett's mother) were also very disappointing, as selfishness, deceit, lust, gossip, and countless other sinful actions and attitudes were prevalent and even justified throughout the pages. I read the last few chapters through tear-filled eyes, severely affected at the mess the characters had made of their lives. I don't think this will be a book I read again in my lifetime.

Having said all that, there were several things that I did enjoy about this book. The first is the fact that because I now live in the metro-Atlanta area, I was super-excited every time I saw the name of a city that I knew of- Marietta, Atlanta, Kennesaw, Jonesboro, etc. The story really came alive for me since I am now familiar with its setting. Secondly, it was nice to review tidbits of history since I haven't had a true history class in years. It has never been a favorite subject of mine, but I do enjoy learning about certain past events when I know there's not going to be a test afterward. The thing I was most intrigued about, however, was that for a book about very worldly (and mostly ungodly) people, there were several Biblical references in the text that I was excited to find- the hanging of Haman, baptism washing away sins, carrying your cross (although, I think this was misused), gaining the world and loosing your soul, etc.

Those are my thoughts on this monster of a novel. Have you ever read Gone with the Wind? What parts did you like/dislike?

MrsDrPoe: Conservation of Momentum, Part IV

Thesis Tuesday is upon us yet again, ladies and gents! As promised, today we will be continuing our discussion of the Conservation of Momentum Equation with an example of how to solve the Navier Stokes equations.

We'll start with fluid flowing between two parallel plates, separated by H meters and infinite in the z-direction. (Note: infinite plates do not exist, but this consideration allows us to approximate the flow as 2D instead of 3D). We will consider the pressure driven flow case where both plates are stationary (I), as well as the Couette Flow case where the upper plate moves in the +x-direction with some velocity U m/s (II).

Given: gap height = H meters
Find: the flow field description (u, v, w, and p)
Assumptions: steady, laminar, incompressible, infinite plates (w = 0, d/dz(...) = 0), constant properties, ignore edge effects, Newtonian fluid, fully developed (d/dx(...) = 0)
Solution:

Step 1: We must first apply the continuity equation.

d/dt(rho) + d/dx(rho*u) + d/dy(rho*v) + d/dz(rho*w) = 0

Since the flow is steady (or because it's incompressible), the first term is equal to zero. For fully developed, laminar flow, d/dx(rho*u) = 0; for infinite plates, d/dz(rho*w) = 0. Because the fluid is incompressible, rho can be pulled out of the derivative in the remaining term. Thus, we are left with:

rho*d/dy(v) = 0 or d/dy(v) = 0

Since the derivative of v is equal to zero, that means that v must be constant (found by integrating both sides of the above equation):

v = const

But we know that because we are dealing with a viscous fluid, the no-slip condition holds. This tells us that the fluid adjacent to either plate has the same velocity as the plate; therefore, since the bottom plate is fixed in both cases, u = v = w = 0 at this location. Putting this boundary information with the result from the continuity equation, v = 0 throughout the domain.

Step 2: Next, we simplify the Navier Stokes equations.

x-momentum:

rho*[d/dt(u) + u*d/dx(u) + v*d/dy(u) + w*d/dz(u)] = -d/dx(p) + rho*gx + mu*[d/dx(d/dx(u)) + d/dy(d/dy(u)) + d/dz(d/dz(u))]

d/dt(u) = 0 since the flow is steady; d/dx(u) = 0 since the flow is fully developed; d/dy(u) = d/dz(u) = 0 because the flow is laminar; thus, the entire left side of the equation equals zero for this case. On the right-hand side, rho*gx = 0 because gravity only acts in the downward (-y) direction; d/dx(d/dx(u)) = 0 also because the flow is fully developed; d/dz(d/dz(u)) = 0 since we are dealing with infinite plates. Rearranging the two terms that are left, we have:

d/dx(p) = mu*d/dy(d/dy(u))

y-momentum:

rho*[d/dt(v) + u*d/dx(v) + v*d/dy(v) + w*d/dz(v)] = -d/dy(p) + rho*gy + mu*[d/dx(d/dx(v)) + d/dy(d/dy(v)) + d/dz(d/dz(v))]

