A velocity field of the two-dimensional, time-dependent fluid flow is given bywhere t is time.Find the material derivative Du/Dt and hence calculate the acceleration of the fluid at anytime t > 0 and any pont x > 0, y > 0.a) Incompressibilitya) Is this flow incompressible (i.e. it has zero divergence)?YesNoдиƏtb) Time derivative of flow fieldCalculate the time derivative of the velocity. Represent your answer in the formi+||33u(t, x, y) =r? (x² + y² ) i− {etxtyjX уј3a =c) Material derivative and accelerationCalculate the material derivative of the velocity and hence the acceleration a. Representyour answer in the formDuDt||ji+j

(a) Write the expression of velocity field. u=t-3/2x1/3+y1/3i-13t-3/2x-2/3yj

A velocity field of the two-dimensional, time-dependent fluid flow is given by 1 3 u(t, x, y) = r² (x² + y² ) i −{rtx‡yj 2 X уј where t is time. Find the material derivative Du/Dt and hence calculate the acceleration of the fluid at any

A velocity field of the two-dimensional, time-dependent fluid flow is given by 1 3 u(t, x, y) = r² (x² + y² ) i −{rtx‡yj 2 X уј where t is time. Find the material derivative Du/Dt and hence calculate the acceleration of the fluid at any

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

See similar textbooks

Related questions

Q: Consider a three-dimensional, steady velocity field given by V = (u, v, w) = (3.2 + 1.4x)ï + (2.4 –…

A: Write the expression of velocity field. V=3.2+1.4xi+2.4-2.1yj+wk Write the components of…

Q: y = 2r'sin20 (2) 0.5 m A 2D inviscid incompressible fluid flow around a corner is In addition, if…

A: Given:- stream function ψ=2r2sin2θ, where ψ has units of ms2,r is in meters. fluid density=103kgm2…

Q: 1 du du, Nij 2 дх, дх 2. The rate-of rotation tensor is defined as - and the vorticity of the…

A: For solution refer below images.

Q: A fluid has a velocity field defined by u = x + 2y and v = 4 - y. In the domain where x and y vary…

A: Since you have posted a multiple independent question, we will solve first question for you. To get…

Q: 3. The two-dimensional velocity field v = (3t- 8xy)i+ 4(y – x')j satisfies the Euler's equation of…

Q: For the following velocity field, determine the constant a for having an incompressible fluid flow.…

Q: QI A/ The inviscid, steady, and incompressible 2D flows are given by (a) o = x-3xy (b) y =x- 2xy-y?…

A: Velocity potential functionφ is defined as a scalar function of space and time such that its…

Q: Derive the Eulerian acceleration of a fluid particle in a flow field given by 1 V = xy²i -y'j + xyk

Q: A two-dimensional flow field has velocities along x2t and the x and y directions given by u = V =…

A: Given data: u=x2tv=-2xyt Need to determine the streamline equation.

Q: The velocity field of an incompressible flow is given by V = (a,x + a,y + azZ)i + (b,x + b,y + b3z)j…

A: Answer: The value of b2 is 2.

Q: In cylindrical polar coordinates (r, theta, z), the velocity components of a flow field are given as…

A: Given details: vr=r2zcosθ r=0.5mvθ=rzsinθ z=1mvz=z2t θ=π4 Formula…

Q: At t = 0, an unstable velocity field V = xy²ti + zxj – t³k, emerges in the 3D plane as a result of a…

A: Given : V=xy2ti+zxj-t3k To find : Equation of streamline

Q: Calculate the material acceleration for the following steady, incompressible velocity field:

A: Given: Velocity fieldV→=0.5+0.8xi^+1.5-0.8yj^

Q: The velocity field of an incompressible flow is given by V = (a1x + a2y + a3z)i + (b1a + b2y + bzz)j…

A: Given, V = ( a1x + a2y + a3z ) i + ( b1x + b2y + b3z ) j + ( c1x + c2y + c3z ) k a1 = 2 c3 = -4…

Q: velocity field V = xy^2ti + zxj −t^3k exists at the 3D plane along a streamline that passes through…

Q: The velocity components for a three dimensional incornpressible flow is given by : u=x'-y-2x,…

Q: A velocity field is given,V=x2 i+2xyt j+2yzt k Find magnitude of velocity at (2,-1,1) and t=1…

A: Given: V=x2 i+2xyt j+2yzt k u= x2 v= 2xyt w= 2yzt x=2, y=-1, z=1, t=1

Q: The stream function is given by y =-4xy. Then the magnitude of velocity at point (2,4) m/s.

