What is the time complexity of finding the residual network in this case? Here are the details:

(1) We are given a number $f$ of units of flow that we wish to transmit from the source node to the sink node. For example, $f=5$ units of flow (data or goods) should be transmitted from the source node to the sink node. 

(2) We are not given any plan or policy to transmit the flow. However, we know that $f$ is less than the maximum flow of the network. So, it is possible to transmit such an amount of flow through the network.

(3) We are given the arcs' capacities.

$\bullet$ Now, it is required to determine the residual network after sending these $f$ units of flow.

Very simple:

Input: A network with the given arcs' capacities and an amount of flow $f$. Also, two nodes are specified as source and sink nodes.

Output: A plan to transmit $f$ units of flow from the source node to the sink node, and the corresponding residual network.

It is clear that there may be more than one plan to send $f$ units of flow through the network, and so the residual network may change. It is not important to reach a special residual network. My main concern is only to find the time complexity of such a job (finding a way to transmit $f$ units of flow through the network, and then determining the residual network).

It is easy to see that if I have the plan to send $f$ units of flow, the residual network can be determined with time complexity of $O(m)$, where $m$ is the number of edges in the network.

$\bullet$ I highly appreciate any help in advance.
The time complexity of this problem is (at most) the time complexity of the network flow problem.  You are given a graph and you want to find a flow of size $f$.  You can use any network flow algorithm for this.  Different algorithms have different running time.  Wikipedia has a list of algorithms, with their running times.  There is not a single optimal algorithm; different algorithms work better in different situations.

You mentioned you want to find a flow of size at least $f$, rather than a max flow.  Many of the algorithms for max flow can be adjusted to stop early when you've found a flow that is large enough.  Or, you can adjust the graph to ensure that a flow of size $f$ is a max flow -- if the original source vertex is $s$, add a new vertex $s_0$ and an edge $s_0 \to s$ of capacity $f$, then treat $s_0$ as the new source vertex and solve the max flow problem.

Once you find the flow itself, you can compute the residual network in $O(m)$ time.  For each edge, you compute how much remaining capacity it has (the capacity of that edge minus the amount sent by the flow); and you add a reverse edge if necessary (the amount of the flow).  This is $O(1)$ work per edge.  All known network flow algorithms take $\Omega(m)$ time.  Therefore, the time to compute the residual network once you find the flow is negligible; the bottleneck is finding the flow itself.