From the result of our continuity equation examination, we know that all the terms containing a v must be equal to zero. We can quickly see that we are only left with two terms:

d/dy(p) = rho*gy

When both sides of this equation are integrated, we obtain the hydrostatic pressure distribution.

z-momentum:

rho*[d/dt(w) + u*d/dx(w) + v*d/dy(w) + w*d/dz(w)] = -d/dz(p) + rho*gz + mu*[d/dx(d/dx(w)) + d/dy(d/dy(w)) + d/dz(d/dz(w))]

Because our flow is laminar, all terms containing a w or d/dz(...) are equal to zero. Also, we know that the term rho*gz = 0, since gravity is only acting in the downward (-y) direction. This equation quickly reduces to:

0 = 0

Step 3: Then we integrate the reduced Navier-Stokes equations and solve for the unknowns.

z-momentum:

0 = 0 tells us nothing

y-momentum:

p = -rho*gy*y + fp(x)
essentially the hydrostatic pressure distribution, the integration constant in this case, fp(x), is constant in the y direction, but may be a function of x

x-momentum:

(1/mu)*d/dx(p) = d/dy(d/dy(u)) rearranging

(1/mu)*d/dx(p)*y + const1 = d/dy(u) integrating once

(0.5/mu)*d/dx(p)*y*y + const2 = u integrating twice

We now have an expression for each of the variables necessary for a full flow field description (v = 0, w = 0, p = ..., u = ...). To completely finish the problem, we must examine our given boundary conditions to determine

Step 4 (I): We need two boundary conditions since there are two constants of integration that need to be determined. For this case, both plates are stationary; therefore, the no-slip condition tells us that u = 0 at y = 0 and u = 0 at y = H (if we consider the bottom plate to be located at y = 0). Applying the first yields:

const2 = 0

Applying the second gives:

const1 = -(0.5/mu)*d/dx(p)*H

So: u = (0.5/mu)*d/dx(p)*(y*y - H*y)

Step 4 (II): We again need two boundary conditions since there are two constants of integration that need to be determined. For this case, the bottom plate is stationary and the top plate is moving with constant velocity, U; therefore, the no-slip condition tells us that u = 0 at y = 0 and u = U at y = H (if we consider the bottom plate to be located at y = 0). Applying the first yields:

const2 = 0

Applying the second gives:

const1 = -(0.5/mu)*d/dx(p)*H + U/H

So: u = (0.5/mu)*d/dx(p)*(y*y - H*y) + (U*y)/H


We can plot the velocity for both of these cases verses y for specified values of d/dx(p), U, mu, and H; this will give us the velocity distribution of the flow between the plates. Here you can see an animation of velocity profiles for various values of U and d/dx(p). If you'd like to play with this program yourself, you can download it (for free I think) from Wolfram Mathematics.

And that's the Navier Stokes equations in a nutshell! Like I mentioned last week, a lot of assumptions need to be made in order to obtain a version of these equations that we can work with; however, once these have been made, the resulting forms of the equations can be solved rather simply.

If you're feeling extra adventurous, try to go through the procedure again by yourself, examining what the velocity profile looks like for the case where the top plate is moving and bottom one is stationary or one plate is moving in the +x-direction and one is moving in the -x-direction or both plates are moving in the +x-direction at the same speed and at different speeds.

MrsDrPoe: JCPenny Deal, 8/21

Another exciting tidbit of information I have for you today involves one of my favorite department stores: JCPenny.

