If the velocity of an incompressible fluid at a point (x,y,z) is given by (image below) prove that the motion is possible and that the velocity potential is −cosθ/r2. Find the equations of the streamlines where r^2 = x^2 +y^2 +z^2 and θ is the angle between r and the z-axis.

Q: y x = r cos 0 V = Or y = r sine r = √x² + y² χ Flow in "solid body rotation" acts like a solid…

A: Solution: The curl of the given velocity vector is zero.

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: 3.4 The stream function in a two dimensional flow field is given by y = x² - y? The magnitude of the…

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Q: unsteady velocity field V = xy'ti + zxj – t°k exists at the 3D plane along a streamline that passes…

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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: Q3) The velocity field for a two dimensional steady and incompressible fluid flow given as: V =(-3x°…

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Q: Consider the velocity field of Problem 2.18. Plot the streakline formed by particles that passed…

A: Given Data: The velocity of flow is, V→=Bx(1+At)i^+Cyj^. The value of A is, A=0.5 s-1. The value of…

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

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Q: A 2D incompressible velocity field is defined by . If both u and v are 0 at point (x = 0, y = 0),…

A: Given Data: The flow is 2D and incompressible. The flow is defined as u=2(x2-y2). The pressure and…

Q: Investigate the complex potential f(z) = A cosh [π(z/a)],and plot the streamlines inside the region…

A: Writing the complex potential function: Where A is a constant.   Splitting the potential function…

Q: 3 Starting with a small fluid element of volume dx dy dz, derive the continuity equation (Eq. 3.4)…

A: Given data  To derive the continuity equation

Q: A two-dimensional flow field is given by u = 5x 2 − 5y 2v = −10xy (i) Find the streamfunction ψ and…

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Q: Sketch the stream lines represented by y = xy. Also find out the velocity and its direction at point…

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Q: coordinates are in meters. Determine whether the flow field is three-, two-, or one-dimensional.…

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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: Q1) One dimensional steady fluid flow along x axis is shown in figure. The velocities at point O and…

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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: f the velocity field, V=3y2 i. Which of the following is NOT TRUE? Select one: The flow is steady…

A: Givenquestion 1V=3y2i^+oj^question 2V=x2+y2-4i^+xy-yj^

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

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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: 1. A flow in the x-y plane is given by the following velocity field: u-3 and v=6m/s for 0<t<20s;…

A: a)In the given problem, for the given particle released at t= 0 s,and u = 3m/s and v = 6 m/sDuring…

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: 2-D unsteady flow field has the velocity components: v = - x²y?t u = - Plot the streamline passing…

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Q: An unsteady velocity field V = xy'ti + zxj – t°k exists at the 3D plane along a streamline that…

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Q: A VISCOUS FLUID FLOWS BETWEEN TWO PARALLEL AND INFINITELY LARGE PLATES KEPT AT A DISTANCE 2H FROM…

A: Write the given data. K=-dPdx

Q: Question 3 : A line source at y-axis is located 1 m above the horizontal plate (x-axis), where the…

A: The objective of the question is to determine the stream function of the potential flow in both…

Q: Problem 3: Obtaining the stream function from velocity components The steady state, incompressible…

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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 flow is described by velocity field, V=ayi+btj, where a=1 s-1and b=0.5 m/s2.At t=n s (Where n is…

A: Given: The expression of velocity is V=ayi+btj. Here, a=1 s-1and b=0.5 m/s2. n = 19

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: or the velocity field: u = 3xy, v = -1.5y²+5; for water, ρ = 1000 kg/m³ If the pressure at (0,0) is…

A: The velocity field is .The density of water is .The pressure at  is .

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

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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: Given the vector equation below for velocity of a given fluid: V=(0.5+1.2x)i + (-2.0-1.2y)j, derive…

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

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Q: Question 4. The stream function is given as w=x³-y³. a. Is this flow possible? b. Is the flow…

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Q: A 2-D, incompressible, flow field has an x-component of flow, u = 2xy² if the velocity at (1,2) is…

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Q: The velocity field of a flow is given by V = aryi + by'j where a = 1 m's' and b = - 0.5 m's'. The…

A: The velocity vector can be compared with  V→= ui^ +vj^  u = axy = xy v= by^2 = -0.5y^2 since both…

Q: If u = ax and v = ay and w = -2az obtain equation of streamline passing through the point (2, 2,4)…

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Q: Establish the equation of continuity for an incompressible fluid in the form V.V = 0 , V(u, v, w)…

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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.