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The velocity ulx; along wall in horizontal ( x )-direction of an incompressible fluid is given by (with (xy) >0u(r,y)-U -{2x-7) mts] We take U = 2 |m / sJ for th...

Question

The velocity ulx; along wall in horizontal ( x )-direction of an incompressible fluid is given by (with (xy) >0u(r,y)-U -{2x-7) mts] We take U = 2 |m / sJ for the characteristic velocity U_(2a) Determine the function of the contour line for the velocity " = 0(2b) Determine the gradient of in (he point (4,1,2)(2c) What is the Interpretation of the direction of the gradient?(2d) Give the (angent plane for u In Ihe point (4,1,2) and simplify as (ar as possible

The velocity ulx; along wall in horizontal ( x )-direction of an incompressible fluid is given by (with (xy) >0 u(r,y)-U - {2x-7) mts] We take U = 2 |m / sJ for the characteristic velocity U_ (2a) Determine the function of the contour line for the velocity " = 0 (2b) Determine the gradient of in (he point (4,1,2) (2c) What is the Interpretation of the direction of the gradient? (2d) Give the (angent plane for u In Ihe point (4,1,2) and simplify as (ar as possible



Answers

The velocity for a steady, incompressible flow in the $x y$ plane is given by $\vec{V}=i A / x+i A y / x^{2},$ where $A=2 \mathrm{m}^{2} / \mathrm{s},$ and the coordinates are measured in meters. Obtain an equation for the streamline that passes through the point $(x, y)=$ (1,3). Calculate the time required for a fluid particle to move from $x=1 \mathrm{m}$ to $x=2 \mathrm{m}$ in this flow field.

Hello, friends for this problem, it is given steady in compressible float an ex by plane having by component off velocity minus B into X and toe by toe. The power three. Yeah, value off we is given 0.2 per meter cubed per second. We have to find the simplest X component off velocity That is you. We have toe calculate Second equation off streamlines, streamlines for the floor and see lot extreme lengths through the point through the points one come up for and toe come up foot. Let us start solving it for in compressible floor Ruiz Constant and in X by plane Having steady flu. Did you upon the legs? Plus Dale be upon Dale by Must have been beautiful, So they'll be upon Dale. Why? We will get data off Dale by minus B X Y cube. So it can be written as minus three. We ex vice square. So from equation one, we can write Dale you upon the legs, Toby three b x by square or dale you Toby three b x y square. Dale affects integrating both the site. You will get three by two. Be access square by square function off by for simplest version, function off. Why must be Jiro? So you will get three way to be access square by square. No divi upon the X can be written as be upon you that is minus beat x y que upon three by to beat access square by square So we will get divi upon the x to be minus two by three by upon X separating the very well and integrating so we can write Do you buy upon why Toby minus two by three d x upon X now integrating you will get loan off by Toby minus two by three Learn off x so we will get the equation off Scream line Toby X and toe by toe The power three by two Constant Now we have toe plot the streamline for the points plotting scream line through the point one comma for X by to the power three by two Constant of constant having the value one in tow. Four to the power three by two That is it on four point to come up for constant having the value two in tow for to the power three by two that is 16. So by ex craft you will get Yeah, that's all for this problem. Thanks for watching it

Hi, everyone. Here it is. Given X component of velocity use called toe mhm in tow. Ex. Keep our fight minus 10 X Q. Why square less five x by Keep our foot here. Value off is given toe. We try to the power minus four Second to the power my husband access measuring meter Fine by component off the velocity and acceleration At one commentary X component of velocity is defined Its dale off by upon Dale by And why is defined us? They'll die upon deluxe So using it be no use cult Oh, a x to the power fight than x Q by square plus five x by to the power for its called to They love sigh upon the love fight So streamline function you will get Zai Toby integration off Okay x to the power fight then x cubed by square five x Keep our by foot Dale a fight So here you can write Okay as I having the value A x to the power five into white 10 by three x cubed by job plus X and toe by toe The power five upon five Bilis Function off X Say you question of bourbon. So by component of velocity is defined us. They rely upon Dale off X so a into X. Keep our five in two by minus 10 by three X cube by cube plus X. Why to the power five upon five less function off X so it can be a little less minus eight five x Keep our for white, then access square Y cube. Why? To the power five plus capital effects for simplest form, functional facts must be zero. The sculpture you will write minus a five X to the power for white uh, extra square by Q plus. Why to the power fight? This is the answer off part A. All right? No acceleration is defined. Its X component off acceleration is you and toe nail you upon the legs. Plus we there, you upon thereby substitute the value. Okay, Okay. X to the power pipe minus tank X cubed by script because fights X by to the power foot Dale off. Dale affects X to the power fight in two. Stan excuse y squared. Let's five x right toe The powerful minus eight fight next to the powerful by minus Chain access square Bike. You by to the power fight deal over Dale by eight x to the power five minus 10 x cubed by square plus five x by to the power foot. So on solving this X component off acceleration, you will get fights a square she two x access square plus by squared for the whole to the powerful. Now finding the Y component off acceleration, which is defined as you. Uh huh. They'll be upon dialects. Yeah, but that's me. Did we upon Dale way substitute the value you is given a in tow X to the power fight minus 10 x cubed by square plus five x Why to the power for Dale over. Dale off X minus eight five times X to the power for into white minus 10. Access squared Y cube plus pi to the power pipe minus eight in tow. Five x to the power for by then access square Y cube plus by to the power fight. Dale over Dale way minus eight five. Exclude the power for white minus 10 X squared y que plus y to the power five on simplifying it. Why component you will get five is square white in tow. access square. Plus why? Square toe the powerful This is equation picked using equations. Foreign five. What acceleration at? Yeah. One commentary so x you will get five in tow. One way to chi square in tow one and toe one chi square plus three chi square. Hold to the powerful. You will get X component. Toby 1.254 and toe 10 to the power four meter per second square. Yeah, by component, off acceleration, you will get fight in tow. Half price square and 23 one kind squared plus three chi square You will get Why component, Toby? Yeah. Three points. 75 10 to the power food meter per second squared. So acceleration can be written as root off x component square. Plus why Component square X component having the value? Yeah. 1.25 can toe the powerful and 3.75 So it would be Yeah. 3.95 Trento The power for Yeah. Meet up or second square. That's all for it. Thanks for watching it