Since we've moved to Atlanta, I've gotten four coupons for $10 off a purchase of $10 or more at Penny's. Needless to say, I've been stoked about it. For my latest purchase, I bought a shirt to go with a hand-me-up skirt a friend passed on to me:


Grand Total OOP: $0.21

While at the store, the clerk told me about JCP Rewards, the rewards program for JCPenny shoppers. Without applying for the store credit card, you can earn rewards points by registering up to three credit cards with JCP online. For each dollar you spend in store or online using one of your registered cards, you earn one point. When you reach 250 points each month, you will get a $10 reward. Plus, by providing your email address, you will be notified about giveaways, discounts, and various other store offers. I'm not sure how much this rewards program will help us, since we don't spend that much at department stores, but I'm hoping we'll see some great deals come out of it around black Friday.

Do any of you have experience with this program? Have you seen significant savings?

MrsDrPoe: Wags and Kroger Deals, 8/21

Happy Money Monday to all! It'd like to share some deals at Walgreens and Kroger that are going on this week:

Wags


Degree Deo: 2/$4, $1 back in ECBs, $1/2 RP 7/31
Dial Body Wash: $3.99, $3.00 back in ECBs
Candles: $1.49 clearance
Extra Dessert Delights gum: $0.65 clearance


Transaction 1
Total Regular Price: $12.36
Total Savings and Coupons: $6.50
ECBs Used: $5.00
Total OOP: $0.86
ECBs Back: $3.00

Transaction 2
Total Regular Price: $7.57
Total Savings and Coupons: $3.62
ECBs Used: $3.00
Total OOP: $0.95
ECBs Back: $1.00

Kroger

Boneless, skinless chicken breasts $1.99/lb
Fresh corn $0.38/ear
Kroger Shredded Cheese B1G1

Check out Mega Event deals here.

MrsDrPoe: Strawberry Huller

Happy Foto Friday to all! This post is definitely not as adventurous as the previous ones since my summer trips (NC, PA, Japan, Gulf Shores) are finished, but I hope that you'll find it helpful.

As we've talked about earlier this week, summer time is coming to a close, and for all of us who love good deals, the time for buying dirt cheap summery items is quickly approaching. With that in mind, I thought I'd share some information an a summer kitchen gadget that I have fallen in love with this summer (and hopefully will be on sale at a store near you sometime soon): the Strawberry Huller.

To use, you simply push down the green top to extend and open the metal claws of the device. Insert the claws into the strawberry over the leaves:


Push in slightly, release the green top, and twist:


Pull the huller out of the strawberry, pushing the the green top again to release the leaves, etc.:


The huller is easy to use, plus it removes the leafy stalk part of the strawberry completely without much waste and allows for a whole strawberry at the end of the process. It does seem a bit flimsy, however, and since I've only had it for one summer, I'm not sure what the life expectancy of the tool is. Occasionally while twisting, I've had to keep my finger and thumb on the body of the device (see second picture) to keep the claw shut.

Despite the negatives, I would definitely recommend snatching this gadget up if you see it on sale. It helps tremendously when you're dealing with fresh strawberries!

MrsDrPoe: Bible Adoption Challenge

Happy Theology Thursday to all! School beginning and summer quickly coming to a close means that the Summer Bible Reading Challenge is almost over as well. How did you do? I hope you were all able to keep up with the reading schedule you picked for yourselves and that going through whatever plan you chose helped you grow spiritually in both your personal study and your study with others!

Since we're now in the habit of reading a portion of God's word daily (whether we were before or not), I propose another challenge: adoption of a Bible book. This idea is not my own; our friend, Mr. Holley, has been suggesting it to serious Bible students for years. In fact, Mr. Holley suggests the eventual adoption of multiple books (one from the OT and one from the NT, to start).

The process of adoption is relatively simple- you choose a book that you wish to adopt, read that book as much as possible (at least once a week), and simultaneously collect and read other books (commentaries, etc.) on your book. By the end of a year, you will have read your adopted book at least 52 times and added several reference books to your theological library. Another suggested "tool" for better understanding your book is to find someone else who has studied your book a lot; this person can help answer your questions, provoke new lines of thought, and suggest materials and references that will help you in your study. It also helps to start a reading-the-Bible-through-in-a-year program while you're "camped out" in your adopted book. This way you'll make connections with your book and the rest of the Bible and vice versa...it's like using the Bible as a commentary on itself!