A: Given Stream function = -4xy Point = 2,4 Find Magnitude of velocity, V

Q: In a 2D dimension incompressible flow , if the fluid velocity components are given by u = x-4y , v =…

A: Given:The fluid velocity components are given as:u=x-4yv=-4x

Q: The velocity potential for a two-dimensional velocity field is given by the relation fi=(7/3)x3-7xy2…

Q: A 2D velocity field is given by V = (u, v) = (2.7 - 1.9x, 0.7 + 1.6y), where the coordinates are in…

Q: The velocity field for an incompressible flow is given as V-›=…

A: Given data: The given velocity vector is V→=5x2i→-20xyj→+100tk→ The given location is (x,y,z)…

Q: the X-Y plane; permanent, 2B, in an uncompressed flow, a, b and C are constant there is a velocity…

Q: A two - dimensional velocity field is given as V = -2yi – 2xj Find the current function 4(x,y) of…

A: Given V = - 2y i - 2x j To determine the current function ψ(x,y) of the velocity field and the…

Q: A velocity field is given as V (x,y,z,t)=(5xy^2+t) I+ (2z+8) j+18 k. Here x,y and z are in metres…

Q: velocity field is given by: A two-dimensional V = (x - 2y) ï- (2x + y) j a. Show that the flow is…

A: V= (x-2y) i^- (2x+y) j^

Q: A two-dimensional fluid element of dimensions dx and dy translates and distorts as shown, during the…

A: In order to proceed with the given problem some assumptions need to be taken. The Flow is two…

Q: eam function is given by u = 3 xy. Determine (i) The velocity function u, eration component at (1,…

Q: (a) Determine the fluid acceleration at x-4, y-5. (b) Determine the rate of volumetric dilatation in…

Q: 4. Show that the velocity field V = 2y(2x + 1)î + [x(x+1) – 2y²1j represents a possible…

Q: A flow is described by velocity V=ay i+bt j ,a=1s-¹ and b =.5m/s².at t=25s,what are the coordinates…

A: we know thatVelocity V=dydtdydt=btdy=bt.dtintegrating both sides we get∫y0ydy=∫t0tbt.dty-yo=br220t

Q: the constants 1, m, and m

A: L , m , n are the constrains of above equations.

Q: onal flow field is give

A: given: φ=x2-y2

Q: An unsteady, incompressible, velocity field is given by the following vector: V = (u, v, w) = (−7t…

A: Velocity vector, V→=u,v,w=-7t+3.5xy+2.2yi→+2t+3x-1.6yj→+5+4zk→Point, x,y,z,t=-1,2,1,2

Q: Ux = -ax Uy = ay Using the streamline equations show the streamlines of the flow as well as stating…

A: Given data ux=-axuy=ay Here, ux and uy is the speed of the flow in X and Y direction. Now we have to…

Q: An unsteady, incompressible, velocity field is given by the following vector: V = (u, v, w) = (t +…

A: Given: Let us consider the given velocity field in vector, V→=u,v,w=t+5x-7yi→+-3yj→+2t-4zk→ To find:…

Concept explainers

Fluid Kinematics

The branch of science that deals with objects which are either in motion or stationary are studied under the branch of classical mechanics. Fluid mechanics is a subcategory of classical mechanics that deals with the behavior of fluids at rest (fluid statics) or in motion (posing some velocity). In fluid kinematics, studies focus on the associated motion of fluid particles or groups of particles, and its associated parameters like displacements, velocity, and acceleration without concerning about the forces that caused the motion. All the laws of classical mechanics are applied in the study of fluid kinematics.

Question

A velocity field of the two-dimensional, time-dependent fluid flow is given by
where t is time.
Find the material derivative Du/Dt and hence calculate the acceleration of the fluid at any
time t > 0 and any pont x > 0, y > 0.
a) Incompressibility
a) Is this flow incompressible (i.e. it has zero divergence)?
Yes
No
ди
Ət
b) Time derivative of flow field
Calculate the time derivative of the velocity. Represent your answer in the form
i+
||
3
3
u(t, x, y) =r? (x² + y² ) i− {etxtyj
X уј
3
a =
c) Material derivative and acceleration
Calculate the material derivative of the velocity and hence the acceleration a. Represent
your answer in the form
Du
Dt
||
j
i+
j

Expert Solution

Step 1

(a)

Write the expression of velocity field.

$u = t^{- 3 / 2} (x^{1 / 3} + y^{1 / 3}) i - \frac{1}{3} t^{- 3 / 2} x^{- 2 / 3} y j$

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

Similar questions

The velocity field of a flow has the following structure V (r, t) = kxi + kyj + (−kx + αt)k, where k, α are known constants, r = (x, y, z) the vector of position and t time. About this: Identify the components of the velocity field. Calculate the speed of the particles moving about the origin of the coordinate system, for every instant t
Given velocity fieldV(x,y,z,t) = (5xy² + t)i + (2z + 8)j +18k m/s, with x,y,z in meters and it in seconds. Calculate V(9,-2, 1,4). What is the magnitude of this velocity?
A 2D velocity field is given by V = (u, v) = (2.8 - 1.6x, 0.7 + 1.6y), where the coordinates are in m and the velocity is in m/s. Find the magnitude of the vorticity.
A 2D velocity field is given by V = (u, v) = (2.5 - 1.9x, 0.65 + 0.9y), where the coordinates are in m and the velocity is in m/s. Find the volumetric strain rate (in s^(-1))
In three-dimensional fluid flow, the velocity component an u = * + y z, v = - (xy + yz + zx). Determine the %3D satisfy the continuity equation.
The velocity components in the x and y directions are given by 3 u = Axy3 - x2y, v = xy2 -- The value of a for a possible flow field involving an incompressible fluid is
Correct answer only