Hello, everyone. Here it is. Given the velocity field the skull toe you white upon X to the power half I kept Bless you, wise sweat full upon X to the power three by two Jacob here is 1 40 but meter toe the power half Andi use cult Oh, 2.40 meter per second. Fine, right. If this described possible in compressible floor are possible in compressible flow. In second part we have to find the acceleration at 0.5 m comma five millimeters in part seat the slope off. Streamline through that point. Let us see here the solution X component off velocity is you by upon route Tex on dwhite component is you. Why square upon for X to the power three by two we will find the value Did you upon the election? Plus they'll be upon Dale by it's called toe You is given you right upon route off X Dale Over. Dale off white A You y squared. Just a moment, please. For X to the power three by two On differentiating it you will get how minus half you buy X to the power minus three by two. So and generally becomes positive plus half you. Why? Upon X To the power three by two It also hands. Flo is compressible. This is the answer off part eight Now for part B, we have to find acceleration on acceleration is defined its you They will be upon Dylex plus B They will be upon deal by. Did we? Upon the legs you will get minus A You buy Toto the power x three by two I kept plus 38 You y square upon it x to the power five by toe Jacob and they'll weigh upon Dale. Why will be you upon X to the power half I kept you bite toe the power x three y toe Jacob, therefore acceleration people get you right root off X in tow minus you white toe X to the power three by two plus three times a u y square it x to the power five by two unless you vice square upon for X to the power three by toe and toe you upon route off X, I kept less Here it is icap and Jacob let me correct it. It will be you upon row Texas called toe a U y upon two x to the power three by two Jacob. So on simplifying, we can write acceleration off that point. Toby is square. You square vice square upon four access square I kept minus is squared. You will square right you upon for X cubed Jacob Value off is given you is given at point 5 m. Uh huh. And sorry at point point 5 m and point to Jiro Jiro. 5 m acceleration. You will get. I'm writing here directly. You can substitute the value and find. Then it would be minus 2.86 Tend to the par minus two I kept plus 10 to the par minus four. Jacob, meet up for second square. So this is the answer off. Be part now. Answer up seat the slope off. Streamline is given. Right. We upon you, That is you. Why sweat upon four x to the power three by two upon you bite X to the power half, so it will be y upon forex value off by is 0.5 X is given 0.5. So you will get the slope off. Streamline Toby 2.5 in tow. 10 to the Power Ministry. That's all for this problem. Thanks for watching it

Mission. We have grown potential function. Phi equals to access square minus by square. And from the continued equation, we know that they'll you by day legs Plus Dell, we buy Dell Y equals two and from the stream function we can, right? You were quite school. Deal side by Dale Y and V equals two. Deal side by deal X minus. Okay, So to determine side, we can interrogate with respect to why To obtain U S. U equals to deal side by Dell. Wide question Dale five by 10 x And by putting values of five, this will be equals two works. So from here, site Dale side integration of Dale side will be equals to integration of two weeks. Amount of data interrogating with respect to what? So from here, Simon could be equals to two x y bless F one function of X. Okay, similarly, we can write equals two minus dale side upon day legs close to they'll fight on Dale wide, which will be equals two minus of Cuba by putting values of five, which is X squared minus y squared. Okay, So differentiating with respect to why we can get this value so again interrogating with respect to X to get value of side so they'll side will be equals to interrogation of the way DX. So from here, side will be equal to who, X way plus function of what? Okay, So to satisfy both equations for Simon and Side do we can right side equals two who works way bless seat two weeks away Bless C So where C is an arbitrary constant since we have given Sequels to geo along the Y axis at Y equals to zero. So from here she will be equal to G. So from here, the stream function side will be equal to two elsewhere. This is the answer for that first part. Okay, where we have to calculate or determine stream function now moving to the next part B in which we will calculate the desires rate. Okay, so part of the and so from here we can right desserts read a Q equals two side B minus side it and four side equals to zero. So eight equals to Joe. We can write q equals two side this and four coordinate x y and y I we can write u equals two two x I Y x eight white I Okay, so this is the value of Q desires rate. Okay, this is the answer for that.


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