Perhaps one of the best places to start is to try to adopt one of the gospels (although you may wish to read through all four multiple times before settling on one- you may set a goal of having a book chosen by Jan 1 and then spending a whole calendar year on it). Mr. Holley's book is Luke; Mr. Ellis' book is Mark; Mr. Poe has also adopted Mark. One benefit to starting with a gospel is that improved study of these books will automatically provide you with a method for sharing the story of Jesus. It's amazing how many questions can be answered from camping out in a gospel, taking the passages in context of the whole book instead of just a paragraph.

On a personal note: for the past few months I've been trying to decide on a gospel that I wanted to be "my" book...I'm currently studying Mark and Luke, but I also really like the idea of starting at the beginning with Matthew, and Mr. Holley suggested John because it bridges Matt and Mark with Luke. I plan to read through them all again before I make my decision...I'll let you know how that goes.

If you'd prefer to start elsewhere, that's cool too- the same procedure is used. The goal of course is to continue to adopt books, spending a year (or two) on in-depth study in each one, slowly building your Bible knowledge, faith, and relationship with Christ, brothers and sisters, and the lost. Just think of how many books you could immerse yourself in in a lifetime! I guess I'm not just challenging you (and myself) to a program for the upcoming year, but for all the years to come...are you up for it?

MrsDrPoe: Easy Ice Cream Cake

Once again we've come to What's for Dinner Wednesday, and once again I've decided to share a recipe with you that's merely a component of dinner...unless you want a stomach ache. The general concept of building an ice cream cake in this fashion came from my mother, and I must say that I greatly prefer an ice cream cake that actually contains cake to one that is just molded and iced ice cream. So without further ado:


Ingredients: 1-2 cakes (baked and unfrosted, any flavor and shape), 1 quart softened ice cream (any flavor), 1 bag powdered sugar, 1 quart heavy whipping cream, 1 stick softened margarine or butter, and food coloring (optional).

To begin the process, I made 2 9" round Cookie and Cream cakes from Williams Sonoma mix that Mama Wolgemuth gave me. If you prefer to have a smaller ice cream cake, you could make a single cake in a 9" round pan, 13x9 pan, or a loaf pan and cut this single cake in half (so that you end up with 2 cake circles half the height of the original cake, etc.). Any flavor* cake works fine, so please make this recipe your own! After the cakes have cooled, wrap them in plastic wrap, and place them in the freezer for an hour (this step can be skipped).

To soften the ice cream, set it out for approximately 15 minutes. Then take one of the cakes (or cake halves) and place it on a plate. Using a spoon and a spatula, scoop the softened ice cream onto the cake, smoothing and leveling as you go:


Use as much of the ice cream as you would like (I used a whole quart of cookies and cream* with my two cakes), again making sure that the top of the ice cream is level when you've finished with this layer. Then place the second cake (or half) on top of the ice cream, pressing down slightly. Immediately cover and place in the freezer for 15-30 minutes so that the ice cream refreezes in form.

In the meantime, make the icing by adding about 1/2 the whipping cream into your mixer bowl. Turn the mixer on med-high and let it run until the cream becomes thicker and fluffy (whipped cream). Then add in the softened butter and mix until incorporated. Add 1 cup of powdered sugar and combine. From here, you can alternately add sugar and the liquid cream by 1/4 cups and tablespoons, respectively, until the icing reaches a consistency and sweetness that you like. (You probably won't use all the cream or sugar- be sure to taste as you go!)

When the icing is how you like it, separate and color as you so desire (I left some white and colored some green). Pull the cake out of the freezer and use a spatula to cover it with a layer of icing:


Pay attention to the ice cream layer, as it may begin to melt around the edges before you finish with the icing. If this happens, carefully place the cake back in the freezer for 5 minutes or so, then pull it out and resume icing. When you're finished with this first layer, put the cake in the freezer for about 10 minutes.

Next comes the fun part- the decorating. Of course the cake shouldn't be out of the freezer for an extended period of time, so if your planned method of decorating takes more than 5-10 minutes, make sure you take plenty of breaks and set the cake in the freezer so it will stay firm. My favorite method for this cake is freehand, silly-string patterns around the top edges and simple dots or scallops around the bottom edge:


I haven't had cake recently, and this easy Ice Cream Cake is perfect for the dog days of summer, so I figured you might like to fix it with me for dessert this week. I hope you enjoy!


*Some other great cake/ice cream flavor combos:
vanilla/vanilla, vanilla/chocolate, vanilla/cookies and cream, chocolate/chocolate, chocolate/rocky road, strawberry/strawberry cheesecake, yellow/sweet corn, lemon/lemon sherbert, etc.

MrsDrPoe: Ms. Looney

For the next installment of Poes' Pals, I'd like to introduce you to our friend...Ms. Looney:


Unlike her name might suggest, Ms. Looney is not crazy. She is, however, a very kind, intelligent, and godly woman. We met her during our time at State when she came to services with another friend of ours. Our friendship has grown over the years; although, I wish we could've spent more time together in Starkville.

One of my favorite memories of Ms. Looney is from the summer I started a knitting circle with all the young women of the local congregation. Of all the ladies that came, I think Ms. Looney is the only one who continued knitting after the summer ended. She was quick to pick up the art, and, although her projects began with scarves, they increasingly grew in difficulty.

Knitting is certainly not all Ms. Looney is about, though. Whenever a Bible study, singing, or general get-together with Christians is happening (and her busy schedule allows it), Ms. Looney is there. She's quick to invite others to Bible studies too- both with the congregation and the Holley's Luke study. Not only that, but she just finished her masters degree in Speech Pathology from the Mississippi University for Women, and with her degree, I know she will help lots of folks in a way that few others could.

We miss Ms. Looney and the rest of the gang from Starkville, and highly suggest that if you're ever near the area, you pop in for a visit!

MrsDrPoe: Conservation of Momentum, Part III

Once again Thesis Tuesday is upon us, so we'll be continuing our discussion of the Conservation of Momentum equation. Today we'll change gears a bit, moving toward examination of the differential form of the momentum equation. When I taught this introductory fluid mechanics class, this form of the equations was the most loved and most hated at the same time by my students- loved because it was much simpler to work with and hated because I always made them start from the full form of the (inviscid or viscous) equation and tell me why they ended up with the final result.

Recall that the linear momentum equation is actually a vector equation that is composed of 3 scalar equations (for a 3D problem). The differential (scalar) form of the linear momentum equation is:

d/dt(rho*u) + u*d/dx(rho*u) + v*d/dy(rho*u) + w*d/dz(rho*u) = rho*gx + d/dx(Sxx) + d/dy(Txy) + d/dz(Txz) + sum(Fx)

d/dt(rho*v) + u*d/dx(rho*v) + v*d/dy(rho*v) + w*d/dz(rho*v) = rho*gy + d/dx(Tyx) + d/dy(Syy) + d/dz(Tyz) + sum(Fy)

d/dt(rho*w) + u*d/dx(rho*w) + v*d/dy(rho*w) + w*d/dz(rho*w) = rho*gz + d/dx(Tzx) + d/dy(Tzy) + d/dz(Szz) + sum(Fz)

We've discussed each of the components in these equations before except for the S and T terms. These terms are ones from the stress tensor/matrix- S represents the normal stresses on a fluid due primarily to pressure forces; T represents the shear stresses on a fluid due primarily to the viscosity of the fluid. These stress terms reduce to different forms depending on whether we are dealing with a viscous or an inviscid fluid. Today we'll examine the viscous fluid equations.

The viscous form of the momentum equations is achieved by applying Stokes' postulates:

1) Each stress component is a linear function of strain rate (Newtonian)
2) The properties of a fluid are isotropic (not varying with direction)
3) when all strain rates are equal to zero, hydrostatic pressure is the only stress present

and Stokes' assumption:

lambda = -(2/3)*mu, where lambda is the bulk viscosity (related to the volumetric dilatation of the fluid) and mu is the viscosity

These allow the stress terms in the equations above to be rewritten as:

d/dx(Sxx) + d/dy(Txy) + d/dz(Txz) = -d/dx(p) + mu*(d/dx(d/dx(u))+d/dy(d/dy(u)) + d/dz(d/dz(u)))

d/dx(Tyx) + d/dy(Syy) + d/dz(Tyz) = -d/dy(p) + mu*(d/dx(d/dx(v))+d/dy(d/dy(v)) + d/dz(d/dz(v)))

d/dx(Tzx) + d/dy(Tzy) + d/dz(Szz) = -d/dz(p) + mu*(d/dx(d/dx(w))+d/dy(d/dy(w)) + d/dz(d/dz(w)))

Placing these expressions in the initial differential form of the momentum equation gives us the Navier Stokes Equations. What these scalar equations physically represents is:

Left side- "mass times acceleration" of a fluid particle (per unit volume)

Right side-
First term- force of gravity of a fluid particle per unit volume; if positive, this terms acts to accelerate the particle in the +x-direction

Second term- pressure difference across the fluid particle; if negative, than this term acts to accelerate the particle in the +x-direction

Third-fifth terms- viscous stresses acting on the fluid particle; if sum is positive, than this group acts to accelerate the particle in the +x-direction

Sixth term- any other forces (support, etc.) per unit volume acting on the fluid; this term is often neglected

By combining these equations with the continuity equation, we can provide a complete mathematical description of the flow of an incompressible, Newtonian fluid: 4 equations allows us to solve for 4 possible unknowns (p, u, v, and w). If we have exactly four unknowns, our problem is well-defined. Fewer and our problem is over-defined; more (unknown external forces, etc.) and our problem is ill-posed and we cannot solve it unless more information is obtained.

These equations may seem a bit intimidating, but remember, we can always employ simplifying assumptions in order to obtain a more manageable set. In fact, that's what we'll look at next week- application of the Navier Stokes Equations. So until next time, happy studying!

MrsDrPoe: Sweet Deals, 8/14

It's Monday Monday here at the blog. There are a couple of deals at my "normal" stores I'd like to share with you today:

Kroger
Milk 1/2 gallons B1G1 (Sunday - Monday)

Check out some Mega Event deals here.

Wags
Pert Shampoo ($3.99) - $3 back in RR, $1 Q in SS 8/14 (makes it free)
Revlon Nail Polish ($3.99) - $3 back in RR, $1 Q in SS 8/14 (makes it free)

CVS
Scan your CVS card at the kiosk this week to get coupons for free stuff- I got bandaids and a Caramello bar. I also got a $3 off $15 in groceries coupon (the free Gatorade GFit this week counts toward the $15 grocery total).


My purchase: 1 Caramello bar, 2 boxes Nature Valley Granola Bars, 2 boxes Fiber One bars, 1 box CVS bandaids 20 ct, 5 GFit 02 drinks.

Total Regular Price: $30.18
Total Sale and Qs: $18.54
Total ECBs used: $7
Total OOP: $4.64
ECBs back: $5

MrsDrPoe: Gulf Shores

It's Foto Friday- Gulf Shores edition! Last weekend, Mama and Papa Poe invited us down for a weekend at the beach. They had gotten a really good deal on a house in Foley that the owners are renting out until they can sell the place. Mama Poe made us some tasty meals, as did the Fish River Grill #2 (Swamp Soup...yum). While we spent a lot of time in the screened in pool at the house, we also made it out to the beach:




It was a nice relaxing weekend, minus a few small sunburn spots!